New Exact Penalty Functions for Nonlinear Constrained Optimization Problems

Abstract

For two kinds of nonlinear constrained optimization problems, we propose two simple penalty functions, respectively, by augmenting the dimension of the primal problem with a variable that controls the weight of the penalty terms. Both of the penalty functions enjoy improved smoothness. Under mild conditions, it can be proved that our penalty functions are both exact in the sense that local minimizers of the associated penalty problem are precisely the local minimizers of the original constrained problem.

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“…Most of these methods transform constrained optimization problems into unconstrained optimization problems to facilitate resolution. This transformation is achieved through the use of penalization functions (see in [3,7,8,12,14,16,17]). Penalization techniques are used in both single objective and multi-objective cases, regardless of the nature of the variables in the problem.…”

“…Penalization techniques are used in both single objective and multi-objective cases, regardless of the nature of the variables in the problem. Among the multiples techniques of penalizing of literature, the one defined in [3], called logarithmic barrier-penalty function, which caught our attention. So far, it has been used only for the single-objective case.…”

“…x ∈ R n ; where X = {x ∈ R n |h j (x) ≤ 0, j = 1, • • • , l} is the admissible set of problem (1.1). The penalty function proposed by Bingzhuang Liu [3] to transform this problem in unconstrained problem is defined as follows:…”

“…where X τ = {x ∈ R n |h j (x) < τ, j = 1, • • • , l} with the operations on X τ can be extended to the admissible domain X of the initial problem (1.1) (see in [3]) and σ a penalty parameter. Furthermore, it does not take into account the differentiation of objective or constrained functions.…”