In certain
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. It is generally divided into two subfi ...
problems the unknown optimal solution might not be a number or a vector, but rather a continuous quantity, for example a
function or the shape of a body. Such a problem is an infinite-dimensional optimization problem, because, a continuous quantity cannot be determined by a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
number of certain
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
.
Examples
* Find the
shortest path
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
The problem of finding the shortest path between ...
between two points in a plane. The variables in this problem are the curves connecting the two points. The optimal solution is of course the line segment joining the points, if the metric defined on the plane is the Euclidean metric.
* Given two cities in a country with many hills and valleys, find the shortest road going from one city to the other. This problem is a generalization of the above, and the solution is not as obvious.
* Given two circles which will serve as top and bottom for a cup of given height, find the shape of the side wall of the cup so that the side wall has
minimal area. The intuition would suggest that the cup must have conical or cylindrical shape, which is false. The actual minimum surface is the
catenoid
In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally descri ...
.
* Find the shape of a bridge capable of sustaining given amount of traffic using the smallest amount of material.
* Find the shape of an airplane which bounces away most of the radio waves from an enemy radar.
Infinite-dimensional optimization problems can be more challenging than finite-dimensional ones. Typically one needs to employ methods from
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s to solve such problems.
Several disciplines which study infinite-dimensional optimization problems are
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
,
optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ...
and
shape optimization.
See also
*
Semi-infinite programming In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a finite number of constraints. In the former case the ...
References
* David Luenberger (1997). ''Optimization by Vector Space Methods.'' John Wiley & Sons. {{ISBN, 0-471-18117-X.
* Edward J. Anderson and Peter Nash, ''Linear Programming in Infinite-Dimensional Spaces'', Wiley, 1987.
* M. A. Goberna and M. A. López, ''Linear Semi-Infinite Optimization'', Wiley, 1998.
* Cassel, Kevin W.: Variational Methods with Applications in Science and Engineering, Cambridge University Press, 2013.
Functional analysis
Optimization in vector spaces