On the generic properties of convex optimization problems in conic form

Abstract

We prove that strict complementarity, primal and dual nondegeneracy of optimal solutions of convex optimization problems in conic form are generic properties. In this paper, we say generic to mean that the set of data possessing the desired property (or properties) has strictly larger Hausdorff dimension than the set of data that does not. Our proof is elementary and it employs an important result due to Larman [7] on the boundary structure of convex bodies.

Generic Optimality Conditions for Semialgebraic Convex Programs

“…The idea of studying optimization from a generic perspective dates back further, at least to Saigal and Simon's 1973 study [36] of the complementarity problem, and has persisted: see for example the studies of generic strict complementarity and primal and dual nondegeneracy for semidefinite programming by Alizadeh, Haeberly and Overton [1] and Shapiro [37], and for general conic convex programs by Pataki and Tunçel [29].…”

“…The results of [40] fixed an objective and constraint functions, allowed linear perturbations to the objective and constant perturbations to the constraints, and proved a measure-theoretic result about the second-order conditions via Sard's Theorem. Both [1] and [37] use rather analogous arguments to prove that strict complementarity and primal and dual nondegeneracy are generic properties of semidefinite programs; using a very different technique based on the boundary behavior of convex sets, Pataki and Tunçel [29] generalized these results to general conic convex programs. Ioffe and Lucchetti [16] adopt a more abstract, topological approach, allowing very general perturbations to the optimization problem but proving a result instead about "well-posedness" [10].…”

“…Ioffe and Lucchetti [16] adopt a more abstract, topological approach, allowing very general perturbations to the optimization problem but proving a result instead about "well-posedness" [10]. Our approach, while notable for the generality of the feasible regions considered, is more concrete, combining some of the spirit of [40] and [29]: we aim to understand the generic nature of second-order sufficient conditions, like [40], but we make no assumption about how the feasible region is presented; instead, we assume only that it is semi-algebraic. In a recent survey [15], Ioffe describes how a semi-algebraic version of Sard's Theorem applies to an analogue of Spingarn and Rockafellar's result on generic optimality conditions.…”

“…Spingarn and Rockafellar's work [40] concerns classical smooth constraint systems, and Ioffe's semi-algebraic version [15] is analogous. Alizadeh, Haeberly and Overton [1] derive results for linear semidefinite programs, extended to general conic convex systems by Pataki and Tunçel [29]; Shapiro and Fan [38] and Shapiro [37] focus on nonlinear semidefinite programs. Such structural approaches reflect the presentation of optimization problems in practice, and are very general: in particular, [3] is a powerful toolkit for semidefinite representation of convex sets (that is, as affine pre-images of the semidefinite cone).…”

“…This transversality condition-called constraint nondegeneracy by Robinson [31], and also discussed in detail in the context of nonlinear semidefinite programming by Sun in [41], is generic, as a consequence of Sard's theorem, providing the problem parametrization is sufficiently rich [4,Section 5.3.1]. Another important ingredient of second-order analysis, the quadratic decay condition, is also generic in semidefinite pro-gramming, since it is equivalent (see [4,Theorem 5.91]) to uniqueness of the optimal solution along with a suitable analogue of the classical "strict complementarity" condition, known to be generic [1,29]. The active manifold also emerges naturally using this approach, via a standard application of the transversality condition, assuming a powerful property of the underlying cone called "cone reducibility" [4,Definition 3.135].…”