Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimization problems of sorts arise in all quantitative disciplines from

computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , , and . Computer science ...

and engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

to operations research and economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science of plant life and a bran ...

, and the development of solution methods has been of interest in mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

for centuries.
In the simplest case, an optimization problem
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

consists of maximizing or minimizing a real function
In mathematical analysis, and applications in geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. ...

by systematically choosing input
Input may refer to:
Computing
* Input (computer science), the act of entering data into a computer or data processing system
* Information, any data entered into a computer or data processing system
* Input device
* Input method
* Input port (disam ...

values from within an allowed set and computing the value
Value or values may refer to:
* Value (ethics) it may be described as treating actions themselves as abstract objects, putting value to them
** Values (Western philosophy) expands the notion of value beyond that of ethics, but limited to Western s ...

of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and be ...

. More generally, optimization includes finding "best available" values of some objective function given a defined domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

(or input), including a variety of different types of objective functions and different types of domains.
Optimization problems

An optimization problem can be represented in the following way: :''Given:'' afunction
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

from some set to the real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s
:''Sought:'' an element such that for all ("minimization") or such that for all ("maximization").
Such a formulation is called an optimization problem
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

or a mathematical programming problem (a term not directly related to computer programming
Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, generating algorithms, Profilin ...

, but still in use for example in linear programming
File:3dpoly.svg, A closed feasible region of a problem with three variables is a convex polyhedron. The surfaces giving a fixed value of the objective function are Plane (geometry), planes (not shown). The linear programming problem is to find a p ...

– see History
History (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approxima ...

below). Many real-world and theoretical problems may be modeled in this general framework.
Since the following is valid
:$f\backslash left(\backslash mathbf\_\backslash right)\backslash geq\; f\backslash left(\backslash mathbf\backslash right)\; \backslash Leftrightarrow\; \backslash tilde\backslash left(\backslash mathbf\_\backslash right)\backslash leq\; \backslash tilde\backslash left(\backslash mathbf\backslash right)$
with
:$\backslash tilde\backslash left(\backslash mathbf\backslash right)\; :=\; -\; f\backslash left(\backslash mathbf\backslash right),\backslash ,\; \backslash tilde\backslash ,\; :\backslash ,\; A\; \backslash rightarrow\; \backslash mathbb$
it is more convenient to solve minimization problems. However, the opposite perspective would be valid, too.
Problems formulated using this technique in the fields of physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

may refer to the technique as ''energy
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

minimization'', speaking of the value of the function as representing the energy of the system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system, surrounded and influenced by its environment, is described by its boundaries, structure and purp ...

being modeled. In machine learning
Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data ...

, it is always necessary to continuously evaluate the quality of a data model by using a cost function where a minimum implies a set of possibly optimal parameters with an optimal (lowest) error.
Typically, is some subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of the Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

, often specified by a set of '' constraints'', equalities or inequalities that the members of have to satisfy. The domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

of is called the ''search space'' or the ''choice set'', while the elements of are called ''candidate solution
In mathematical optimization, a feasible region, feasible set, search space, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's Constraint (mathemati ...

s'' or ''feasible solutions''.
The function is called, variously, an ''objective function'', a ''loss function In mathematical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimizat ...

'' or ''cost function'' (minimization), a ''utility function'' or ''fitness function'' (maximization), or, in certain fields, an ''energy function'' or ''energy functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
In architecture
File:Plan d'exécution du second étage de l'hôtel de Brionne (dessin) De Cotte 2503c – Gallica 2011 (adjusted).jpg, upright=1.45, alt=Pl ...

''. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an ''optimal solution''.
In mathematics, conventional optimization problems are usually stated in terms of minimization.
A ''local minimum'' is defined as an element for which there exists some such that
:$\backslash forall\backslash mathbf\backslash in\; A\; \backslash ;\; \backslash text\; \backslash ;\backslash left\backslash Vert\backslash mathbf-\backslash mathbf^\backslash right\backslash Vert\backslash leq\backslash delta,\backslash ,$
the expression holds;
that is to say, on some region around all of the function values are greater than or equal to the value at that element.
Local maxima are defined similarly.
While a local minimum is at least as good as any nearby elements, a global minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function (mathematics), function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, ei ...

is at least as good as every feasible element.
Generally, unless the objective function is convex
Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to:
Science and technology
* Convex lens
A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...

in a minimization problem, there may be several local minima.
In a convex problem, if there is a local minimum that is interior (not on the edge of the set of feasible elements), it is also the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima.
A large number of algorithms proposed for solving the nonconvex problems – including the majority of commercially available solvers – are not capable of making a distinction between locally optimal solutions and globally optimal solutions, and will treat the former as actual solutions to the original problem. Global optimization
Global optimization is a branch of applied mathematics and numerical analysis that attempts to find the global minima or maxima of a function or a set of functions on a given set. It is usually described as a minimization problem because the max ...

is the branch of applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and be ...

and numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...

that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a nonconvex problem.
Notation

Optimization problems are often expressed with special notation. Here are some examples:Minimum and maximum value of a function

Consider the following notation: :$\backslash min\_\backslash ;\; \backslash left(x^2\; +\; 1\backslash right)$ This denotes the minimumvalue
Value or values may refer to:
* Value (ethics) it may be described as treating actions themselves as abstract objects, putting value to them
** Values (Western philosophy) expands the notion of value beyond that of ethics, but limited to Western s ...

of the objective function , when choosing from the set of real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s . The minimum value in this case is 1, occurring at .
Similarly, the notation
:$\backslash max\_\backslash ;\; 2x$
asks for the maximum value of the objective function , where may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "infinity
Infinity is that which is boundless, endless, or larger than any number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything ...

