In this paper, we establish new bounds for restricted isometry constants (RIC) in low-rank matrix recovery. Let $\A$ be a linear transformation from $\R^{m \times n}$ into $\R^p$, and $r$ the rank of recovered matrix $X\in \R^{m \times n}$. Our main result is that if the condition on RIC satisfies $\delta_{2r+k}+2(\frac{r}{k})^{1/2}\delta_{\max\{r+\frac{3}{2}k,2k\}}
In this paper, we propose an efficient approach for solving a class of convex optimization problems in Hilbert spaces. Our feasible region is a (possibly infinite-dimensional) simple convex set, i.e. we assume that projections on this set are computationally easy to compute. The problem we consider is the minimization of a convex function over this … Read more
We present a bundle method for convex nondifferentiable minimization where the model is a piecewise quadratic convex approximation of the objective function. Unlike standard bundle approaches, the model only needs to support the objective function from below at a properly chosen (small) subset of points, as opposed to everywhere. We provide the convergence analysis for … Read more
We extend the target map, together with the weighted barriers and the notions of weighted analytic centers, from linear programming to general convex conic programming. This extension is obtained from a novel geometrical perspective of the weighted barriers, that views a weighted barrier as a weighted sum of barriers for a strictly decreasing sequence of … Read more
We consider the linearly constrained separable convex programming whose objective function is separable into m individual convex functions without crossed variables. The alternating direction method (ADM) has been well studied in the literature for the special case m=2. But the convergence of extending ADM to the general case m>=3 is still open. In this paper, … Read more
We introduce the notion of inexact first-order oracle and analyze the behaviour of several first-order methods of smooth convex optimization used with such an oracle. This notion of inexact oracle naturally appears in the context of smoothing techniques, Moreau-Yosida regularization, Augmented Lagrangians and many other situations. We derive complexity estimates for primal, dual and fast … Read more
The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness. It is the union of the various central paths for minimizing or maximizing the cost function over any region in the associated hyperplane arrangement. We determine the degree, arithmetic genus and defining prime ideal … Read more
We give a proximal bundle method for minimizing a convex function $f$ over $\mathbb{R}_+^n$. It requires evaluating $f$ and its subgradients with a possibly unknown accuracy $\epsilon\ge0$, and maintains a set of free variables $I$ to simplify its prox subproblems. The method asymptotically finds points that are $\epsilon$-optimal. In Lagrangian relaxation of convex programs, it … Read more
We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic … Read more
The structural properties of graphs are usually characterized in terms of invariants, which are functions of graphs that do not depend on the labeling of the nodes. In this paper we study convex graph invariants, which are graph invariants that are convex functions of the adjacency matrix of a graph. Some examples include functions of … Read more