Finding Optimal Solutions to a Class of Parametric Optimization Problems in Terms of Parameter Values by using Multilayer Neural Networks

Document Type : Original Paper

Authors

1 Department of Mathematice, Tabriz Branch, Islamic Azad University, Tabriz, Iran.

2 Department of Mathematics, Tabriz Branch, Islamic azad University, Tabriz, Iran.

3 Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.

Abstract

‎‎In this paper, parametric optimization problems are investigated. ‎In a‎ ‎parametric ‎optimization ‎problem ‎we ‎assume ‎‏‎$‎‎‎‎‎‏‎‎lambda‎in‎mathbb{R}^n‎$‎‎ ‎is ‎the ‎vector ‎of ‎the ‎parameters ‎and ‎‎$‎‎x^*$ ‎is ‎the ‎optimal ‎answer ‎corresponding ‎to ‎it. ‎The ‎purpose ‎of ‎this ‎paper ‎is ‎to ‎determine a‎ ‎function ‎such ‎as ‎‎$‎‎psi$ ‎so ‎that ‎we ‎have ‎‎$‎‎psi(‎lambda‎)=x^*$.‎ To do this, first for each ‎$‎‎‎lambda‎$‎, the corresponding optimal answer is calculated. In this way, a set of data bases consisting of parameters and the corresponding optimal values are obtained. A multilayer network of data base is trained to obtain the function ‎$‎‎psi$‎ in a domain. In fact, the function ‎$‎‎psi$‎ for each value of the parameter specifies the corresponding answer by the trained multilayer network.‎‎ Finally, we conduct several numerical examples to test our method.

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