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Exactness of the absolute value penalty function method for nonsmooth ( Φ , ρ ) ‐invex optimization problems
Author(s) -
Antczak Tadeusz
Publication year - 2019
Publication title -
international transactions in operational research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.032
H-Index - 52
eISSN - 1475-3995
pISSN - 0969-6016
DOI - 10.1111/itor.12374
Subject(s) - mathematics , penalty method , lagrange multiplier , saddle point , mathematical optimization , optimization problem , stationary point , equivalence (formal languages) , augmented lagrangian method , function (biology) , convex function , constrained optimization , convex optimization , regular polygon , mathematical analysis , discrete mathematics , geometry , evolutionary biology , biology
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 multiplier such that, for every penalty parameter exceeding this lower bound, there is the equivalence between an optimal solution in the original constrained minimization problem with ( Φ , ρ ) ‐invex functions and a minimizer in its associated penalized optimization problem with the exact l 1 penalty function. Further, under nondifferentiable ( Φ , ρ ) ‐invexity assumptions, a characterization of a saddle point of the Lagrange function, defined for the considered constrained optimization problem in terms of minimizers of its associated exact penalized optimization problem with the exact l 1 penalty function, is presented.