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Exact Global Constrained Optimization

Adam Strzebonski
Organization: Wolfram Research, Inc.
Department: Kernel Technology

2005 Wolfram Technology Conference
Conference location

Champaign IL

Mathematica functions Minimize and Maximize allow users to compute exact global extrema of functions on sets constrained by systems of equations and inequalities. When the objective function and the constraints are real algebraic functions, methods based on cylindrical algebraic decomposition allow users to always compute global extrema (or extremal values, if they are not attained). If parameters are present, the developmental version of Mathematica can compute the extrema as functions of parameters. In addition to cylindrical algebraic decomposition, in certain special cases, like linear objective and constraints, bounded constraints, or univariate objective and constraints, faster methods can be used. A method based on Lagrange multipliers, used for (sufficiently regular) bounded constraints, can also succeed for some transcendental problems, thanks to the transcendental equation and inequality solving functionality. Mathematica can also solve some optimization problems over the integers. Bounded linear problems are solved by integer linear programming methods. Several other integer optimization problems can be solved by a combination of real optimization methods and integer solution finding. In my talk I will show examples and discuss the algorithms used.

*Applied Mathematics > Optimization
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Talk0510Optim.nb (1.4 MB) - Mathematica Notebook [for Mathematica 5.2]