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Extended Mathematical Programming (EMP)
Algebraic modeling languages like AIMMS, AMPL, GAMS, MPL and others have been developed to facilitate the description of a problem in mathematical terms and to link the abstract formulation with data-management systems on the one hand and appropriate algorithms for solution on the other. Robust algorithms and modeling language interfaces have been developed for a large variety of mathematical programming problems such as linear programs (LPs), nonlinear programs (NPs), mixed integer programs (MIPs), mixed complementarity programs (MCPs) and others. Researchers are constantly updating the types of problems and algorithms that they wish to use to model in specific domain applications. Extended Mathematical Programming (EMP) is an extension to algebraic modeling languages that facilitates the automatic reformulation of new model types by converting the EMP model into established mathematical programming classes to solve by mature solver algorithms. A number of important problem class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Modeling Language
Algebraic modeling languages (AML) are high-level computer programming languages for describing and solving high complexity problems for large scale mathematical computation (i.e. large scale optimization type problems). 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. An AML does not solve those problems directly; instead, it calls appropriate external algorithms to obtain a solution. These algorithms are called solvers and can handle certain kind of mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bilevel Optimization
Bilevel optimization is a special kind of optimization where one problem is embedded (nested) within another. The outer optimization task is commonly referred to as the upper-level optimization task, and the inner optimization task is commonly referred to as the lower-level optimization task. These problems involve two kinds of variables, referred to as the upper-level variables and the lower-level variables. Mathematical formulation of the problem A general formulation of the bilevel optimization problem can be written as follows: : \min\limits_\;\; F(x,y) subject to: : G_i(x,y) \leq 0, for i \in \ : y \in \arg \min \limits_ \ where : F,f: R^ \times R^ \to R : G_i,g_j: R^ \times R^ \to R : X \subseteq R^ : Y \subseteq R^. In the above formulation, F represents the upper-level objective function and f represents the lower-level objective function. Similarly x represents the upper-level decision vector and y represents the lower-level decision vector. G_i and g_j represent the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |