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Second-order Cone Programming
A second-order cone program (SOCP) is a convex optimization problem of the form :minimize \ f^T x \ :subject to ::\lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i,\quad i = 1,\dots,m ::Fx = g \ where the problem parameters are f \in \mathbb^n, \ A_i \in \mathbb^, \ b_i \in \mathbb^, \ c_i \in \mathbb^n, \ d_i \in \mathbb, \ F \in \mathbb^, and g \in \mathbb^p. x\in\mathbb^n is the optimization variable. \lVert x \rVert_2 is the Euclidean norm and ^T indicates transpose. The "second-order cone" in SOCP arises from the constraints, which are equivalent to requiring the affine function (A x + b, c^T x + d) to lie in the second-order cone in \mathbb^. SOCPs can be solved by interior point methods and in general, can be solved more efficiently than semidefinite programming (SDP) problems. Some engineering applications of SOCP include filter design, antenna array weight design, truss design, and grasping force optimization in robotics. Applications in quantitative finance include p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard. Convex optimization has applications in a wide range of disciplines, such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, finance, statistics (optimal experimental design), and structural optimization, where the approximation concept has proven to be efficient. With recent advancements in computing and optimization algorithms, convex programming is nearly as straightforward as linear programming. Definition A convex optimization problem is an optimization problem in which the objective function is a convex function and the feasible set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semialgebraic Set
In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n) = 0) and inequalities (of the form Q(x_1,...,x_n) > 0), or any finite union of such sets. A semialgebraic function is a function with a semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers. Properties Similarly to algebraic subvarieties, finite unions and intersections of semialgebraic sets are still semialgebraic sets. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski–Seidenberg theorem says that they are also closed under the projection operation: in other words a semialgebraic set projected onto a linear subspace yields another such (as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NAG Numerical Library
The NAG Numerical Library is a software product developed and sold by The Numerical Algorithms Group Ltd. It is a software library of numerical analysis routines, containing more than 1,900 mathematical and statistical algorithms. Areas covered by the library include linear algebra, optimization, quadrature, the solution of ordinary and partial differential equations, regression analysis, and time series analysis. Users of the NAG Library call its routines from within their applications in order to incorporate its mathematical or statistical functionality and to solve numerical problems - for example, finding the minimum or maximum of a function, fitting a curve or surface to data, or solving a differential equation. The Library is available in the many forms, but namely the NAG C Library, the NAG Fortran Library, and the NAG Library for .NET. Its contents are accessible from several computing environments, including standard languages such as C, C++, Fortran, Visual Basic, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MOSEK
MOSEK is a software package for the solution of linear, mixed-integer linear, quadratic, mixed-integer quadratic, quadratically constraint, conic and convex nonlinear mathematical optimization problems. The applicability of the solver varies widely and is commonly used for solving problems in areas such as engineering, finance and computer science. The emphasis in MOSEK is on solving large scale sparse problems, in particular the interior-point optimizer for linear, conic quadratic (a.k.a. Second-order cone programming) and semi-definite (aka. semidefinite programming), which the software is considerably efficient solving. A special feature of the solver, is its interior-point optimizer, based on the so-called homogeneous model. This implies that MOSEK can reliably detect a primal and/or dual infeasible status as documented in several published papers. In addition to the interior-point optimizer MOSEK includes: * Primal and dual simplex optimizer for linear problems. * Mixed-int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. As of 2020, MATLAB has more than 4 million users worldwide. They come from various backgrounds of engineering, science, and economics. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was based on his 1960s PhD thesis. Moler became a math professor at the University of New Mexico an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FICO Xpress
The FICO Xpress optimizer is a commercial optimization solver for linear programming (LP), mixed integer linear programming (MILP), convex quadratic programming (QP), convex quadratically constrained quadratic programming (QCQP), second-order cone programming (SOCP) and their mixed integer counterparts. Xpress includes a general purpose non-linear solver, Xpress NonLinear, including a successive linear programming algorithm (SLP, first-order method), and Artelys Knitro (second-order methods). Xpress was originally developed by Dash Optimization, and was acquired by FICO in 2008. Its initial authors were Bob Daniel and Robert Ashford. The first version of Xpress could only solve LPs; support for MIPs was added in 1986. Being released in 1983, Xpress was the first commercial LP and MIP solver running on PCs. In 1992, an Xpress version for parallel computing was published, which was extended to distributed computing five years later. Xpress was the first MIP solver to cross the billi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CPLEX
IBM ILOG CPLEX Optimization Studio (often informally referred to simply as CPLEX) is an optimization software package. In 2004, the work on CPLEX earned the first INFORMS Impact Prize. History The CPLEX Optimizer was named for the simplex method as implemented in the C programming language, although today it also supports other types of mathematical optimization and offers interfaces other than C. It was originally developed by Robert E. Bixby and sold commercially from 1988 by CPLEX Optimization Inc. This was acquired by ILOG in 1997 and ILOG was subsequently acquired by IBM in January 2009. CPLEX continues to be actively developed by IBM. Features The IBM ILOG CPLEX Optimizer solves integer programming problems, very large linear programming problems using either primal or dual variants of the simplex method or the barrier interior point method, convex and non-convex quadratic programming problems, and convex quadratically constrained problems (solved via second- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artelys Knitro
Artelys Knitro is a commercial software package for solving large scale nonlinear mathematical optimization problems. KNITRO – (the original solver name) short for "Nonlinear Interior point Trust Region Optimization" (the "K" is silent) – was co-created by Richard Waltz, Jorge Nocedal, Todd Plantenga and Richard Byrd. It was first introduced in 2001, as a derivative of academic research at Northwestern University (through Ziena Optimization LLC), and has since been continually improved by developers at Artelys. Optimization problems must be presented to Knitro in mathematical form, and should provide a way of computing function derivatives using sparse matrices (Knitro can compute derivatives approximation but in most cases providing the exact derivatives is beneficial). An often easier approach is to develop the optimization problem in an algebraic modeling language. The modeling environment computes function derivatives, and Knitro is called as a "solver" from within the en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
AMPL
AMPL (A Mathematical Programming Language) is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing (i.e., large-scale optimization and scheduling-type problems). It was developed by Robert Fourer, David Gay, and Brian Kernighan at Bell Laboratories. AMPL supports dozens of solvers, both open source and commercial software, including CBC, CPLEX, FortMP, MINOS, IPOPT, SNOPT, KNITRO, and LGO. Problems are passed to solvers as nl files. AMPL is used by more than 100 corporate clients, and by government agencies and academic institutions. One advantage of AMPL is the similarity of its syntax to the mathematical notation of optimization problems. This allows for a very concise and readable definition of problems in the domain of optimization. Many modern solvers available on the NEOS Server (formerly hosted at the Argonne National Laboratory, currently hosted at the University of Wisconsin, Madison) accept AMPL inpu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Cumulative Distribution Function
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distrib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Linear Program
In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. A stochastic program is an optimization problem in which some or all problem parameters are uncertain, but follow known probability distributions. This framework contrasts with deterministic optimization, in which all problem parameters are assumed to be known exactly. The goal of stochastic programming is to find a decision which both optimizes some criteria chosen by the decision maker, and appropriately accounts for the uncertainty of the problem parameters. Because many real-world decisions involve uncertainty, stochastic programming has found applications in a broad range of areas ranging from finance to transportation to energy optimization. Two-stage problems The basic idea of two-stage stochastic programming is that (optimal) decisions should be based on data available at the time the decisions are made and cannot depend on future ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