" or "undefined".
Optimal input arguments

Consider the following notation: :$\backslash underset\; \backslash ;\; x^2\; +\; 1,$ or equivalently :$\backslash underset\; \backslash ;\; x^2\; +\; 1,\; \backslash ;\; \backslash text\; \backslash ;\; x\backslash in(-\backslash infty,-1].$ This represents the value (or values) of the Argument of a function, argument in the interval that minimizes (or minimize) the objective function (the actual minimum value of that function is not what the problem asks for). In this case, the answer is , since is infeasible, that is, it does not belong to thefeasible set
A closed feasible region of a linear programming problem with three variables is a convex polyhedron.">polyhedron.html" ;"title="linear programming problem with three variables is a convex polyhedron">linear programming problem with three variabl ...

.
Similarly,
:$\backslash underset\; \backslash ;\; x\backslash cos\; y,$
or equivalently
:$\backslash underset\; \backslash ;\; x\backslash cos\; y,\; \backslash ;\; \backslash text\; \backslash ;\; x\backslash in;\; href="/html/ALL/s/5,5.html"\; ;"title="5,5">5,5$
represents the pair (or pairs) that maximizes (or maximize) the value of the objective function , with the added constraint that lie in the interval (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form and , where ranges over all integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s.
Operators and are sometimes also written as and , and stand for ''argument of the minimum'' and ''argument of the maximum''.
History

Fermat
Pierre de Fermat (; between 31 October and 6 December 1607
– 12 January 1665) was a French lawyer at the '' Parlement'' of Toulouse
Toulouse ( , ; oc, Tolosa ; la, Tolosa ) is the capital of the French departments of France, department ...

and Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaRoman_Forum.html" ;"title="Curia Julia in the Roman Forum">Curia Julia in the Roman Forum
A senate is a deliberative assembly, often the upper house or Debating chamber, chamber of a bicame ...

found calculus-based formulae for identifying optima, while Newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* Newton (film), ''Newton'' (film), a 2017 Indian fil ...

and Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of m ...

proposed iterative methods for moving towards an optimum.
The term "linear programming
File:3dpoly.svg, A closed feasible region of a problem with three variables is a convex polyhedron. The surfaces giving a fixed value of the objective function are Plane (geometry), planes (not shown). The linear programming problem is to find a p ...

" for certain optimization cases was due to George B. Dantzig, although much of the theory had been introduced by Leonid Kantorovich
Leonid Vitaliyevich Kantorovich ( rus, Леони́д Вита́льевич Канторо́вич, , p=lʲɪɐˈnʲit vʲɪˈtalʲjɪvʲɪtɕ kəntɐˈrovʲɪtɕ, a=Ru-Leonid_Vitaliyevich_Kantorovich.ogg; 19 January 19127 April 1986) was a Soviet ...

in 1939. (''Programming'' in this context does not refer to computer programming
Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, generating algorithms, Profilin ...

, but comes from the use of ''program'' by the United States military to refer to proposed training and logistics
Logistics is generally the detailed organization and implementation of a complex operation. In a general business sense, logistics is the management of the flow of things between the point of origin and the point of consumption to meet the requ ...

schedules, which were the problems Dantzig studied at that time.) Dantzig published the Simplex algorithm
In mathematical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimi ...

in 1947, and John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American
Hungarian Americans (Hungarian language, Hungarian: ''amerikai magyarok'') are United States, Americans of Hungarian p ...

developed the theory of in the same year.
Other notable researchers in mathematical optimization include the following:
* Richard Bellman
Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics.
Biography
Bellman was born in 1 ...

* Roger Fletcher
* Ronald A. Howard Ronald Arthur Howard (born August 27, 1934) is an emeritus professor in the Department of Engineering-Economic Systems (now the Department of Management Science
Management science (MS) is the broad interdisciplinary study of problem solving and dec ...

* Fritz John
Fritz John (14 June 1910 – 10 February 1994) was a German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For citizens of Germany, see also ...

* Narendra Karmarkar
Narendra Krishna Karmarkar (born circa 1956) is an India
India (Hindi: ), officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and dependencies by population, second-most populous country, the ...

* William Karush
William Karush (1 March 1917 – 22 February 1997) was an American professor
Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other post-secondary education and research institutions in ...

* Leonid Khachiyan
Leonid Genrikhovich Khachiyan (; russian: Леонид Генрихович Хачиян; May 3, 1952April 29, 2005) was a Soviet and American mathematician and computer scientist.
He was most famous for his ellipsoid algorithm (1979) for linear ...

* Bernard Koopman
Bernard Osgood Koopman (January 19, 1900 – August 18, 1981) was a French-born American mathematician, known for his work in ergodic theory, the foundations of probability, statistical theory and operations research.
Education and work
Afte ...

* Harold Kuhn
Harold William Kuhn (July 29, 1925 – July 2, 2014) was an American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

* László Lovász
László Lovász (; born March 9, 1948) is a Hungarian-American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quanti ...

* Arkadi Nemirovski
Arkadi Nemirovski (born March 14, 1947) is a professor at the H. Milton Stewart School of Industrial and Systems Engineering at the Georgia Institute of Technology
The Georgia Institute of Technology, commonly referred to as Georgia Tech or, ...

