Felipe Alvarez

On the Minimizing Property of a Second Order Dissipative System in Hilbert Spaces

We study the asymptotic behavior at infinity of solutions of a second order evolution equation with linear damping and convex potential. The differential system is defined in a real Hilbert space. It is proved that if the potential is bounded from below, then the solution trajectories are minimizing for it and converge weakly towards a minimizer of

An Inertial Proximal Method for Maximal Monotone Operators via Discretization of a Nonlinear Oscillator with Damping

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Weak Convergence of a Relaxed and Inertial Hybrid Projection-Proximal Point Algorithm for Maximal Monotone Operators in Hilbert Space

if one exists; this convergence is strong when

A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

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HESSIAN RIEMANNIAN GRADIENT FLOWS IN CONVEX PROGRAMMING

is even or when the optimal set has a nonempty interior. We introduce a second order proximal-like iterative algorithm for the minimization of a convex function. It is defined by an implicit discretization of the continuous evolution problem and is valid for any closed proper convex function. We find conditions on some parameters of the algorithm in order to have a convergence result similar to the continuous case.

The heavy ball with friction dynamical system for convex constrained minimization problems

The ‘heavy ball with friction’ dynamical system x + γx + ∇f(x)=0 is a nonlinear oscillator with damping (γ>0). It has been recently proved that when H is a real Hilbert space and f: H→R is a differentiable convex function whose minimal value is achieved, then each solution trajectory t→x(t) of this system weakly converges towards a solution of ∇f(x)=0. We prove a similar result in the discrete setting for a general maximal monotone operator A by considering the following iterative method: xk+1−xk−αk(xk−xk−1)+λkA(xk+1)∋0, giving conditions on the parameters λk and αk in order to ensure weak convergence toward a solution of 0∈A(x) and extending classical convergence results concerning the standard proximal method.

Interior proximal algorithm with variable metric for second-order cone programming: applications to structural optimization and support vector machines

This paper introduces a general implicit iterative method for finding zeros of a maximal monotone operator in a Hilbert space which unifies three previously studied strategies: relaxation, inertial type extrapolation and projection step. The first two strategies are intended to speed up the convergence of the standard proximal point algorithm, while the third permits one to perform inexact proximal iterations with fixed relative error tolerance. The paper establishes the global convergence of the method for the weak topology under appropriate assumptions on the algorithm parameters.

Primal and dual convergence of a proximal point exponential penalty method for linear programming

We consider the problem of finding a singularity of a differentiable vector field X defined on a complete Riemannian manifold. We prove a unified result for theexistence and local uniqueness of the solution, and for the local convergence of a Riemannian version of Newtons method. Our approach relies on Kantorovichs majorant principle: under suitable conditions, we construct an auxiliary scalar equation φ(r) = 0 which dominates the original equation X(p) = 0 in the sense that the Riemannian-Newton method for the latter inherits several features of the real Newton method applied to the former. The majorant φ is derived from an adequate radial parametrization of a Lipschitz-type continuity property of the covariant derivative of X, a technique inspired by the previous work of Zabrejko and Nguen on Newtons method in Banach spaces. We show how different specializations of the main result recover Riemannian versions of Kantorovichs theorem and Smales α-theorem, and, at least partially, the Euclidean self-concordant theory of Nesterov and Nemirovskii. In the specific case of analytic vector fields, we improve recent developments inthis area by Dedieu et al. (IMA J. Numer. Anal., Vol. 23, 2003, pp. 395-419). Some Riemannian techniques used here were previously introduced by Ferreira and Svaiter (J. Complexity, Vol. 18, 2002, pp. 304-329) in the context of Kantorovichs theorem for vector fields with Lipschitz continuous covariant derivatives.

Asymptotic expansions for interior penalty solutions of control constrained linear-quadratic problems

In view of solving theoretically constrained minimization problems, we investigate the properties of the gradient flows with respect to Hessian Riemannian metrics induced by Legendre functions. The first result characterizes Hessian Riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities, giving a new motivation for the introduction of Bregman-type distances. Then, the general evolution problem is introduced, and global convergence is established under quasi-convexity conditions, with interesting refinements in the case of convex minimization. Some explicit examples of these gradient flows are discussed. Dual trajectories are identified, and sufficient conditions for dual convergence are examined for a convex program with positivity and equality constraints. Some convergence rate results are established. In the case of a linear objective function, several optimality characterizations of the orbits are given: optimal path of viscosi...

A continuous framework for open pit mine planning

The “heavy ball with friction” dynamical system