Generalized Newton methods for the 2D-Signorini contact problem with friction in function space

International audienceThe 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented Lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

A trust region-type normal map-based semismooth Newton method for nonsmooth nonconvex composite optimization

We propose a novel trust region method for solving a class of nonsmooth and nonconvex composite-type optimization problems. The approach embeds inexact semismooth Newton steps for finding zeros of a normal map-based stationarity measure for the problem in a trust region framework. Based on a new merit function and acceptance mechanism, global convergence and transition to fast local q-superlinear convergence are established under standard conditions. In addition, we verify that the proposed trust region globalization is compatible with the Kurdyka-{\L}ojasiewicz (KL) inequality yielding finer convergence results. We further derive new normal map-based representations of the associated second-order optimality conditions that have direct connections to the local assumptions required for fast convergence. Finally, we study the behavior of our algorithm when the Hessian matrix of the smooth part of the objective function is approximated by BFGS updates. We successfully link the KL theory, properties of the BFGS approximations, and a Dennis-Mor{\'e}-type condition to show superlinear convergence of the quasi-Newton version of our method. Numerical experiments on sparse logistic regression and image compression illustrate the efficiency of the proposed algorithm.Comment: 56 page

Algorithms for Large Scale Nuclear Norm Minimization and Convex Quadratic Semidefinite Programming Problems

Ph.DDOCTOR OF PHILOSOPH

Optimization and Applications

Proceedings of a workshop devoted to optimization problems, their theory and resolution, and above all applications of them. The topics covered existence and stability of solutions; design, analysis, development and implementation of algorithms; applications in mechanics, telecommunications, medicine, operations research

Sur la résolution du problème de frottement tridimensionnel : Formulations et comparaisons des méthodes numériques

In this report, we review several formulations of the discrete frictional contact problemthat arises in space and time discretized mechanical systems with unilateral contact andthree-dimensional Coulomb’s friction. Most of these formulations are well–known concepts in theoptimization community, or more generally, in the mathematical programming community. Tocite a few, the discrete frictional contact problem can be formulated as variational inequalities,generalized or semi–smooth equations, second–order cone complementarity problems, or as optimizationproblems such as quadratic programming problems over second-order cones. Thanks tothese multiple formulations, various numerical methods emerge naturally for solving the problem.We review the main numerical techniques that are well-known in the literature and we also proposenew applications of methods such as the fixed point and extra-gradient methods with self-adaptivestep rules for variational inequalities or the proximal point algorithm for generalized equations.All these numerical techniques are compared over a large set of test examples using performanceprofiles. One of the main conclusion is that there is no universal solver. Nevertheless, we are ableto give some hints to choose a solver with respect to the main characteristics of the set of testsDans ce rapport, plusieurs formulations du problème discret de contact frottant qui apparaîtdans les systèmes mécaniques avec du contact unilatéral et du frottement de Coulomb, sont présentées.La plupart de ces formulations sont des objets bien connus dans la communauté de l’optimisation, etplus généralement, de la programmation mathématique. Pour en citer quelques uns, le problème decontact frottant peut être formulé comme une inégalité variationnelle, comme une équation non-régulièreou semi–lisse, comme un problème de complémentarité sur des cônes, ou encore comme des problèmesd’optimisation par exemple des problèmes quadratiques sur des cônes du second ordre. Grâce à cesmultiples formulations, de nombreuses méthodes numériques de résolutions émergent naturellement. Ondétaille dans ce rapport les principales techniques numériques bien connues dans la littérature et nousproposons aussi des nouvelles méthodes comme les méthodes de point fixe et d’extra-gradient pour lesinégalités variationnelles avec une règle d’adaptation automatique du pas, ainsi que l’application del’algorithme du point optimal pour les équations généralisées. Toutes ces techniques sont comparées surun grand ensemble de problème–tests en utilisant des profils de performance. Une des conclusions est qu’iln’existe pas de méthode universelle. Néanmoins, on peut donner des conseils pour choisir une méthodeparticulière la mieux adaptée aux caractéristiques d’un problème donné