* Yurii Nesterov
Yurii Nesterov is a Russian mathematician, an internationally recognized expert in convex optimization, especially in the development of efficient algorithms and numerical optimization analysis. He is currently a professor at the Université cathol ...

* Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet
The Soviet Union,. officially the Union of Soviet Socialist Republi ...

* R. Tyrrell Rockafellar
* Naum Z. Shor
* Albert Tucker
Major subfields

*Convex programming
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimizati ...

studies the case when the objective function is convex
Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to:
Science and technology
* Convex lens
A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...

(minimization) or concave
Concave means curving in or hollowed inward, as opposed to convex.
Concave may refer to:
* Concave function
In mathematics, a concave function is the additive inverse, negative of a convex function. A concave function is also synonymously called ...

(maximization) and the constraint set is convex
Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to:
Science and technology
* Convex lens
A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...

. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.
** Linear programming
Linear programming (LP, also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model
A mathematical model is a description of a system
A system is a group of ...

(LP), a type of convex programming, studies the case in which the objective function ''f'' is linear and the constraints are specified using only linear equalities and inequalities. Such a constraint set is called a polyhedron
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

or a polytope
In elementary geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

if it is bounded.
** Second-order cone programming (SOCP) is a convex program, and includes certain types of quadratic programs.
** Semidefinite programmingSemidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize)
over the intersection of the cone of positi ...

(SDP) is a subfield of convex optimization where the underlying variables are semidefinite matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object.
Fo ...

. It is a generalization of linear and convex quadratic programming.
** Conic programming is a general form of convex programming. LP, SOCP and SDP can all be viewed as conic programs with the appropriate type of cone.
** Geometric programmingA geometric program (GP) is an optimization (mathematics), optimization problem of the form
:
\begin
\mbox & f_0(x) \\
\mbox & f_i(x) \leq 1, \quad i=1, \ldots, m\\
& g_i(x) = 1, \quad i=1, \ldots, p,
\end
where f_0,\dots,f_m are posynomials and g_1 ...

is a technique whereby objective and inequality constraints expressed as posynomials and equality constraints as monomials can be transformed into a convex program.
* Integer programming
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquiall ...

studies linear programs in which some or all variables are constrained to take on integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

values. This is not convex, and in general much more difficult than regular linear programming.
* Quadratic programming
Quadratic programming (QP) is the process of solving certain mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their ...

allows the objective function to have quadratic terms, while the feasible set must be specified with linear equalities and inequalities. For specific forms of the quadratic term, this is a type of convex programming.
* Fractional programming studies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem.
* Nonlinear programming
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

studies the general case in which the objective function or the constraints or both contain nonlinear parts. This may or may not be a convex program. In general, whether the program is convex affects the difficulty of solving it.
* Stochastic programming
In the field of mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) val ...

studies the case in which some of the constraints or parameters depend on random variable
A random variable is a variable whose values depend on outcomes of a random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...

s.
* Robust optimizationRobust optimization is a field of optimization (mathematics), optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented as deterministic variability in the ...

is, like stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem. Robust optimization aims to find solutions that are valid under all possible realizations of the uncertainties defined by an uncertainty set.
* Combinatorial optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of Candidate solution, feasible solutions is Discrete set, discrete or can be reduced t ...

is concerned with problems where the set of feasible solutions is discrete or can be reduced to a discrete
Discrete in science is the opposite of :wikt:continuous, continuous: something that is separate; distinct; individual.
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic c ...

one.
* Stochastic optimization
Stochastic optimization (SO) methods are optimization method
Method ( grc, μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a presc ...

is used with random (noisy) function measurements or random inputs in the search process.
* Infinite-dimensional optimization studies the case when the set of feasible solutions is a subset of an infinite-dimension
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

al space, such as a space of functions.
* Heuristics
A heuristic (; ), or heuristic technique, is any approach to problem solving
Problem solving consists of using generic or ad hoc
Ad hoc is a Latin phrase
__NOTOC__
This is a list of Wikipedia articles of Latin phrases and their transla ...

and metaheuristic
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , , a ...

s make few or no assumptions about the problem being optimized. Usually, heuristics do not guarantee that any optimal solution need be found. On the other hand, heuristics are used to find approximate solutions for many complicated optimization problems.
* Constraint satisfaction In artificial intelligence
Artificial intelligence (AI) is intelligence
Intelligence has been defined in many ways: the capacity for abstraction
Abstraction in its main sense is a conceptual process where general rules and concept
Con ...

studies the case in which the objective function ''f'' is constant (this is used in artificial intelligence
Artificial intelligence (AI) is intelligence
Intelligence has been defined in many ways: the capacity for abstraction
Abstraction in its main sense is a conceptual process where general rules and concept
Concepts are defined as abstra ...

, particularly in automated reasoning
Automation describes a wide range of technologies that reduce human intervention in processes. Human intervention is reduced by predetermining decision criteria, subprocess relationships, and related actions — and embodying those predeterm ...

).
** Constraint programming
Constraint programming (CP) is a paradigm for solving combinatorial
Combinatorics is an area of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, ...

is a programming paradigm wherein relations between variables are stated in the form of constraints.
* Disjunctive programming is used where at least one constraint must be satisfied but not all. It is of particular use in scheduling.
* Space mapping
The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation and design Mathematical optimization, optimiza ...

is a concept for modeling and optimization of an engineering system to high-fidelity (fine) model accuracy exploiting a suitable physically meaningful coarse or surrogate modelA surrogate model is an engineering method used when an outcome of interest cannot be easily directly measured, so a model of the outcome is used instead. Most engineering design problems require experiments and/or simulations to evaluate design obje ...

