33 research outputs found

    An interior penalty method for a finite-dimensional linear complementarity problem in financial engineering

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    In this work we study an interior penalty method for a finite-dimensional large-scale linear complementarity problem (LCP) arising often from the discretization of stochastic optimal problems in financial engineering. In this approach, we approximate the LCP by a nonlinear algebraic equation containing a penalty term linked to the logarithmic barrier function for constrained optimization problems. We show that the penalty equation has a solution and establish a convergence theory for the approximate solutions. A smooth Newton method is proposed for solving the penalty equation and properties of the Jacobian matrix in the Newton method have been investigated. Numerical experimental results using three non-trivial test examples are presented to demonstrate the rates of convergence, efficiency and usefulness of the method for solving practical problems

    Polyhedral Newton-min algorithms for complementarity problems

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    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

    A full approximation scheme multilevel method for nonlinear variational inequalities

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    We present the full approximation scheme constraint decomposition (FASCD) multilevel method for solving variational inequalities (VIs). FASCD is a common extension of both the full approximation scheme (FAS) multigrid technique for nonlinear partial differential equations, due to A.~Brandt, and the constraint decomposition (CD) method introduced by X.-C.~Tai for VIs arising in optimization. We extend the CD idea by exploiting the telescoping nature of certain function space subset decompositions arising from multilevel mesh hierarchies. When a reduced-space (active set) Newton method is applied as a smoother, with work proportional to the number of unknowns on a given mesh level, FASCD V-cycles exhibit nearly mesh-independent convergence rates, and full multigrid cycles are optimal solvers. The example problems include differential operators which are symmetric linear, nonsymmetric linear, and nonlinear, in unilateral and bilateral VI problems.Comment: 25 pages, 9 figure

    Reformulation semi-lisse appliquée au problème de complémentarité

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    Ce mémoire fait une revue des notions élémentaires concernant le problème de complé- mentarité. On y fait aussi un survol des principales méthodes connues pour le résoudre. Plus précisément, on s’intéresse à la méthode de Newton semi-lisse. Un article proposant une légère modification à cette méthode est présenté. Cette nouvelle méthode compétitive est démontrée convergente. Un second article traitant de la complexité itérative de la méthode de Harker et Pang est aussi introduit

    Adaptive inexact smoothing Newton method for a nonconforming discretization of a variational inequality

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    International audienceWe develop in this work an adaptive inexact smoothing Newton method for a nonconforming discretization of a variational inequality. As a model problem, we consider the contact problem between two membranes. Discretized with the finite volume method, this leads to a nonlinear algebraic system with complementarity constraints. The non-differentiability of the arising nonlinear discrete problem a priori requests the use of an iterative linearization algorithm in the semismooth class like, e.g., the Newton-min. In this work, we rather approximate the inequality constraints by a smooth nonlinear equality, involving a positive smoothing parameter that should be drawn down to zero. This makes it possible to directly apply any standard linearization like the Newton method. The solution of the ensuing linear system is then approximated by any iterative linear algebraic solver. In our approach, we carry out an a posteriori error analysis where we introduce potential reconstructions in discrete subspaces included in H1 (Ω), as well as H (div, Ω)-conforming discrete equilibrated flux reconstructions. With these elements, we design an a posteriori estimate that provides guaranteed upper bound on the energy error between the unavailable exact solution of the continuous level and a postprocessed, discrete, and available approximation, and this at any resolution step. It also offers a separation of the different error components, namely, discretization, smoothing, linearization, and algebraic. Moreover, we propose stopping criteria and design an adaptive algorithm where all the iterative procedures (smoothing, linearization, algebraic) are adaptively stopped; this is in particular our way to fix the smoothing parameter. Finally, we numerically assess the estimate and confirm the performance of the proposed adaptive algorithm, in particular in comparison with the semismooth Newton method

    Solving variational inequalities and cone complementarity problems in nonsmooth dynamics using the alternating direction method of multipliers

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    This work presents a numerical method for the solution of variational inequalities arising in nonsmooth flexible multibody problems that involve set-valued forces. For the special case of hard frictional contacts, the method solves a second order cone complementarity problem. We ground our algorithm on the Alternating Direction Method of Multipliers (ADMM), an efficient and robust optimization method that draws on few computational primitives. In order to improve computational performance, we reformulated the original ADMM scheme in order to exploit the sparsity of constraint jacobians and we added optimizations such as warm starting and adaptive step scaling. The proposed method can be used in scenarios that pose major difficulties to other methods available in literature for complementarity in contact dynamics, namely when using very stiff finite elements and when simulating articulated mechanisms with odd mass ratios. The method can have applications in the fields of robotics, vehicle dynamics, virtual reality, and multiphysics simulation in general

    Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks

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    This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush-Kuhn-Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer-Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time

    Nash Equilibria with Piecewise Quadratic Costs

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    A New Mathematical Programming Framework for Facility Layout Design

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    We present a new framework for efficiently finding competitive solutions for the facility layout problem. This framework is based on the combination of two new mathematical programming models. The first model is a relaxation of the layout problem and is intended to find good starting points for the iterative algorithm used to solve the second model. The second model is an exact formulation of the facility layout problem as a non-convex mathematical program with equilibrium constraints (MPEC). Aspect ratio constraints, which are frequently used in facility layout methods to restrict the occurrence of overly long and narrow departments in the computed layouts, are easily incorporated into this new framework. Finally, we present computational results showing that both models, and hence the complete framework, can be solved efficiently using widely available optimization software. This important feature of the new framework implies that it can be used to find competitive layouts with relatively little computational effort. This is advantageous for a user who wishes to consider several competitive layouts rather than simply using the mathematically optimal layout

    Un algoritmo cuasi Newton inexacto global para problemas de complementariedad no lineal.

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    En este trabajo de investigación proponemos y desarrollamos un método cuasi Newton inexacto global para resolver el problema de complementariedad no lineal (PCNL) de una manera indirecta: en primer lugar, reescribiremos el PCNL como un problema de Complementariedad Horizontal (PCH) y posteriormente, reescribiremos el PCH como un problema de minimización. Cabe destacar que abordar el PCNL de esta manera nos permitirá trabajar con reformulaciones diferenciales de la versión original del problema. De igual forma, proponemos una leve modificación al algoritmo para resolver problemas de complementariedad no lineal, con el fin de obtener un método que permita encontrar las raíces positivas de sistemas de ecuaciones no lineales de gran tamaño
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