An Iterative Algorithm for the Nonlinear MC2 Model with Variational Inequality Method

Chao Min,Zhaozhong Yang,Feifei Fan,Xiaogang Li

doi:10.3390/math7060514

Abstract

The multiple criteria and multiple constraint level (MC 2 ) model is a useful tool to deal with the decision programming problems, which concern multiple decision makers and uncertain resource constraint levels. In this paper, by regarding the nonlinear MC 2 problems as a class of mixed implicit variational inequalities, we develop an iterative algorithm to solve the nonlinear MC 2 problems through the resolvent operator technique. The convergence of the generated iterative sequence is analyzed and discussed by a calculation example, and the stability of Algorithm 1 is also verified by error propagation. By comparing with two other MC 2 -algorithms, Algorithm 1 performs well in terms of number of iterations and computation complexity.

Highlights

In practical applications, the problem involving multi criteria decision-making has become a research hotspot
In [3], a nonlinear programming (NP) model based on the technique for order preference by similarity to ideal solution (TOPSIS) was developed to solve decision-making problems
Traditional methods for solving nonlinear programming include the steepest descent algorithm, Newton method, feasible direction method, function approximation method and trust region method. Aside from those methods, the enhanced Lagrange method is to solve the problem by replacing the original constraint problem with a series of unconstrained sub-problems, [19] proposed an algorithm for the infeasible constrained nonlinear programming problem based on the large-scale augmented Lagrangian function, and analyzed the global convergence considering the possibility of not being feasible

Summary

Introduction

The problem involving multi criteria decision-making has become a research hotspot. Traditional methods for solving nonlinear programming include the steepest descent algorithm, Newton method, feasible direction method, function approximation method and trust region method Aside from those methods, the enhanced Lagrange method is to solve the problem by replacing the original constraint problem with a series of unconstrained sub-problems, [19] proposed an algorithm for the infeasible constrained nonlinear programming problem based on the large-scale augmented Lagrangian function, and analyzed the global convergence considering the possibility of not being feasible. Algorithms for feasible SQP (Sequential Quadratic Programming) were designed by Craig and André [21] to solve optimization problems with nonlinear constraints. Motivated and inspired by [26,27], the purpose of this paper is to develop a new iterative algorithm for MC2 NLP problems by employing the theory of variational inequalities and the resolvent operator technique. (Lagrange multiplier rule) in [28]

Preliminaries

Application and Comparison

Conclusions