.
In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):
* Calculus of variations
The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions
and functional (mathematics), functionals, to find maxima and minima of functionals: Map (mathemati ...

seeks to optimize an action integral over some space to an extremum by varying a function of the coordinates.
* Optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a Control (optimal control theory), control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applicatio ...

theory is a generalization of the calculus of variations which introduces control policies.
* Dynamic programming
Dynamic programming is both a mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lo ...

is the approach to solve the stochastic optimization
Stochastic optimization (SO) methods are optimization method
Method ( grc, μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a presc ...

problem with stochastic, randomness, and unknown model parameters. It studies the case in which the optimization strategy is based on splitting the problem into smaller subproblems. The equation that describes the relationship between these subproblems is called the Bellman equation
A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical Optimization (mathematics), optimization method known as dynamic programming. It writes the "value" of a decision problem ...

.
* Mathematical programming with equilibrium constraintsMathematical programming with equilibrium constraints (MPEC) is the study of
constrained optimization problems where the constraints include variational inequalities or complementarities. MPEC is related to the Stackelberg game.
MPEC is used ...

is where the constraints include variational inequalities or complementarities.
Multi-objective optimization

Adding more than one objective to an optimization problem adds complexity. For example, to optimize a structural design, one would desire a design that is both light and rigid. When two objectives conflict, a trade-off must be created. There may be one lightest design, one stiffest design, and an infinite number of designs that are some compromise of weight and rigidity. The set of trade-off designs that improve upon one criterion at the expense of another is known as the Pareto set. The curve created plotting weight against stiffness of the best designs is known as the Pareto frontier. A design is judged to be "Pareto optimal" (equivalently, "Pareto efficient" or in the Pareto set) if it is not dominated by any other design: If it is worse than another design in some respects and no better in any respect, then it is dominated and is not Pareto optimal. The choice among "Pareto optimal" solutions to determine the "favorite solution" is delegated to the decision maker. In other words, defining the problem as multi-objective optimization signals that some information is missing: desirable objectives are given but combinations of them are not rated relative to each other. In some cases, the missing information can be derived by interactive sessions with the decision maker. Multi-objective optimization problems have been generalized further into vector optimization problems where the (partial) ordering is no longer given by the Pareto ordering.Multi-modal or global optimization

Optimization problems are often multi-modal; that is, they possess multiple good solutions. They could all be globally good (same cost function value) or there could be a mix of globally good and locally good solutions. Obtaining all (or at least some of) the multiple solutions is the goal of a multi-modal optimizer. Classical optimization techniques due to their iterative approach do not perform satisfactorily when they are used to obtain multiple solutions, since it is not guaranteed that different solutions will be obtained even with different starting points in multiple runs of the algorithm. Common approaches toglobal optimization
Global optimization is a branch of applied mathematics and numerical analysis that attempts to find the global minima or maxima of a function or a set of functions on a given set. It is usually described as a minimization problem because the max ...

problems, where multiple local extrema may be present include evolutionary algorithm
In computational intelligence (CI), an evolutionary algorithm (EA) is a subset of evolutionary computation, a generic population-based metaheuristic optimization (mathematics), optimization algorithm. An EA uses mechanisms inspired by biological ev ...

s, Bayesian optimizationBayesian optimization is a sequential analysis, sequential design strategy for global optimization of Black box, black-box functions that does not assume any functional forms. It is usually employed to optimize expensive-to-evaluate functions.
Histo ...

and simulated annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema ( ...

.
Classification of critical points and extrema

Feasibility problem

The '' satisfiability problem'', also called the ''feasibility problem'', is just the problem of finding anyfeasible solution
A closed feasible region of a linear programming problem with three variables is a convex polyhedron.">polyhedron.html" ;"title="linear programming problem with three variables is a convex polyhedron">linear programming problem with three variabl ...

at all without regard to objective value. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal.
Many optimization algorithms need to start from a feasible point. One way to obtain such a point is to relax
Relax may refer to:
Aviation
* Roland Z-120 Relax, a German ultralight aircraft design for the 120 kg class
Music Albums
* ''Relax'' (Blank & Jones album), 2003
* ''Relax'' (Das Racist album), 2011
*'' Cool Relax'', a 1997 album by Jon B.
Songs ...

the feasibility conditions using a slack variableIn an optimization problem, a slack variable is a variable that is added to an inequality constraint to transform it into an equality. Introducing a slack variable replaces an inequality constraint with an equality constraint and a non-negativity c ...

; with enough slack, any starting point is feasible. Then, minimize that slack variable until the slack is null or negative.
Existence

Theextreme value theorem
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of ...

of Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) incl ...

states that a continuous real-valued function on a compact set attains its maximum and minimum value. More generally, a lower semi-continuous function on a compact set attains its minimum; an upper semi-continuous function on a compact set attains its maximum point or view.
Necessary conditions for optimality

One of Fermat's theorems states that optima of unconstrained problems are found atstationary point
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...

s, where the first derivative or the gradient of the objective function is zero (see first derivative testIn calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...

