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Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize a multivariate quadratic function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming.
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome in a mathematical model whose requirements and objective are represented by linear relationships. Linear programming is a special case of mathematical programming.
Mathematical optimization or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.
In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms. Quadratus is Latin for square.
AMPL is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing . 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, MOSEK, 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.
Computational Infrastructure for Operations Research (COIN-OR), is a project that aims to "create for mathematical software what the open literature is for mathematical theory." The open literature provides the operations research (OR) community with a peer-review process and an archive. Papers in operations research journals on mathematical theory often contain supporting numerical results from computational studies. The software implementations, models, and data used to produce the numerical results are typically not published. The status quo impeded researchers needing to reproduce computational results, make fair comparisons, and extend the state of the art.
The TOMLAB Optimization Environment is a modeling platform for solving applied optimization problems in MATLAB.
Algebraic modeling languages (AML) are high-level computer programming languages for describing and solving high complexity problems for large scale mathematical computation. One particular advantage of some algebraic modeling languages like AIMMS, AMPL, GAMS, Gekko, MathProg, Mosel, and OPL is the similarity of their syntax to the mathematical notation of optimization problems. This allows for a very concise and readable definition of problems in the domain of optimization, which is supported by certain language elements like sets, indices, algebraic expressions, powerful sparse index and data handling variables, constraints with arbitrary names. The algebraic formulation of a model does not contain any hints how to process it.
nl is a file format for presenting and archiving mathematical programming problems. Initially, this format has been invented for connecting solvers to AMPL. It has also been adopted by other systems such as COIN-OR, FortSP, and Coopr.
The Zuse Institute Berlin is a research institute for applied mathematics and computer science on the campus of Freie Universität Berlin in Dahlem, Berlin, Germany.
BARON is a computational system for solving non-convex optimization problems to global optimality. Purely continuous, purely integer, and mixed-integer nonlinear problems can be solved by the solver. Linear programming (LP), nonlinear programming (NLP), mixed integer programming (MIP), and mixed integer nonlinear programming (MINLP) are supported. In a comparison of different solvers, BARON solved the most benchmark problems and required the least amount of time per problem.
MOSEK is a software package for the solution of linear, mixed-integer linear, quadratic, mixed-integer quadratic, quadratically constrained, 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.
AIMMS is a prescriptive analytics software company with offices in the Netherlands, United States and Singapore.
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, and Artelys Knitro.
Parametric programming is a type of mathematical optimization, where the optimization problem is solved as a function of one or multiple parameters. Developed in parallel to sensitivity analysis, its earliest mention can be found in a thesis from 1952. Since then, there have been considerable developments for the cases of multiple parameters, presence of integer variables as well as nonlinearities.
Deterministic global optimization is a branch of numerical optimization which focuses on finding the global solutions of an optimization problem whilst providing theoretical guarantees that the reported solution is indeed the global one, within some predefined tolerance. The term "deterministic global optimization" typically refers to complete or rigorous optimization methods. Rigorous methods converge to the global optimum in finite time. Deterministic global optimization methods are typically used when locating the global solution is a necessity, when it is extremely difficult to find a feasible solution, or simply when the user desires to locate the best possible solution to a problem.
Artelys Knitro is a commercial software package for solving large scale nonlinear mathematical optimization problems.
The quadratic knapsack problem (QKP), first introduced in 19th century, is an extension of knapsack problem that allows for quadratic terms in the objective function: Given a set of items, each with a weight, a value, and an extra profit that can be earned if two items are selected, determine the number of items to include in a collection without exceeding capacity of the knapsack, so as to maximize the overall profit. Usually, quadratic knapsack problems come with a restriction on the number of copies of each kind of item: either 0, or 1. This special type of QKP forms the 0-1 quadratic knapsack problem, which was first discussed by Gallo et al. The 0-1 quadratic knapsack problem is a variation of knapsack problems, combining the features of unbounded knapsack problem, 0-1 knapsack problem and quadratic knapsack problem.
ANTIGONE, is a deterministic global optimization solver for general Mixed-Integer Nonlinear Programs (MINLP).
References
- ↑ Neumaier, Arnold; Oleg Shcherbina (March 2004). "Safe bounds in linear and mixed-integer linear programming". Mathematical Programming. 99 (2): 283–296. CiteSeerX 10.1.1.373.508 . doi:10.1007/s10107-003-0433-3. ISSN 0025-5610.