LIBOPT - An environment for testing solvers on heterogeneous collections of problems - Version 1.0

The Libopt environment is both a methodology and a set of tools that can be used for testing, comparing, and profiling solvers on problems belonging to various collections. These collections can be heterogeneous in the sense that their problems can have common features that differ from one collection to the other. Libopt brings a unified view on this composite world by offering, for example, the possibility to run any solver on any problem compatible with it, using the same Unix/Linux command. The environment also provides tools for comparing the results obtained by solvers on a specified set of problems. Most of the scripts going with the Libopt environment have been written in Perl

Algorithmes de Newton-min polyédriques pour les problèmes de complémentarité

The semismooth Newton method is a very efficient approach for computing a zero of a large class of nonsmooth equations. When the initial iterate is sufficiently close to a regular zero and the function is strongly semismooth, the generated sequence converges quadratically to that zero, while the iteration only requires to solve a linear system.If the first iterate is far away from a zero, however, it is difficult to force its convergence using linesearch or trust regions because a semismooth Newton direction may not be a descent direction of the associated least-square merit function, unlike when the function is differentiable. We explore this question in the particular case of a nonsmooth equation reformulation of the nonlinear complementarity problem, using the minimum function. We propose a globally convergent algorithm using a modification of a semismooth Newton direction that makes it a descent direction of the least-square function. Instead of requiring that the direction satisfies a linear system, it must be a feasible point of a convex polyhedron; hence, it can be computed in polynomial time. This polyhedron is defined by the often very few inequalities, obtained by linearizing pairs of functions that have close negative values at the current iterate; hence, somehow, the algorithm feels the proximity of a "bad kink" of the minimum function and acts accordingly.In order to avoid as often as possible the extra cost of having to find a feasible point of a polyhedron, a hybrid algorithm is also proposed, in which the Newton-min direction is accepted if a sufficient-descent-like criterion is satisfied, which is often the case in practice. Global convergence to regular points is proved; the notion of regularity is associated with the algorithm and is analysed with care.L'algorithme de Newton semi-lisse est très efficace pour calculer un zéro d'une large classe d'équations non lisses. Lorsque le premier itéré est suffisamment proche d'un zéro régulier et si la fonction est fortement semi-lisse, la suite générée converge quadratiquement vers ce zéro, alors que l'itération ne requière que la résolution d'un système linéaire.Cependant, si le premier itéré est éloigné d'un zéro, il est difficile de forcer sa convergence par recherche linéaire ou régions de confiance, parce que la direction de Newton semi-lisse n'est pas nécessairement une direction de descente de la fonction de moindres-carrés associée, contrairement au cas où la fonction à annuler est différentiable. Nous explorons cette question dans le cas particulier d'une reformulation par équation non lisse du problème de complémentarité non linéaire, en utilisant la fonction minimum. Nous proposons un algorithme globalement convergent, utilisant une direction de Newton semi-lisse modifiée, qui est de descente pour la fonction de moindres-carrés. Au lieu de requérir la satisfaction d'un système linéaire, cette direction doit être intérieur à un polyèdre convexe, ce qui peut se calculer en temps polynomial. Ce polyèdre est défini par souvent très peu d'inégalités, obtenus en linéarisant des couples de fonctions qui ont des valeurs négatives proches à l'itéré courant; donc, d'une certaine manière, l'algorithme est capable d'estimer la proximité des "mauvais plis" de la fonction minimum et d'agir en conséquence.De manière à éviter au si souvent que possible le coût supplémentaire lié au calcul d'un point admissible de polyèdre, un algorithme hybride est également proposé, dans lequel la direction de Newton-min est acceptée si un critère de décroissance suffisante est vérifié, ce qui est souvent le cas en pratique. La convergence globale vers des points régulier est démontrée; la notion de régularité est associée à l'algorithme et est analysée avec soin

A dedicated constrained optimization method for 3D reflexion tomography

International audienceSeismic reflection tomography is a method for the determination of a subsurface velocity model from the traveltimes of seismic waves reflecting on geological interfaces. From an optimization viewpoint, the problem consists in minimizing a nonlinear least-squares function measuring the mismatch between observed traveltimes and those calculated by raytracing in this model. The introduction of a priori information on the model is crucial to reduce the under-determination. The contribution of this paper is to introduce a technique able to take into account geological a priori information in the reflection tomography problem expressed as constraints in the optimization problem. This constrained optimization is based on a Gauss-Newton Sequential Quadratic Programming approach. At each Gauss-Newton step, a solution to a strictly convex quadratic optimization problem subject to linear constraints is computed thanks to an augmented Lagrangian relaxation method. Our choice for this optimization method is motivated and its original aspects are described. The efficiency of the method is shown on applications on a 2D OBC real data set and on a 3D real data set: the introduction of constraints coming both from well logs and from geological knowledge allows to reduce the under-determination of the 2 inverse problems. Introduction Reflection tomography allows to determine a velocity model from the traveltimes of seismic waves reflecting on geological interfaces. This inverse problem is formulated as a nonlinear least-squares function which measures the mismatch between observed traveltimes and traveltimes computed by ray tracing method. This method has been successfully applied to real data sets (Ehinger et al, 2001, Broto et al, 2003). Nevertheless, the under-determination of the inverse problem generally requires the introduction of additional information on the model to reduce the number of admissible models. Penalization terms modelling this information can be added to the seismic terms in the objective functions but the tuning of the penalization weights may be tricky. In this paper, we propose to handle the a priori information by the introduction of equality and inequality constraints in the optimization process. This approach allows to introduce lot of constraints of different types, provided we have at our disposal an adequate constrained optimization method. We developed an original method designed for the tomographic inverse problem which presents the following characteristics: it is a large scale problem (10000-50000 unknowns), the forward operator is nonlinear and its computation may be expensive (large number of source-receiver couples, up to 500000), the problem is ill-conditioned. In the first part of this paper, the chosen method is motivated and its original aspects are shortly described (for further details, refer to Delbos et al, 2004). Applications on a 2D PP/PS real data set and on a 3D PP real data set are presented in a second part