). More generally, they may be found at critical points, where the first derivative or gradient of the objective function is zero or is undefined, or on the boundary of the choice set. An equation (or set of equations) stating that the first derivative(s) equal(s) zero at an interior optimum is called a 'first-order condition' or a set of first-order conditions.
Optima of equality-constrained problems can be found by the Lagrange multiplier
In mathematical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimiza ...

method. The optima of problems with equality and/or inequality constraints can be found using the 'Karush–Kuhn–Tucker conditions In mathematical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimizat ...

'.
Sufficient conditions for optimality

While the first derivative test identifies points that might be extrema, this test does not distinguish a point that is a minimum from one that is a maximum or one that is neither. When the objective function is twice differentiable, these cases can be distinguished by checking the second derivative or the matrix of second derivatives (called theHessian matrix
In mathematic
Mathematics (from Greek: ) includes the study of such topics as quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in t ...

) in unconstrained problems, or the matrix of second derivatives of the objective function and the constraints called the bordered Hessian
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a function of many variables. The Hessi ...

in constrained problems. The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' (see 'Second derivative testIn calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...

'). If a candidate solution satisfies the first-order conditions, then the satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.
Sensitivity and continuity of optima

Theenvelope theoremIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

describes how the value of an optimal solution changes when an underlying parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified wh ...

changes. The process of computing this change is called comparative statics
In economics
Economics () is a social science
Social science is the branch
A branch ( or , ) or tree branch (sometimes referred to in botany
Botany, also called , plant biology or phytology, is the science of plant ...

.
The maximum theorem
The maximum theorem provides conditions for the continuous function, continuity of an Optimization (mathematics), optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959 ...

of Claude Berge
Claude Jacques Berge (5 June 1926 – 30 June 2002) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), f ...

(1963) describes the continuity of an optimal solution as a function of underlying parameters.
Calculus of optimization

For unconstrained problems with twice-differentiable functions, some critical points can be found by finding the points where thegradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

of the objective function is zero (that is, the stationary points). More generally, a zero subgradient
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

certifies that a local minimum has been found for minimization problems with convex functions
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

and other locallyIn mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighbourhood (mathematics), ne ...

Lipschitz function
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function
In mathematics
Mathematics (from Ancient Greek, Greek: ) i ...

s.
Further, critical points can be classified using the definiteness
In linguistics
Linguistics is the scientific study of language
A language is a structured system of communication
Communication (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic ...

of the Hessian matrix
In mathematic
Mathematics (from Greek: ) includes the study of such topics as quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in t ...

: If the Hessian is ''positive'' definite at a critical point, then the point is a local minimum; if the Hessian matrix is negative definite, then the point is a local maximum; finally, if indefinite, then the point is some kind of saddle point
In mathematics, a saddle point or minimax point is a Point (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), ...

.
Constrained problems can often be transformed into unconstrained problems with the help of Lagrange multiplier
In mathematical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimiza ...

s. Lagrangian relaxationIn the field of mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value ...

can also provide approximate solutions to difficult constrained problems.
When the objective function is a convex function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

, then any local minimum will also be a global minimum. There exist efficient numerical techniques for minimizing convex functions, such as interior-point methodInterior-point methods (also referred to as barrier methods or IPMs) are a certain class of algorithm
of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and ...

s.
Computational optimization techniques

To solve problems, researchers may usealgorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

s that terminate in a finite number of steps, or iterative method
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...

s that converge to a solution (on some specified class of problems), or heuristics
A heuristic (; ), or heuristic technique, is any approach to problem solving
Problem solving consists of using generic or ad hoc
Ad hoc is a Latin phrase
__NOTOC__
This is a list of Wikipedia articles of Latin phrases and their transla ...

that may provide approximate solutions to some problems (although their iterates need not converge).
Optimization algorithms

*Simplex algorithm
In mathematical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimi ...

of George Dantzig
George Bernard Dantzig (; November 8, 1914 – May 13, 2005) was an American mathematical scientist who made contributions to industrial engineering
Industrial Engineering is an engineering profession that is concerned with the optimization ...

, designed for linear programming
File:3dpoly.svg, A closed feasible region of a problem with three variables is a convex polyhedron. The surfaces giving a fixed value of the objective function are Plane (geometry), planes (not shown). The linear programming problem is to find a p ...

* Extensions of the simplex algorithm, designed for quadratic programming
Quadratic programming (QP) is the process of solving certain mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their ...

and for linear-fractional programming
* Variants of the simplex algorithm that are especially suited for network optimization.
* Combinatorial algorithms
The following is a list of algorithms along with one-line descriptions for each.
Automated planning
Combinatorial algorithms
General combinatorial algorithms
* Cycle detection#Brent's algorithm, Brent's algorithm: finds a cycle in function val ...

* Quantum optimization algorithms
Iterative methods

Theiterative methods
In computational mathematics
Computational mathematics involves mathematics, mathematical research in mathematics as well as in areas of science where computation, computing plays a central and essential role, and emphasizes algorithms, numerical ...

used to solve problems of nonlinear programming
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

differ according to whether they evaluate
Evaluation is a
systematic determination of a subject's merit, worth and significance, using criteria governed by a set of Standardization, standards. It can assist an organization, program, design, project or any other intervention or initiative ...

