Relating Optimization Problems to Systems of Inequalities and Equalities ()
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.
Share and Cite:
Corley, H. and Dwobeng, E. (2020) Relating Optimization Problems to Systems of Inequalities and Equalities.
American Journal of Operations Research,
10, 284-298. doi:
10.4236/ajor.2020.106016.