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Stonyakin, F., Gasnikov, A., Tyurin, A., Pasechnyuk, D., Agafonov, A., Dvurechensky, P., Dvinskikh, D., Kroshnin, A. and Piskunova, V. (2020) Inexact Model: A Framework for Optimization and Variational Inequalities. Cornell University, New York.

has been cited by the following article:

• 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.