Hessians, gradients, or only function values. While evaluating Hessians (H) and gradients (G) improves the rate of convergence, for functions for which these quantities exist and vary sufficiently smoothly, such evaluations increase the computational complexity (or computational cost) of each iteration. In some cases, the computational complexity may be excessively high.
One major criterion for optimizers is just the number of required function evaluations as this often is already a large computational effort, usually much more effort than within the optimizer itself, which mainly has to operate over the N variables. The derivatives provide detailed information for such optimizers, but are even harder to calculate, e.g. approximating the gradient takes at least N+1 function evaluations. For approximations of the 2nd derivatives (collected in the Hessian matrix), the number of function evaluations is in the order of N². Newton's method requires the 2nd-order derivatives, so for each iteration, the number of function calls is in the order of N², but for a simpler pure gradient optimizer it is only N. However, gradient optimizers need usually more iterations than Newton's algorithm. Which one is best with respect to the number of function calls depends on the problem itself.
* Methods that evaluate Hessians (or approximate Hessians, using finite difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...

s):
** Newton's method
In numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathem ...

** Sequential quadratic programming: A Newton-based method for small-medium scale ''constrained'' problems. Some versions can handle large-dimensional problems.
** Interior point methodsInterior-point methods (also referred to as barrier methods or IPMs) are a certain class of algorithms that solve linear and nonlinear convex optimization problems.
John von Neumann suggested an interior-point method of linear programming, which was ...

: This is a large class of methods for constrained optimization, some of which use only (sub)gradient information and others of which require the evaluation of Hessians.
* Methods that evaluate gradients, or approximate gradients in some way (or even subgradients):
** Coordinate descentCoordinate descent is an optimization algorithm that successively minimizes along coordinate directions to find the minimum of a function. At each iteration, the algorithm determines a Coordinate system, coordinate or coordinate block via a coordinat ...

methods: Algorithms which update a single coordinate in each iteration
** Conjugate gradient method
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s: Iterative method
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...

s for large problems. (In theory, these methods terminate in a finite number of steps with quadratic objective functions, but this finite termination is not observed in practice on finite–precision computers.)
** Gradient descent
Gradient descent (also often called steepest descent) is a first-order iterative
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, ...

(alternatively, "steepest descent" or "steepest ascent"): A (slow) method of historical and theoretical interest, which has had renewed interest for finding approximate solutions of enormous problems.
** Subgradient methodSubgradient methods are iterative method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which t ...

s: An iterative method for large locallyIn mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighbourhood (mathematics), ne ...

Lipschitz functions
In mathematical analysis, Lipschitz continuity, named after Germany, German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics), functions. Intuitively, a Lipschitz continuous function is limited in h ...

using generalized gradients. Following Boris T. Polyak, subgradient–projection methods are similar to conjugate–gradient methods.
** Bundle method of descent: An iterative method for small–medium-sized problems with locally Lipschitz functions, particularly for convex minimization
Convex optimization is a subfield of mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having ...

problems (similar to conjugate gradient methods).
** Ellipsoid method
In mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt=
Mat ...

: An iterative method for small problems with objective functions and of great theoretical interest, particularly in establishing the polynomial time complexity of some combinatorial optimization problems. It has similarities with Quasi-Newton methods.
** Conditional gradient method (Frank–Wolfe) for approximate minimization of specially structured problems with linear constraints
Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to Proportionality (mathema ...

, especially with traffic networks. For general unconstrained problems, this method reduces to the gradient method, which is regarded as obsolete (for almost all problems).
** Quasi-Newton methodQuasi-Newton methods are methods used to either find zeroes or local maxima and minima of functions, as an alternative to Newton's method. They can be used if the Jacobian or Hessian is unavailable or is too expensive to compute at every iteration. ...

s: Iterative methods for medium-large problems (e.g. N<1000).
** Simultaneous perturbation stochastic approximationSimultaneous perturbation stochastic approximation (SPSA) is an algorithmic method for optimizing systems with multiple unknown parameters. It is a type of stochastic approximation algorithm. As an optimization method, it is appropriately suited to l ...

(SPSA) method for stochastic optimization; uses random (efficient) gradient approximation.
* Methods that evaluate only function values: If a problem is continuously differentiable, then gradients can be approximated using finite differences, in which case a gradient-based method can be used.
** Interpolation
In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantitie ...

methods
** Pattern search methods, which have better convergence properties than the Nelder–Mead heuristic (with simplices), which is listed below.
Global convergence

More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies online search
In optimization (mathematics), optimization, the line search strategy is one of two basic iteration, iterative approaches to find a maxima and minima, local minimum \mathbf^* of an objective function f:\mathbb R^n\to\mathbb R. The other approach is ...

es, which optimize a function along one dimension. A second and increasingly popular method for ensuring convergence uses trust regionIn mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt=
Math ...

s. Both line searches and trust regions are used in modern methods of non-differentiable optimization. Usually, a global optimizer is much slower than advanced local optimizers (such as BFGS), so often an efficient global optimizer can be constructed by starting the local optimizer from different starting points.
Heuristics

Besides (finitely terminating)algorithm
In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specifications for performing s, , , and other ...

s and (convergent) iterative method
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...

s, there are heuristics
A heuristic (; ), or heuristic technique, is any approach to problem solving
Problem solving consists of using generic or ad hoc
Ad hoc is a Latin phrase
__NOTOC__
This is a list of Wikipedia articles of Latin phrases and their transla ...

. A heuristic is any algorithm which is not guaranteed (mathematically) to find the solution, but which is nevertheless useful in certain practical situations. List of some well-known heuristics:
* Mayfly Optimization Algorithm
* Memetic algorithm
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of Algor ...

