The exact absolute value penalty function method for identifying strict global minima of order m in nonconvex nonsmooth programming

Tadeusz Antczak

doi:10.1007/s11590-015-0967-3

Tadeusz Antczak

Open Access

https://doi.org/10.1007/s11590-015-0967-3

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Journal: Optimization Letters	Publication Date: Nov 14, 2015
Citations: 21	License type: CC BY 4.0

Affiliation: University of Łódź

Abstract

In this paper, it is demonstrated that the exact absolute value penalty function method is useful for identifying the special sort of minimizers in nonconvex nonsmooth optimization problems with both inequality and equality constraints. The equivalence between the sets of strict global minima of order m in nonsmooth minimization problem and of its associated penalized optimization problem with the exact \(l_{1}\) penalty function is established under nondifferentiable \(\left( F,\rho \right) \)-convexity assumptions imposed on the involved functions. The threshold of the penalty parameter, above which this result holds, is also given.

Highlights

The notion of a strict local minimizer of order m plays an important role in the convergence analysis of iterative numerical methods and in stability analysis
Most of the results established on the nondifferentiable exact l1 penalty function is devoted to the study of conditions ensuring that an optimal solution in the given convex optimization problem is an unconstrained solution of the penalty function
We present a new characterization of the exact penalty method with the absolute value penalty function used to solve a class of nonconvex nondifferentiable optimization problems involving both inequality and equality constraints in which the functions constituting them are locally Lipschitz (F, ρ)-convex functions of order m, not necessarily with respect to the same ρ

Summary

Introduction

The notion of a strict local minimizer of order m plays an important role in the convergence analysis of iterative numerical methods (see, for example, [1]) and in stability analysis (see, for example, [2,3]). We use the exact l1 penalty function method to find a strict global minimizer of order m in the considered nonconvex nondifferentiable optimization problem involving both inequality and equality constraints.

Results

Conclusion