Non-Smooth Optimization by Abs-Linearization in Reflexive Function Spaces

Nichtglatte Optimierungsprobleme in reflexiven Banachräumen treten in vielen Anwendungen auf. Häufig wird angenommen, dass alle vorkommenden Nichtdifferenzierbarkeiten durch Lipschitz-stetige Operatoren wie abs, min und max gegeben sind. Bei solchen Problemen kann es sich zum Beispiel um optimale Steuerungsprobleme mit möglicherweise nicht glatten Zielfunktionen handeln, welche durch partielle Differentialgleichungen (PDG) eingeschränkt sind, die ebenfalls nicht glatte Terme enthalten können. Eine effiziente und robuste Lösung erfordert eine Kombination numerischer Simulationen und spezifischer Optimierungsalgorithmen. Lokal Lipschitz-stetige, nichtglatte Nemytzkii-Operatoren, welche direkt in der Problemformulierung auftreten, spielen eine wesentliche Rolle in der Untersuchung der zugrundeliegenden Optimierungsprobleme. In dieser Dissertation werden zwei spezifische Methoden und Algorithmen zur Lösung solcher nichtglatter Optimierungsprobleme in reflexiven Banachräumen vorgestellt und diskutiert. Als erste Lösungsmethode wird in dieser Dissertation die Minimierung von nichtglatten Operatoren in reflexiven Banachräumen mittels sukzessiver quadratischer Überschätzung vorgestellt, SALMIN. Ein neuartiger Optimierungsansatz für Optimierungsprobleme mit nichtglatten elliptischen PDG-Beschränkungen, welcher auf expliziter Strukturausnutzung beruht, stellt die zweite Lösungsmethode dar, SCALi. Das zentrale Merkmal dieser Methoden ist ein geeigneter Umgang mit Nichtglattheiten. Besonderes Augenmerk liegt dabei auf der zugrundeliegenden nichtglatten Struktur des Problems und der effektiven Ausnutzung dieser, um das Optimierungsproblem auf angemessene und effiziente Weise zu lösen.Non-smooth optimization problems in reflexive Banach spaces arise in many applications. Frequently, all non-differentiabilities involved are assumed to be given by Lipschitz-continuous operators such as abs, min and max. For example, such problems can refer to optimal control problems with possibly non-smooth objective functionals constrained by partial differential equations (PDEs) which can also include non-smooth terms. Their efficient as well as robust solution requires numerical simulations combined with specific optimization algorithms. Locally Lipschitz-continuous non-smooth non-linearities described by appropriate Nemytzkii operators which arise directly in the problem formulation play an essential role in the study of the underlying optimization problems. In this dissertation, two specific solution methods and algorithms to solve such non-smooth optimization problems in reflexive Banach spaces are proposed and discussed. The minimization of non-smooth operators in reflexive Banach spaces by means of successive quadratic overestimation is presented as the first solution method, SALMIN. A novel structure exploiting optimization approach for optimization problems with non-smooth elliptic PDE constraints constitutes the second solution method, SCALi. The central feature of these methods is the appropriate handling of non-differentiabilities. Special focus lies on the underlying structure of the problem stemming from the non-smoothness and how it can be effectively exploited to solve the optimization problem in an appropriate and efficient way

Numerical Analysis of Algorithms for Infinitesimal Associated and Non-Associated Elasto-Plasticity

The thesis studies nonlinear solution algorithms for problems in infinitesimal elastoplasticity and their numerical realization within a parallel computing framework. New algorithms like Active Set and Augmented Lagrangian methods are proposed and analyzed within a semismooth Newton setting. The analysis is often carried out in function space which results in stable algorithms. Large scale computer experiments demonstrate the efficiency of the new algorithms

On alternating direction methods for monotropic semidefinite programming

Ph.DDOCTOR OF PHILOSOPH