* Differential evolution
* Evolutionary algorithms
In computational intelligence (CI), an evolutionary algorithm (EA) is a subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (a ...

* Dynamic relaxation
* Genetic algorithms
spacecraft antenna. This complicated shape was found by an evolutionary computer design program to create the best radiation pattern. It is known as an evolved antenna.
In computer science
Computer science deals with the theoretical found ...

* Hill climbing
In numerical analysis, hill climbing is a Optimization (mathematics), mathematical optimization technique which belongs to the family of Local search (optimization), local search. It is an iterative algorithm that starts with an arbitrary solut ...

with random restart
* Nelder–Mead simplicial heuristic: A popular heuristic for approximate minimization (without calling gradients)
* Particle swarm optimization
In , particle swarm optimization (PSO) is a computational method that a problem by trying to improve a with regard to a given measure of quality. It solves a problem by having a population of candidate solutions, here dubbed s, and moving the ...

* Gravitational search algorithm
* Simulated annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema ( ...

* Stochastic tunneling
* Tabu searchTabu search is a metaheuristic search method employing local search methods used for mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex verti ...

* Reactive Search Optimization (RSO) implemented in LIONsolver
* Forest Optimization Algorithm
A forest is an area of land dominated by tree
In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, supporting branches and leaves in most species. In some usages, the definition of a tr ...

Applications

Mechanics

Problems inrigid body dynamics
In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external force
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (e ...

(in particular articulated rigid body dynamics) often require mathematical programming techniques, since you can view rigid body dynamics as attempting to solve an ordinary differential equation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

on a constraint manifold; the constraints are various nonlinear geometric constraints such as "these two points must always coincide", "this surface must not penetrate any other", or "this point must always lie somewhere on this curve". Also, the problem of computing contact forces can be done by solving a linear complementarity problemIn mathematical optimization (mathematics), optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It was proposed by Cottle a ...

, which can also be viewed as a QP (quadratic programming) problem.
Many design problems can also be expressed as optimization programs. This application is called design optimization. One subset is the engineering optimizationEngineering optimization is the subject which uses optimization techniques to achieve design goals in engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, ...

, and another recent and growing subset of this field is multidisciplinary design optimization
Multi-disciplinary design optimization (MDO) is a field of engineering that uses optimization (mathematics), optimization methods to solve design problems incorporating a number of disciplines. It is also known as multidisciplinary system design opt ...

, which, while useful in many problems, has in particular been applied to aerospace engineering
Aerospace engineering is the primary field of engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipl ...

problems.
This approach may be applied in cosmology and astrophysics.
Economics and finance

Economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interact ...

is closely enough linked to optimization of agents that an influential definition relatedly describes economics ''qua'' science as the "study of human behavior as a relationship between ends and means" with alternative uses. Modern optimization theory includes traditional optimization theory but also overlaps with game theory
Game theory is the study of mathematical model
A mathematical model is a description of a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
...

and the study of economic equilibria. The ''Journal of Economic Literature
The ''Journal of Economic Literature'' is a peer-reviewed
Peer review is the evaluation of work by one or more people with similar competencies as the producers of the work ( peers). It functions as a form of self-regulation by qualified mem ...

'' codes
In communication
Communication (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Lati ...

classify mathematical programming, optimization techniques, and related topics under JEL:C61-C63.
In microeconomics, the utility maximization problem
Utility maximization was first developed by utilitarian philosophers Jeremy Bentham
Jeremy Bentham (; 15 February 1748 Old_Style_and_New_Style_dates">O.S._4_February_1747.html" ;"title="Old_Style_and_New_Style_dates.html" ;"title="nowiki/ ...

and its dual problemIn mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a low ...

, the expenditure minimization problem
In microeconomics, the expenditure minimization problem is the Duality (optimization), dual of the utility maximization problem: "how much money do I need to reach a certain level of happiness?". This question comes in two parts. Given a consumer' ...

, are economic optimization problems. Insofar as they behave consistently, consumer
A consumer is a person or a group who intends to order, orders, or uses purchased goods, products, or Service (economics), services primarily for personal, social, family, household and similar needs, not directly related to entrepreneurial or bu ...

s are assumed to maximize their utility
As a topic of economics
Economics () is a social science
Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within thos ...

, while firm
A company, abbreviated as co., is a legal entity
In law
Law is a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system, surro ...

s are usually assumed to maximize their profit
Profit may refer to:
Business and law
* Profit (accounting)
Profit, in accounting
Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information about economic entity, economic ...

. Also, agents are often modeled as being , thereby preferring to avoid risk. Asset prices are also modeled using optimization theory, though the underlying mathematics relies on optimizing stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a Indexed family, family of random variables. Stochastic processes are widely used as mathematical models of systems and phen ...

es rather than on static optimization. International trade theory
International trade theory is a sub-field of economics
Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and ...

also uses optimization to explain trade patterns between nations. The optimization of portfolios is an example of multi-objective optimization in economics.
Since the 1970s, economists have modeled dynamic decisions over time using control theory
Control theory deals with the control of dynamical system
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contai ...

