Lagrange Multiplier Local Necessary and Global Sufficiency Criteria for Some Non-Convex Programming Problems

TL;DR

This paper develops new necessary and sufficient optimality conditions for three types of non-convex programming problems, enabling the identification of global minimizers among local solutions.

Contribution

It introduces Lagrange multiplier-based global sufficiency criteria for quadratic, ρ-convex, and quadratic fractional problems with mixed variables, extending existing optimality theory.

Findings

Derived local necessary optimality conditions for all three problem types.

Established Lagrangian-based global sufficiency criteria to distinguish global minimizers.

Provided illustrative examples demonstrating the effectiveness of the proposed conditions.

Abstract

In this paper we consider three minimization problems, namely quadratic, $\rho$-convex and quadratic fractional programing problems. The quadratic problem is considered with quadratic inequality constraints with bounded continuous and discrete mixed variables. The $\rho$-convex problem is considered with $\rho$-convex inequality constraints in mixed variables. The quadratic fractional problem is studied with quadratic fractional constraints in mixed variables. For all three problems we reformulate the problem as a mathematical programming problem and apply standard Karush Kuhn Tucker necessary conditions. Then, for each problem, we provide local necessary optimality condition. Further, for each problem a Lagrangian multiplier sufficient optimality condition is provided to identify global minimizer among the local minimizers. For the quadratic problem underestimation of a Lagrangian was employed to obtain the desired sufficient conditions. For the $\rho$-convex problem we obtain two sufficient optimality conditions to distinguish a global minimizer among the local minimizers, one with an underestimation of a Lagrangian and the other with a different technique. A global sufficient optimality condition for the quadratic fractional problem is obtained by reformulating the problem as a quadratic problem and then utilizing the results of the quadratic problem. Examples are provided to illustrate the significance of the results obtained.