Asymptotic Analysis for Penalty and Barrier Methods in Convex and Linear Programming

Abstract

We consider a wide class of penalty and barrier methods for convex programming which includes a number of specific functions proposed in the literature. We provide a systematic way to generate penalty and barrier functions in this class, and we analyze the existence of primal and dual optimal paths generated by these penalty methods, as well as their convergence to the primal and dual optimal sets. For linear programming we prove that these optimal paths converge to single points.

Penalty and Smoothing Methods for Convex Semi-Infinite Programming

“…Then, since it is nonconstant, θ ∞ (1) > 0. The following result was proved in [3] in a more general setting.…”

“…This smoothing approach has been also proposed by Polak-Royset-Womersley [20], by Sheu-Wu [27] for finite min-max problems subject to infinitely many linear constraints and, more recently, by Sheu-Lin [26] for continuous min-max problems, motivated by the global approach of Fang-Wu [12] using an integral analog. We must also smooth the function δ D k and to do that we consider the smoothing approach by penalty and barrier functions introduced, for ordinary convex programs, by Auslender-Cominetti-Haddou [3]. These authors exploited the notion of recession functions to provide a wide class of penalty and barrier methods for usual convex programs, with a finite number of inequalities.…”

“…So we consider here two subclasses of penalty functions introduced in [3] (not all can be used). They are composed by those functions θ : R → R + which are C 1 , convex, nondecreasing, and satisfy some additional properties, and we choosẽ…”

Examples of ill-behaved central paths in convex optimization

“…For this extremely simple feasible set, most penalty functions (e.g., those studied in [2]) give the same sets of optimizers, those described by…”

“…Let k, m, n ∈ N be fixed, q = n + m. For q = 0, the fact that g 0 k is bounded by 2 on readily implies (47) with K 0 = θ (2). On the other hand g ∞ k is clearly continuous on .…”