. For example, dynamic search models
In microeconomics, search theory studies buyers or sellers who cannot instantly find a trading partner, and must therefore search for a partner prior to transacting.
Search theory has been influential in many areas of economics. It has been appli ...

are used to study labor-market behavior. A crucial distinction is between deterministic and stochastic models. Macroeconomists build dynamic stochastic general equilibrium (DSGE) models that describe the dynamics of the whole economy as the result of the interdependent optimizing decisions of workers, consumers, investors, and governments.
Electrical engineering

Some common applications of optimization techniques inelectrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics
The field of electronics is a branch of physics and electrical enginee ...

include active filter
An active filter is a type of analog circuit
Analogue electronics ( en-US, analog electronics) are electronics, electronic systems with a continuous function, continuously variable signal, in contrast to digital electronics where signals usuall ...

design, stray field reduction in superconducting magnetic energy storage systems, space mapping
The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation and design Mathematical optimization, optimiza ...

design of microwave
Microwave is a form of electromagnetic radiation
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space a ...

structures, handset antennas, electromagnetics-based design. Electromagnetically validated design optimization of microwave components and antennas has made extensive use of an appropriate physics-based or empirical surrogate modelA surrogate model is an engineering method used when an outcome of interest cannot be easily directly measured, so a model of the outcome is used instead. Most engineering design problems require experiments and/or simulations to evaluate design obje ...

and space mapping
The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation and design Mathematical optimization, optimiza ...

methodologies since the discovery of space mapping
The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation and design Mathematical optimization, optimiza ...

in 1993.
Civil engineering

Optimization has been widely used in civil engineering.Construction management
Construction management (CM) is a professional service that uses specialized, project management techniques to oversee the planning, design, and construction of a project, from its beginning to its end. The purpose of Construction management is to c ...

and transportation engineering
The engineering of this roundabout in Bristol, England, attempts to make traffic flow free-moving">Bristol.html" ;"title="roundabout in Bristol">roundabout in Bristol, England, attempts to make traffic flow free-moving
Transportation engineeri ...

are among the main branches of civil engineering that heavily rely on optimization. The most common civil engineering problems that are solved by optimization are cut and fill of roads, life-cycle analysis of structures and infrastructures, resource levelingIn project management, resource leveling is defined by ''A Guide to the Project Management Body of Knowledge'' (''PMBOK Guide'') as "A technique in which start and finish dates are adjusted based on resource limitation with the goal of balancing dem ...

, water resource allocation, traffic
Traffic on roads
A road is a wide way leading from one place to another, typically one with a specially prepared surface which vehicles and bikes can use.
Roads consist of one or two roadway
A carriageway (British English
Briti ...

management and schedule optimization.
Operations research

Another field that uses optimization techniques extensively is operations research. Operations research also uses stochastic modeling and simulation to support improved decision-making. Increasingly, operations research usesstochastic programming
In the field of mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) val ...

to model dynamic decisions that adapt to events; such problems can be solved with large-scale optimization and stochastic optimization
Stochastic optimization (SO) methods are optimization method
Method ( grc, μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a presc ...

methods.
Control engineering

Mathematical optimization is used in much modern controller design. High-level controllers such as model predictive control (MPC) or real-time optimization (RTO) employ mathematical optimization. These algorithms run online and repeatedly determine values for decision variables, such as choke openings in a process plant, by iteratively solving a mathematical optimization problem including constraints and a model of the system to be controlled.Geophysics

Optimization techniques are regularly used ingeophysical
Geophysics () is a subject of natural science
Natural science is a Branches of science, branch of science concerned with the description, understanding and prediction of Phenomenon, natural phenomena, based on empirical evidence from observa ...

parameter estimation problems. Given a set of geophysical measurements, e.g. seismic recordings, it is common to solve for the physical properties
A physical property is any Property (philosophy), property that is Measurement, measurable, whose value describes a state of a physical system. The changes in the physical properties of a system can be used to describe its changes between momenta ...

and geometrical shapes of the underlying rocks and fluids. The majority of problems in geophysics are nonlinear with both deterministic and stochastic methods being widely used.
Molecular modeling

Nonlinear optimization methods are widely used in conformational analysis.Computational systems biology

Optimization techniques are used in many facets of computational systems biology such as model building, optimal experimental design, metabolic engineering, and synthetic biology.Linear programming
Linear programming (LP, also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model
A mathematical model is a description of a system
A system is a group of ...

has been applied to calculate the maximal possible yields of fermentation products, and to infer gene regulatory networks from multiple microarray datasets as well as transcriptional regulatory networks from high-throughput data. Nonlinear programming
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

has been used to analyze energy metabolism and has been applied to metabolic engineering and parameter estimation in biochemical pathways.
Machine learning

Solvers

See also

*Brachistochrone
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

* Curve fitting
Curve fitting is the process of constructing a curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve m ...

* Deterministic global optimization
* Goal programming
Goal programming is a branch of multiobjective optimization, which in turn is a branch of multi-criteria decision analysis (MCDA). It can be thought of as an extension or generalisation of linear programming to handle multiple, normally conflicting ...

*
* Least squares
The method of least squares is a standard approach in regression analysis
In ing, regression analysis is a set of statistical processes for the relationships between a (often called the 'outcome' or 'response' variable) and one or more s ...

* Mathematical Optimization SocietyThe Mathematical Optimization Society (MOS), known as the Mathematical Programming Society until 2010,Mathematical optimization algorithms
* :Mathematical optimization software, Mathematical optimization software
* Process optimization
* Simulation-based optimization
* Test functions for optimization
* Variational calculus
* Vehicle routing problem

Notes

Further reading

* * * * *External links

* Links to optimization source codes * * * {{Authority control Mathematical optimization, Operations research Mathematical and quantitative methods (economics), Optimization