Complementarity and nondegeneracy in semidefinite programming

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].…”

“…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).…”

“…They consider smooth pre-images of general closed convex cones, and, like [38,37,1], present in [4, Section 4.6.1] an appropriate generalization of the classical full rank condition. 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].…”

“…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].…”