Patrick Redont

Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality

We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x,y)=f(x)+Q(x,y)+g(y), where f and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L. We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Łojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to “metrically regular” problems. Our main result can be stated as follows: If L has the Kurdyka-Łojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to

A Dynamical Approach to an Inertial Forward-Backward Algorithm for Convex Minimization

Q(x,y)=\Vert x-y \Vert ^2

A New Class of Alternating Proximal Minimization Algorithms with Costs-to-Move

and to f, g indicator functions, the algorithm is an alternating projection mehod (a variant of von Neumanns) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with “regular” intersection. To illustrate our results with concrete problems, we provide a convergent proximal reweighted l1 algorithm for compressive sensing and an application to rank reduction problems.

Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity

We introduce a new class of forward-backward algorithms for structured convex minimization problems in Hilbert spaces. Our approach relies on the time discretization of a second-order differential system with two potentials and Hessian-driven damping, recently introduced in [H. Attouch, P.-E. Mainge, and P. Redont, Differ. Equ. Appl., 4 (2012), pp. 27--65]. This system can be equivalently written as a first-order system in time and space, each of the two constitutive equations involving one (and only one) of the two potentials. Its time dicretization naturally leads to the introduction of forward-backward splitting algorithms with inertial features. Using a Liapunov analysis, we show the convergence of the algorithm under conditions enlarging the classical step size limitation. Then, we specialize our results to gradient-projection algorithms and give some illustrations of sparse signal recovery and feasibility problems.

Semi-deterministic versus genetic algorithms for global optimisation of multichannel optical filters

Given two objective functions

Singular Riemannian barrier methods and gradient-projection dynamical systems for constrained optimization

f:\mathcal X\mapsto\R\cup\{+\infty\}

Global Convergence of a Closed-Loop Regularized Newton Method for Solving Monotone Inclusions in Hilbert Spaces

and

THEME-SEER: a multidimensional exploratory technique to analyze a structural model using an extended covariance criterion

g:\mathcal Y\mapsto\R\cup\{+\infty\}

Multidimensional Exploratory Analysis of a Structural Model Using a Class of Generalized Covariance Criteria

on abstract spaces

A second-order gradient-like dissipative dynamical system with Hessian-driven damping.: Application to optimization and mechanics

\mathcal X