Solving variational inequalities with Stochastic Mirror-Prox algorithm

In this paper we consider iterative methods for stochastic variational inequalities (s.v.i.) with monotone operators. Our basic assumption is that the operator possesses both smooth and nonsmooth components. Further, only noisy observations of the problem data are available. We develop a novel Stochastic Mirror-Prox (SMP) algorithm for solving s.v.i. and show that with the convenient stepsize strategy it attains the optimal rates of convergence with respect to the problem parameters. We apply the SMP algorithm to Stochastic composite minimization and describe particular applications to Stochastic Semidefinite Feasability problem and Eigenvalue minimization

Acceleration of the path-following method for optimization over the cone of positive semidefinite matrices

Projet META2The paper is devoted to acceleration of the path-following interior point polynomial time method for optimization over the cone of positive semidefinite matrices, with applications to quadratically constrained problems and extensions onto the general self-concordant case. In particular, we demonstrate that in a problem involving m of general type m x m linear matrix inequalities with n 3 m scalar control variables the conjugate-gradient-based acceleration allows to reduce the arithmetic cost of an e-solution by a factor of order of max {n1/3 m-1/6, n1/5}, for the Karmarkar-type acceleration this factor is of order of min {n, m1/2}. The conjugate-gradient-based acceleration turns out to be efficient also in the case of several specific "structured" problems coming from applications in control and graph theory

Interior point polynomial algorithms in convex programming

Written for specialists working in optimization, mathematical programming, or control theory. The general theory of path-following and potential reduction interior point polynomial time methods, interior point methods, interior point methods for linear and quadratic programming, polynomial time methods for nonlinear convex programming, efficient computation methods for control problems and variational inequalities, and acceleration of path-following methods are covered. In this book, the authors describe the first unified theory of polynomial-time interior-point methods. Their approach provides a simple and elegant framework in which all known polynomial-time interior-point methods can be explained and analyzed; this approach yields polynomial-time interior-point methods for a wide variety of problems beyond the traditional linear and quadratic programs. The book contains new and important results in the general theory of convex programming, e.g., their "conic" problem formulation in which duality theory is completely symmetric. For each algorithm described, the authors carefully derive precise bounds on the computational effort required to solve a given family of problems to a given precision. In several cases they obtain better problem complexity estimates than were previously known. Several of the new algorithms described in this book, e.g., the projective method, have been implemented, tested on "real world" problems, and found to be extremely efficient in practice. Contents : Chapter 1: Self-Concordant Functions and Newton Method; Chapter 2: Path-Following Interior-Point Methods; Chapter 3: Potential Reduction Interior-Point Methods; Chapter 4: How to Construct Self- Concordant Barriers; Chapter 5: Applications in Convex Optimization; Chapter 6: Variational Inequalities with Monotone Operators; Chapter 7: Acceleration for Linear and Linearly Constrained Quadratic Problem

On sequential hypotheses testing via convex optimization

International audienceWe propose a new approach to sequential testing which is an adaptive (on-line) extension of the (off-line) framework developed in [10]. It relies upon testing of pairs of hypotheses in the case where each hypothesis states that the vector of parameters underlying the distribution of observations belongs to a convex set. The nearly optimal under appropriate conditions test is yielded by a solution to an efficiently solvable convex optimization problem. The proposed methodology can be seen as a computationally friendly reformulation of the classical sequential testing

Hypothesis testing by convex optimization

International audienceWe discuss a general approach to handling "multiple hypotheses" testing in the case when a particular hypothesis states that the vector of parameters identifying the distribution of observations belongs to a convex compact set associated with the hypothesis. With our approach, this problem reduces to testing the hypotheses pairwise. Our central result is a test for a pair of hypotheses of the outlined type which, under appropriate assumptions, is provably nearly optimal. The test is yielded by a solution to a convex programming problem, so that our construction admits computationally efficient implementation. We further demonstrate that our assumptions are satisfied in several important and interesting applications. Finally, we show how our approach can be applied to a rather general detection problem encompassing several classical statistical settings such as detection of abrupt signal changes, cusp detection and multi-sensor detection

Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators

International audienceThe majority of first-order methods for large-scale convex–concave saddle point problems and variational inequalities with monotone operators are proximal algorithms. To make such an algorithm practical, the problem’s domain should be proximal-friendly—admit a strongly convex function with easy to minimize linear perturbations. As a by-product, this domain admits a computationally cheap linear minimization oracle (LMO) capable to minimize linear forms. There are, however, important situations where a cheap LMO indeed is available, but the problem domain is not proximal-friendly, which motivates search for algorithms based solely on LMO. For smooth convex minimization, there exists a classical algorithm using LMO—conditional gradient. In contrast, known to us similar techniques for other problems with convex structure (nonsmooth convex minimization, convex–concave saddle point problems, even as simple as bilinear ones, and variational inequalities with monotone operators, even as simple as affine) are quite recent and utilize common approach based on Fenchel-type representations of the associated objectives/vector fields. The goal of this paper was to develop alternative (and seemingly much simpler) decomposition techniques based on LMO for bilinear saddle point problems and for variational inequalities with affine monotone operators

Rejoinder of “Hypothesis testing by convex optimization”

International audienc

Hypothesis testing by convex optimization

International audienceWe discuss a general approach to handling "multiple hypotheses" testing in the case when a particular hypothesis states that the vector of parameters identifying the distribution of observations belongs to a convex compact set associated with the hypothesis. With our approach, this problem reduces to testing the hypotheses pairwise. Our central result is a test for a pair of hypotheses of the outlined type which, under appropriate assumptions, is provably nearly optimal. The test is yielded by a solution to a convex programming problem, so that our construction admits computationally efficient implementation. We further demonstrate that our assumptions are satisfied in several important and interesting applications. Finally, we show how our approach can be applied to a rather general detection problem encompassing several classical statistical settings such as detection of abrupt signal changes, cusp detection and multi-sensor detection

New variants of bundle methods

SIGLEAvailable at INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.1991 n.1508 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc