44,259 research outputs found

    Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems

    Full text link
    Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies which jointly brings into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and non-smooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primal-dual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primal-dual algorithms both for solving large-scale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness

    Discrete Approximations of a Controlled Sweeping Process

    Get PDF
    The paper is devoted to the study of a new class of optimal control problems governed by the classical Moreau sweeping process with the new feature that the polyhe- dral moving set is not fixed while controlled by time-dependent functions. The dynamics of such problems is described by dissipative non-Lipschitzian differential inclusions with state constraints of equality and inequality types. It makes challenging and difficult their anal- ysis and optimization. In this paper we establish some existence results for the sweeping process under consideration and develop the method of discrete approximations that allows us to strongly approximate, in the W^{1,2} topology, optimal solutions of the continuous-type sweeping process by their discrete counterparts

    How revealing is revealed preference?

    Get PDF
    This lecture address the following two key criticisms of the empirical application of revealed preference theory: When the RP conditions do not reject, they do not provide precise predictions; and when they do reject, they do not help characterize the nature of irrationality or the degree/direction of changing tastes. Recent developments in the application of RP theory are shown to have rendered these criticisms unfounded. A powerful test of rationality is available that also provides a natural characterization of changing tastes. Tight bounds on demand responses and on the welfare costs of relative price and tax changes are also available and are shown to work well in practice
    corecore