A descent algorithm for solving monotone variational inequalities and monotone complementarity problems

The paper provides a descent algorithm for solving certain monotone variational inequalities and shows how this algorithm may be used for solving certain monotone complementarity problems. Convergence is proved under natural monotonicity and smoothness conditions; neither symmetry nor strict monotonicity is required.The author is grateful to two anonymous referees for their very valuable comments on an earlier draft of this paper.     

Regularized gap functions for nonsmooth variational inequality problems

Under the condition that the involved function F is locally Lipschitz, but not necessarily differentiable, we investigate the regularized gap function defined by a generalized distance function for the variational inequality problem (VIP). First, we compute exactly the Clarke-Rockafellar directional derivatives of the regularized gap functions (and of some modified ones). Second, using these results, we show that, under the strongly monotonicity assumption, the regularized gap functions have fractional exponent error bounds, and thereby we provide an algorithm of Armijo type to solve the VIP.     

A descent method with inexact linear search for nonsmooth equilibrium problems

A descent method with respect to the gap function is formulated and justified for the nonsmooth equilibrium problem. It uses the procedure of inexact linear search of the Armijo type. The proposed method converges under the same assumptions as the methods with exact linear search.     

A feasible descent SQP algorithm for general constrained optimization without strict complementarity

In this paper, a class of optimization problems with equality and inequality constraints is discussed. Firstly, the original problem is transformed to an associated simpler problem with only inequality constraints and a parameter. The later problem is shown to be equivalent to the original problem if the parameter is large enough (but finite), then a feasible descent SQP algorithm for the simplified problem is presented. At each iteration of the proposed algorithm, a master direction is obtained by solving a quadratic program (which always has a feasible solution). With two corrections on the master direction by two simple explicit formulas, the algorithm generates a feasible descent direction for the simplified problem and a height-order correction direction which can avoid the Maratos effect without the strict complementarity, then performs a curve search to obtain the next iteration point. Thanks to the new height-order correction technique, under mild conditions without the strict complementarity, the globally and superlinearly convergent properties are obtained. Finally, an efficient implementation of the numerical experiments is reported.     

A globally convergent descent method for nonsmooth variational inequalities

We propose a descent method via gap functions for solving nonsmooth variational inequalities with a locally Lipschitz operator. Assuming monotone operator (not necessarily strongly monotone) and bounded domain, we show that the method with an Armijo-type line search is globally convergent. Finally, we report some numerical experiments. This work has been supported by the National Research Program PRIN/2005017083 “Innovative Problems and Methods in Nonlinear Optimization”.     

A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems

A new algorithm for the solation of large-scale nonlinear complementarity problems is introduced. The algorithm is based on a nonsmooth equation reformulation of the complementarity problem and on an inexact Levenberg-Marquardt-type algorithm for its solution. Under mild assumptions, and requiring only the approximate solution of a linear system at each iteration, the algorithm is shown to be both globally and superlinearly convergent, even on degenerate problems. Numerical results for problems with up to 10 000 variables are presented. Partially supported by Agenzia Spaziale Italiana, Roma, Italy.     

Merit Functions for Complementarity and Related Problems: A Survey

Merit functions have become important tools for solving various mathematical problems arising from engineering sciences and economic systems. In this paper, we are surveying basic principles and properties of merit functions and some of their applications. As a particular case we will consider the nonlinear complementarity problem (NCP) and present a collection of different merit functions. We will also introduce and study a class of smooth merit functions for the NCP.     

A polynomial-time algorithm for a class of linear complementarity problems

Given ann × n matrixM and ann-dimensional vectorq, the problem of findingn-dimensional vectorsx andy satisfyingy = Mx + q, x 0,y 0,x _i y _i = 0 (i = 1, 2,,n) is known as a linear complementarity problem. Under the assumption thatM is positive semidefinite, this paper presents an algorithm that solves the problem in O(n ³ L) arithmetic operations by tracing the path of centers,{(x, y) S: x _i y _i = (i = 1, 2,,n) for some > 0} of the feasible regionS = {(x, y) 0:y = Mx + q}, whereL denotes the size of the input data of the problem.     

Feasible descent algorithms for mixed complementarity problems

In this paper we consider a general algorithmic framework for solving nonlinear mixed complementarity problems. The main features of this framework are: (a) it is well-defined for an arbitrary mixed complementarity problem, (b) it generates only feasible iterates, (c) it has a strong global convergence theory, and (d) it is locally fast convergent under standard regularity assumptions. This framework is applied to the PATH solver in order to show viability of the approach. Numerical results for an appropriate modification of the PATH solver indicate that this framework leads to substantial computational improvements. Received April 9, 1998 / Revised version received November 23, 1998?Published online March 16, 1999     

A linearly convergent derivative-free descent method for the second-order cone complementarity problem

We consider a class of derivative-free descent methods for solving the second-order cone complementarity problem (SOCCP). The algorithm is based on the Fischer–Burmeister (FB) unconstrained minimization reformulation of the SOCCP, and utilizes a convex combination of the negative partial gradients of the FB merit function ψ_FB as the search direction. We establish the global convergence results of the algorithm under monotonicity and the uniform Jordan P-property, and show that under strong monotonicity the merit function value sequence generated converges at a linear rate to zero. Particularly, the rate of convergence is dependent on the structure of second-order cones. Numerical comparisons are also made with the limited BFGS method used by Chen and Tseng (An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104(2005), pp. 293–327), which confirm the theoretical results and the effectiveness of the algorithm.     