QPAL -- A solver of convex quadratic optimization problems, using an augmented Lagrangian approach

QPAL is a piece of software that aims at solving a convex quadratic optimization problem with linear equality and inequality constraints. The implemented algorithm uses an augmented Lagrangian approach, which relaxes the equality constraints and deals explicitely with the bound constraints on the original and slack variables. The generated quadratic functions are minimized on the activated faces by a truncated conjugate gradient algorithm, interspersed with gradient projection steps. When the optimal value is finite, convergence occurs at a linear speed that can be prescribed by the user. Matrices can be strored in dense or sparse structures; in addition, the Hessian of the quadratic objective function may have a direct of inverse $\ell$-BFGS form. QPAL is written in Fortran-2003

Fragments d'Optimisation Différentiable - Théories et Algorithmes

MasterLecture Notes (in French) of optimization courses given at ENSTA (Paris, next Saclay), ENSAE (Paris) and at the universities Paris I, Paris VI and Paris Saclay (979 pages).Syllabus d’enseignements délivrés à l’ENSTA (Paris, puis Saclay), à l’ENSAE (Paris) et aux universités Paris I, Paris VI et Paris Saclay (979 pages)

Conception pas à pas d'un solveur par points intérieurs en optimisation conique auto-duale, avec applications

MasterThese notes present a project in numerical optimization dealing with the implementation of an interior-point method for solving a self-dual conic optimization (SDCO) problem. The cone is the Cartesian product of cones of positive semidefinite matrices of various dimensions (imposing to matrices to be positive semidefinite) and of a positive orthant. Therefore, the solved problem encompasses semidefinite and linear optimization.The project was given in a course entitled 'Advanced Continuous Optimization II' at the University Paris-Saclay, in 2016-2020. The solver is designed step by step during a series of 5 sessions of 4 hours each. Each session corresponds to a chapter of these notes (or a part of it). The correctness of the SDCO solver is verified during each session on small academic problems, having diverse properties. During the last session, the developed piece of software is used to minimize a univariate polynomial on an interval and to solve a few small size rank relaxations of QCQO (quadratically constrained quadratic optimization) problems, modeling various instances of the OPF (optimal power flow) problem. The student has to master not ony the implementation of the interior-point solver, but is also asked to understand the underliying theory by solving exercises consisting in proving some properties of the implemented algorithms.The goal of the project is not to design an SDCO solver that would beat the best existing solver but to help the students to understand and demystify what there is inside such a piece of software. As a side outcome, this course also shows that a rather performent SDCO solver can be realized in a relatively short time.Ces notes présentent un projet d'optimisation numérique dans lequel on implémente une méthode de points intérieurs pour résoudre un problème d'optimisation conique auto-duale (OCAD). Le cône est le produit cartésien de cônes de matrices semi-définies positives de dimensions variables et d'un orthant positif. Dès lors, le problème contient l'optimisation semi-défiinie et l'optimisation linéaire.Ce projet a été proposé dans un cours intitulé 'Advanced Continuous Optimization II' à l'université Paris-Saclay, en 2016-1020. Le solveur est conçu pas à pas durant une suite de 5 leçons de 4 heures chacune. Chaque session fait l'objet d'un chapitre de ces notes. La bonne marche du solveur OCAD est vérifiée à chaque session sur de petits problèmes académiques, ayant diverses propriétés. Durant la dernière session, le code développé est utilisé pour minimiser un polynôme d'une variable sur un intervalle et pour résoudre la relaxation de rang de la formulation QCQP (quadratically constrained quadratic programming) de quelques problèmes d'optimisation de flux d'énergie (OPF) dans de petits réseaux de distribution d'électricité. L'étudiant doit maîtriser non seulement l'implémentation du solveur de points-intérieurs, mais aussi la théorie sous-jacente de manière à pouvoir résoudre des exercices qui consistent à démontrer des propriétés des algorithmes implémentés.Le but de ce cours n'est pas de concevoir un solveur OCAD qui surpasserait le meilleur solveur existant, mais d'aider l'étudiant à comprendre et à démythifier ce que contient un tel solveur. Une conséquence secondaire de cet exercice est de montrer qu'un code OCAD assez performant peut être réalisé en très peu de temps

Superlinear convergence of a reduced BFGS method with piecewise line-search and update criterion

Projet PROMATHWe show the q-superlinear convergence of a reduced BFGS method for equality constrained problems, using eventually only one constraint linearization per iteration. The local method is globalized either with a standard arc-search or when an update criterion is satisfied, with a piecewise line-search. The aim of the latter technique is to realize generalized Wolfe conditions, which allow the algorithm to maintain naturally the positive definiteness of the generated matrices. We show that if the sequence of iterates converges, the convergence is q-superlinear. No assumption is made on the speed of convergence of the sequence of iterates or on the boundedness of the sequence of generated matrices. The main difficulty is to show that the ideal step-size is accepted after finitely many steps