Relating Optimization Problems to Systems of Inequalities and Equalities

摘要

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.