In this paper we present a variant of the conjugate gradient (CG) algorithm in which we invoke a subspace minimization subproblem on each iteration. We call this algorithm CGSO for “conjugate gradient with subspace optimization”. It is related to earlier work by Nemirovsky and Yudin. We apply the algorithm to solve unconstrained strictly convex problems. … Read more
Finding clusters, or communities, in a graph, or network is a very important problem which arises in many domains. Several models were proposed for its solution. One of the most studied and exploited is the maximization of the so called modularity, which represents the sum over all communities of the fraction of edges within these … Read more
In this paper, we are interested in the development of efficient first-order methods for convex optimization problems in the simultaneous presence of smoothness of the objective function and stochasticity in the first-order information. First, we consider the Stochastic Primal Gradient method, which is nothing else but the Mirror Descent SA method applied to a smooth … Read more
Recently, a worst-case O(1/t) convergence rate was established for the Douglas-Rachford alternating direction method of multipliers in an ergodic sense. This note proposes a novel approach to derive the same convergence rate while in a non-ergodic sense. Article Download View On non-ergodic convergence rate of Douglas-Rachford alternating direction method of multipliers
The contributions of the first half of this thesis are on the computational and algebraic aspects of convexity in polynomial optimization. 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 … Read more
This article uses classical notions of convex analysis over euclidean spaces, like Gale & Klee’s boundary rays and asymptotes of a convex set, or the inner aperture directions defined by Larman and Brøndsted for the same class of sets, to provide a new zero duality gap criterion for ordinary convex programs. On this ground, we … Read more
This article addresses a general criterion providing a zero duality gap for convex programs in the setting of the real locally convex spaces. The main theorem of our work is formulated only in terms of the constraints of the program, hence it holds true for any objective function fulfilling a very general qualification condition, implied … Read more
Given x, a point of a convex subset C of an Euclidean space, the two following statements are proven to be equivalent: (i) any convex function f : C → R is upper semi-continuous at x, and (ii) C is polyhedral at x. In the particular setting of closed convex mappings and Fσ domains, we … Read more
In this paper, we derive a randomized version of the Mirror-Prox method for solving some structured matrix saddle-point problems, such as the maximal eigenvalue minimization problem. Deterministic first-order schemes, such as Nesterov’s Smoothing Techniques or standard Mirror-Prox methods, require the exact computation of a matrix exponential at every iteration, limiting the size of the problems … Read more
We show that the Hedge algorithm, a method that is widely used in Machine Learning, can be interpreted as a particular instance of Dual Averaging schemes, which have recently been introduced by Nesterov for regret minimization. Based on this interpretation, we establish three alternative methods of the Hedge algorithm: one in the form of the … Read more