nonsingularity – Optimization Online

Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

Published: 2007/10/15, Updated: 2008/05/19

Semi-definite Programming (strictly) quasi-concavity, convex hull, convex set, irredundancy, linear matrix inequality (lmi), nonsingularity, positive curvature, semialgebraic set, semidefinite programming, singularity, smoothness, sos-concavity, sum of squares (sos)

The article proves sufficient conditions and necessary conditions for SDP representability of convex sets and convex hulls by proposing a new approach to construct SDP representations. The contributions of this paper are: (i) For bounded SDP representable sets $W_1,\cdots,W_m$, we give an explicit construction of an SDP representation for $conv(\cup_{k=1}^mW_k)$. This provides a technique for … Read more

Constraint Nondegeneracy, Strong Regularity and Nonsingularity in Semidefinite Programming

Published: 2007/01/29

Zi Xian Chan

Defeng Sun

Convex Optimization, Linear, Cone and Semidefinite Programming constraint nondegeneracy, nonsingularity, quadratic convergence, semidefinite programming, strong regularity, variational analysis

It is known that the Karush-Kuhn-Tucker (KKT) conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices. We show in this paper that the primal and dual constraint nondegeneracies, the strong regularity, the nonsingularity of the B-subdifferential of this nonsmooth system, … Read more