Deflation for semismooth equations

Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea is the combination of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, deflation constructs a new problem for which a semismooth Newton method will not converge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth Newton method. We demonstrate its utility on various finite- and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics.Comment: 24 pages, 3 figure

Polyhedral Newton-min algorithms for complementarity problems

Abstract : 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 “negative 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

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

Using exact penalties to derive a new equation reformulation of KKT systems associated to variational inequalities

Abstract In this paper, we present a new reformulation of the KKT system associated to a variational inequality as a semismooth equation. The reformulation is derived from the concept of differentiable exact penalties for nonlinear programming. The best results are presented for nonlinear complementarity problems, where simple, verifiable, conditions ensure that the penalty is exact. We also develop a semismooth Newton method for complementarity problems based on the reformulation. We close the paper showing some preliminary computational tests comparing the proposed method with classical reformulations, based on the minimum or on the Fischer-Burmeister function