A Dynamical Approach to Convex Minimization Coupling Approximation with the Steepest Descent Method

AbstractWe study the asymptotic behavior of the solutions to evolution equations of the form 0∈u(t)+∂f(u(t), ε(t));  u(0)=u0, where {f(·, ε):ε>0} is a family of strictly convex functions whose minimum is attained at a unique pointx(ε). Assuming thatx(ε) converges to a pointx* as ε tends to 0, and depending on the behavior of the optimal trajectoryx(ε), we derive sufficient conditions on the parametrization ε(t) which ensure that the solutionu(t) of the evolution equation also converges tox* whent→+∞. The results are illustrated on three different penalty and viscosity-approximation methods for convex minimization

Approximation and Convergence in Nonlinear Optimization

We show that the theory of e-convergence, originally developed to study approximation techniques, is also useful in the analysis of the convergence properties of algorithmic procedures for nonlinear optimization problems

Quantitative Stability of Variational Systems: I. The Epigraphical Distance

This paper proposes a global measure for the distance between the elements of a variational system (parametrized families of optimization problems)

Pointwise Sum of two Maximal Monotone Operators

∗ Cette recherche a été partiellement subventionnée, en ce qui concerne le premier et le dernier auteur, par la bourse OTAN CRG 960360 et pour le second auteur par l’Action Intégrée 95/0849 entre les universités de Marrakech, Rabat et Montpellier.The primary goal of this paper is to shed some light on the maximality of the pointwise sum of two maximal monotone operators. The interesting purpose is to extend some recent results of Attouch, Moudafi and Riahi on the graph-convergence of maximal monotone operators to the more general setting of reflexive Banach spaces. In addition, we present some conditions which imply the uniform Brézis-Crandall-Pazy condition. Afterwards, we present, as a consequence, some recent conditions which ensure the Mosco-epiconvergence of the sum of convex proper lower semicontinuous functions

Quantitative Stability of Variational Systems: II. A Framework for Nonlinear Conditioning

It is shown that for well-conditioned problems (local) optima are holderian with respect to the epi-distance

A Convergence of Bivariate Functions aimed at the Convergence of Saddle Functions

Epi/hypo-convergence is introduced from a variational viewpoint. The known topological properties are reviewed and extended. Finally, it is shown that the (partial) Legendre-Fenchel transform is bicontinuous with respect to the topology induced by epi/hypoconvergence on the space of convex-concave bivariate functions

Damage as Gamma-limit of microfractures in anti-plane linearized elasticity

A homogenization result is given for a material having brittle inclusions arranged in a periodic structure. <br/> According to the relation between the softness parameter and the size of the microstructure, three different limit models are deduced via Gamma-convergence. <br/> In particular, damage is obtained as limit of periodically distributed microfractures

Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators

In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator $S(\cdot + p)+T(\cdot)$, where $p\in X$ and $S$ and $T$ are maximal monotone operators on the reflexive Banach space $X$. Then, this is used to obtain sufficient conditions for the surjectivity of $S+T$ and for the situation when $0$ belongs to the range of $S+T$. Several special cases are discussed, some of them delivering interesting byproducts.Comment: 11 pages, no figure