TITLE:
Relating Optimization Problems to Systems of Inequalities and Equalities
AUTHORS:
H. W. Corley, E. O. Dwobeng
KEYWORDS:
Optimization, Inequalities and Equalities, Goal Programming, Games, Diophantine Equations
JOURNAL NAME:
American Journal of Operations Research,
Vol.10 No.6,
November
10,
2020
ABSTRACT: In quantitative decision analysis, an analyst
applies mathematical models to make decisions. Frequently these models involve
an optimization problem to determine the
values of the decision variables, a system S of possibly non- linear inequalities and
equalities to restrict these variables, or both. In this note, we relate a general nonlinear
programming problem to such a system S in such a way as to provide a solution
of either by solving the other—with certain limitations. We first start
with S and generalize phase 1 of the
two-phase simplex method to either solve S or establish that a solution does not exist. A conclusion is reached by trying
to solve S by minimizing a sum of
artificial variables subject to the system S as constraints. Using examples, we illustrate how
this approach can give the core of a cooperative game and an equilibrium
for a noncooperative game, as well as solve both linear and nonlinear goal
programming problems. Similarly, we start with a general nonlinear programming
problem and present an algorithm to solve it as a series of systems S by generalizing the “sliding objective function method” for two-dimensional
linear programming. An example is presented to illustrate the geometrical
nature of this approach.