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
This paper is devoted to optimality conditions for nonsmooth quasiconcave programming. Arrow and Enthoven (1961) formulate several economic problems into quasiconcave programming, and give a sufficient condition for smooth quasiconcave programming in their epoch-making and comprehensive paper. In this paper, generalized necessary and sufficient conditions for nonsmooth quasiconcave programming have been derived in terms of … Read more
A convex variational framework is proposed for solving inverse problems in Hilbert spaces with a priori information on the representation of the target solution in a frame. The objective function to be minimized consists of a separable term penalizing each frame coefficient individually and of a smooth term modeling the data formation model as well … Read more
When is the linear image of a closed convex cone closed? We present very simple, and intuitive necessary conditions, which 1) unify, and generalize seemingly disparate, classical sufficient conditions: polyhedrality of the cone, and “Slater” type conditions; 2) are necessary and sufficient, when the dual cone belongs to a class, that we call nice cones. … Read more
In this paper we explore properties of a family of probability density functions, called norm-induced densities, defined as $$f_t(x) = \left\{ \begin{array}{ll} \displaystyle \frac{ e^{-t\|x\|^p}dx}{\int_K e^{-t\|y\|^p}dy}, & x \in K \\ 0, & x \notin K,\\ \end{array}\right. $$ where $K$ is a $n$-dimensional convex set that contains the origin, parameters $t > 0$ and $p … Read more
We provide a sufficient condition on a class of compact basic semialgebraic sets K for their convex hull to have a lifted semidefinite representation (SDr). This lifted SDr is explicitly expressed in terms of the polynomials that define K. Examples are provided. For convex and compact basic semi-algebraic sets K defined by concave polynomials, we … Read more
In this paper we consider the general cone programming problem, and propose primal-dual convex (smooth and/or nonsmooth) minimization reformulations for it. We then discuss first-order methods suitable for solving these reformulations, namely, Nesterov’s optimal method \cite{Nest83-1,Nest05-1}, Nesterov’s smooth approximation scheme \cite{Nest05-1}, and Nemirovski’s prox-method \cite{Nem05-1}, and propose a variant of Nesterov’s optimal method which has … Read more
We present a two phase interior point decomposition framework for solving semidefinite (SDP) relaxations of sparse maxcut, stable set, and box constrained quadratic programs. In phase 1, we suitably modify the {\em matrix completion} scheme of Fukuda et al. \cite{fukuda_et_al} to preprocess an existing SDP into an equivalent SDP in the block-angular form. In phase … Read more
Three primal-dual interior-point algorithms for homogeneous cone programming are presented. They are a short-step algorithm, a large-update algorithm, and a predictor-corrector algorithm. These algorithms are described and analyzed based on a characterization of homogeneous cone via T-algebra. The analysis show that the algorithms have polynomial iteration complexity. CitationDivision of Mathematical Sciences, Nanyang Technological University, December … Read more
It is not straightforward to find a new feasible solution when several conic constraints are added to a conic optimization problem. Examples of conic constraints include semidefinite constraints and second order cone constraints. In this paper, a method to slightly modify the constraints is proposed. Because of this modification, a simple procedure to generate strictly … Read more