Optimization Using Mathematica
Asghar Bhatti
University of Iowa
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Optimization problems arise very naturally in many different disciplines. A structural
engineer designing a multistory building must choose materials and proportions for
different structural components in the building in order to have a safe structure that is as economical as possible. A portfolio manager for a large mutual fund company must choose
investments that generate the largest possible rate of return for its investors while
keeping the risk of major losses to acceptably low levels. A plant manager in a
manufacturing facility must schedule the plant operations such that the plant produces
products that maximize the company's revenues while meeting customer demands for
different products and staying within available resource limitations. A scientist in a
research laboratory may be interested in finding a mathematical function that best
describes an observed physical phenomenon.
Over the last four decades or so, a large number of algorithms have been developed to
solve these types of optimization problems. Most of these algorithms are natural for
implementation within Mathematica. However, builtin support in Mathematica
for solving optimization problems is limited and consists of following functions.
FindMinimum : solves a nonlinear unconstrained problems
using a variety of methods, including conjugate gradient and BFGS quasiNewton
methods.
ConstrainedMin and ConstrainedMax :
These functions solve linear programming problems. All variables are restricted to be
positive.
LinearProgramming : Solves the same class of problems as the
ConstrainedMin and the ConstrainedMax ;
however, the problem must be specified in the form of matrices of coefficients of linear
functions.
Currently there is no builtin support for solving nonlinearly constrained optimization
problems. For teaching optimization, the builtin functions have another drawback in that
they typically return only the optimum solution. Intermediate results cannot be seen to
gain an understanding of the process.
For the past several years the author has been working on developing Mathematica
functions for optimization that overcome these limitations. A number of functions have been developed to solve quadratic programming problems and general nonlinear constrained
optimization problems. In addition to supplementing Mathematica's builtin support
for optimization, these functions optionally provide details of all intermediate results.
In this way they can be used by students and others who are interested in the learning theory
on which these algorithms are based. For small problems, functions have been developed for
graphical solution and direct solution using optimality conditions. Several functions,
duplicating the functionality of the builtin functions, have also been developed to meet this
educational mission. The paper will demonstrate the use of these functions. Implementation
of some of these algorithms will be discussed.