An algorithm for a class of nonlinear complementarity problems with non-Lipschitzian functions

In this paper, we focus on solving a class of nonlinear complementarity problems with non-Lipschitzian functions. We first introduce a generalized class of smoothing functions for the plus function. By combining it with Robinson's normal equation, we reformulate the complementarity problem as a family of parameterized smoothing equations. Then, a smoothing Newton method combined with a new nonmonotone line search scheme is employed to compute a solution of the smoothing equations. The global and local superlinear convergence of the proposed method is proved under mild assumptions. Preliminary numerical results obtained applying the proposed approach to nonlinear complementarity problems arising in free boundary problems are reported. They show that the smoothing function and the nonmonotone line search scheme proposed in this paper are effective.     

Multiple subgradient descent bundle method for convex nonsmooth multiobjective optimization

The aim of this paper is to propose a new multiple subgradient descent bundle method for solving unconstrained convex nonsmooth multiobjective optimization problems. Contrary to many existing multiobjective optimization methods, our method treats the objective functions as they are without employing a scalarization in a classical sense. The main idea of this method is to find descent directions for every objective function separately by utilizing the proximal bundle approach, and then trying to form a common descent direction for every objective function. In addition, we prove that the method is convergent and it finds weakly Pareto optimal solutions. Finally, some numerical experiments are considered.     

The largest step path following algorithm for monotone linear complementarity problems

Path-following algorithms take at each iteration a Newton step for approaching a point on the central path, in such a way that all the iterates remain in a given neighborhood of that path. This paper studies the case in which each iteration uses a pure Newton step with the largest possible reduction in complementarity measure (duality gap). This algorithm is known to converge superlinearly in objective values. We show that with the addition of a computationally trivial safeguard it achieves Q-quadratic convergence, and show that this behaviour cannot be proved by usual techniques for the original method. Research done while visiting Delft University of Technology, and supported in part by CAPES-Brazil.     

Coordinate relaxation methods for multivalued complementarity problems

Methods of the Jacobi and Gauss-Seidel type with underrelaxation and a combined method of the splitting type are proposed for complementarity problems with multivalued mappings. The convergence of these methods to the solution is proved under the conditions that the basic mapping is upper off-diagonal antitone and the feasible set is nonempty. The numerical results obtained for test examples are presented.     

Infeasible-interior-point algorithm for a class of nonmonotone complementarity problems and its computational complexity

This paper presents an infeasible-interior-point algorithm for a class of nonmonotone complementarity problems, and analyses its convergence and computational complexity. The results indicate that the proposed algorithm is a polynomial-time one.     

Merit functions for semi-definite complemetarity problems

Merit functions such as the gap function, the regularized gap function, the implicit Lagrangian, and the norm squared of the Fischer-Burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors. We study the extension of these merit functions to complementarity problems defined over the cone of block-diagonal symmetric positive semi-definite real matrices. The extension suggests new solution methods for the latter problems. This research is supported by National Science Foundation Grant CCR-9311621.     

A semismooth equation approach to the solution of nonlinear complementarity problems

In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions. Other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration. We include numerical tests which show the viability of the approach.     

A descent algorithm for nonsmooth convex optimization

This paper presents a new descent algorithm for minimizing a convex function which is not necessarily differentiable. The algorithm can be implemented and may be considered a modification of the ε-subgradient algorithm and Lemarechal's descent algorithm. Also our algorithm is seen to be closely related to the proximal point algorithm applied to convex minimization problems. A convergence theorem for the algorithm is established under the assumption that the objective function is bounded from below. Limited computational experience with the algorithm is also reported.     

Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems

Whether or not the general asymmetric variational inequality problem can be formulated as a differentiable optimization problem has been an open question. This paper gives an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequality problem, and show that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem. We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.     

Properties of a family of merit functions and a merit function method for the NCP

A family of merit functions are proposed, which are the generalization of several existing merit functions. A number of favorable properties of the proposed merit functions are established. By using these properties, a merit function method for solving nonlinear complementarity problem is investigated, and the global convergence of the proposed algorithm is proved under some standard assumptions. Some preliminary numerical results are given.