A modified exact smooth penalty function for nonlinear constrained optimization

Abstract

In this paper, a modified simple penalty function is proposed for a constrained nonlinear programming problem by augmenting the dimension of the program with a variable that controls the weight of the penalty terms. This penalty function enjoys improved smoothness. Under mild conditions, it can be proved to be exact in the sense that local minimizers of the original constrained problem are precisely the local minimizers of the associated penalty problem. MSC: 47H20; 35K55; 90C30

Smooth exact penalty functions II: a reduction to standard exact penalty functions

“…Observe that the penalty function F λ (x, ε) depends on the additional parameter ε ≥ 0, and F λ (x, ε) is smooth for any ε ∈ (0, ε) and x such that 0 < ∆(x, ε) < a. Therefore one can apply standard algorithms of smooth optimization to the penalty function (2) in order to find a globally/locally optimal solution of penalized problem (3), which under natural assumptions (namely, constraint qualification) has the form (x * , 0), where x * is a globally/locally optimal solution of problem (1). However, it should be noted that the standard proofs of the exactness of the smooth penalty function (2) (i.e.…”

A unifying theory of exactness of linear penalty functions II: parametric penalty functions

“…The conditions under which a penalty function does not have any stationary points outside the set of feasible solutions of the original problem are very important for applications, and they have been studied by different researchers (see, e.g., [9,[11][12][13]50]). It should also be noted that such conditions were the main tool for the study of singular exact penalty functions [6,23,44]. Despite all attention and importance, the property of a penalty function to not have any infeasible stationary points has never been named.…”

“…Throughout this article, we refer to the penalty function proposed in [23] as a singular exact penalty function, since one achieves smoothness of this exact penalty function via the introduction of a singular term into the definition of this function. Recently, singular exact penalty functions have attracted a lot of attention of researchers [6,14,15,44], and were successfully applied to various constrained optimization problems [25,27,31,32,35,51,52]. It should be noted that the main feature of both smoothing approximations of exact penalty functions and singular exact penalty functions is the fact that they depend on some additional parameters apart from the penalty parameter.…”

Smooth exact penalty functions: a general approach

“…where λ ≥ 0 is the penalty parameter, ∆(x, ε) = F (x) − εw 2 is the constraint violation measure, β : [0, ε] → [0, +∞) with β(0) = 0 is the penalty term, q > 0 and ε > 0 are some prespecified thresholds. Finally, one replaces the augmented problem (2) with the penalized problem min…”

“…Consequently, one can apply methods of smooth unconstrained minimization to penalized problem (4) in order to find a solution of initial constrained optimization problem (1). Later on, Huyer and Neumaier's approach was generalized [18,2] and successfully applied to various constrained optimization problems [15,13], including some optimal control problems [12,10,14]. However, it should be noted that the existing proofs of the exactness of the smooth penalty function (3) and its various generalizations are quite complicated, and overburdened by technical details that overshadow the understanding of the technique of smooth exact penalty functions.…”