Exactness of the absolute value penalty function method for nonsmooth (Φ,ρ)‐invex optimization problems

• Date: 01 July 2019
• Author: Tadeusz Antczak
• Published date: 01 July 2019
• DOI: http://doi.org/10.1111/itor.12374

Intl. Trans. in Op. Res. 26 (2019) 1504–1526

DOI: 10.1111/itor.12374

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Exactness of the absolute value penalty function method

for nonsmooth (, ρ)-invex optimization problems

Tadeusz Antczak

Faculty of Mathematics and Computer Science, University of Ł´

od´

z, Ł´

od´

z, Poland

E-mail: antczak@math.uni.lodz.pl [Antczak]

Received 20 November2015; received in revised form 2 July 2016; accepted 29 October 2016

Abstract

In this paper, the classical exact absolute value penalty function method is used for solving a new class

of nonconvex nonsmooth optimization problems. Nonconvex nondifferentiable optimization problems with

both inequality and equality constraints are considered here,in which not all functions constituting them have

the fundamental property of convex functions and most classes of generalized convex functions—namely,

a stationary point of such a function is its global minimum. It is proved for such nonconvex optimization

problems that there exists a finite threshold of penalty parameters equal to the largest absolute value of a

Lagrange multipliersuch that, for every penalty parameter exceeding this lowerbound, there is the equivalence

between an optimal solution in the original constrained minimization problem with (,ρ )-invex functions

and a minimizer in its associated penalized optimization problem with the exact l1penalty function. Further,

under nondifferentiable (,ρ )-invexity assumptions, a characterization of a saddle point of the Lagrange

function, defined for the considered constrained optimizationproblem in terms of minimizers of its associated

exact penalized optimization problem with the exact l1penalty function, is presented.

Keywords: exact absolute value penalty function method; penalized optimization problem; generalized Karush–Kuhn–

Tucker necessary optimality conditions; saddlepoint criteria; locally Lipschitz (, ρ)-invex function

1. Introduction

Penalty methods are well-known techniques to handle constrained optimization problems. In the

past few years, considerable attention has been given to such penalty methods that use exact

penalty functions. These methods transform the constrained nonlinear programming problem into

an unconstrained problem by adding a penalty term to the original objective function. Thus, exact

penalty function methods avoid the sequence of unconstrained minimization problems, which is

characteristic for the usual strategy of penalty function methods (see, e.g., Fiacco and McCormick,

1968).

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2017 The Authors.

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2017 International Federation ofOperational Research Societies

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T. Antczak / Intl. Trans. in Op. Res. 26 (2019) 1504–1526 1505

Nondifferentiable exact penalty function methods were introduced for the first time by Eremin

(1967) and Zangwill (1967). The most popular nondifferentiable exact penalty function is the

absolute value penalty function, also called the exact l1penalty function. The nondifferentiable

exact absolute penalty function method has been investigated in Bazaraa et al. (1991), Bertsekas

and Koksal-Ozdaglar (2000), Charalambous (1978), Di Pillo and Grippo (1989), Dolecki and

Rolewicz (1979), Fletcher (1973), Han and Mangasarian (1979), Luenberger (1970), Mangasarian

(1985), Peressini et al. (1988), Wang and Liu (2010), Yang and Ralph (2005), and others.

Most of the papers on the nondifferentiable exact l1penalty function method are devoted to

the study of conditions ensuring that an optimal solution in the convex original optimization

problem is also an unconstrained solution of the penalty function (see, e.g., Bazaraa et al., 1991;

Charalambous, 1978; Mangasarian, 1985; Peressini et al., 1988). Namely, it has been established for

convexoptimization problems that, for anypenalty parameter exceeding some suitable threshold, an

optimal solution of a convex minimizationproblem occurs at a minimizer of an exact absolute value

penalty function being the objective function in the associated penalized optimization problem (see,

e.g., Bazaraa et al., 1991). Thus, the lower bound of the penalty parameter has been found such

that, for all penalty parameters exceeding this value, any optimal solution of the original convex

minimization problem is also a minimizer of its penalized optimization problem. However, it is

also of great interest to investigate properties that ensure an optimal solution or stationary point

of the penalty function is an optimal solution or a stationary point in the nonconvex constrained

optimization problem. Recently, some results on the exactness property for the exact l1penalty

function method used for solving nonconvex optimization problems have been established in the

literature (see Antczak, 2009, 2013).

In this paper, we analyze the exactness property of the classical exact l1penalty function method

that we use for solving a new class of nonconvex nondifferentiable optimization problems with

both inequality and equality constraints. Namely, we use the exact absolute value penalty function

method for solving nondifferentiable optimization problems with locally Lipschitz (, ρ)-invex

functions. Thus, we associate a global optimal solution in the original nonsmooth optimization

problem involving locally Lipschitz (, ρ)-invex functions with a global minimizer in its associated

penalized optimization problem with the absolute value penalty function. This result is established

for all penalty parameters exceeding the threshold equal to the largest absolute value of a Lagrange

multiplier. Further, it appears to be of interest to also study converse properties ensuring that a

global minimizer in the penalized optimization problem with the absolute value penalty function

is also a global optimal solution in the original constrained optimization problem. Hence, under

nondifferentiable (,ρ )-invexity hypotheses, we prove the equivalence between the sets of optimal

solutions in the original nonconvex constrained optimization problem and its associated penalized

optimization problem with the absolute value penalty function for all penalty parameters exceed-

ing the threshold equal to the largest absolute value of a Lagrange multiplier. Since the concept

of nondifferentiable (,ρ )-invexity extends several notions of generalized convex functions, this

threshold of penalty parameters, above which there is the mentioned equivalence between the sets

of optimal solutions, has been found not only for a larger class of optimization problems than

convex extremum problems, but also than invex ones. In fact, it has been found for a significantly

larger class of nonconvex optimization problems in comparison to various classes of nonconvex

optimization problems with other generalized convex functions, previously defined in the litera-

ture (see, e.g., Antczak, 2013). Thus, the equivalence between the sets of optimal solutions in the

C

2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation of OperationalResearch Societies