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
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
The regularization of a convex program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. For a general convex program, we show that the regularization is exact if and only if a certain selection problem has a … Read more
We show the linear convergence of a simple first-order algorithm for the minimum-volume enclosing ellipsoid problem and its dual, the D-optimal design problem of statistics. Computational tests confirm the attractive features of this method. CitationOptimization Methods and Software 23 (2008), 5–19. ArticleDownload View PDF
Dini derivative on Riemannian manifold setting is studied in this paper. In addition, a characterization for Lipschitz and convex functions defined on Riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the Dini derivative are given. ArticleDownload View PDF
It is a matter of common knowledge that traditional Markowitz optimization based on sample means and covariances performs poorly in practice. For this reason, diverse attempts were made to improve performance of portfolio optimization. In this paper, we investigate three popular portfolio selection models built upon classical mean-variance theory. The first model is an extension … Read more
The notion of soft thresholding plays a central role in problems from various areas of applied mathematics, in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis. Using convex-analytical tools, we extend this notion to that of proximal thresholding and investigate its properties, providing in particular several characterizations of … Read more