Error bounds for proximal point subproblems and associated inexact proximal point algorithms

We study various error measures for approximate solution of proximal point regularizations of the variational inequality problem, and of the closely related problem of finding a zero of a maximal monotone operator. A new merit function is proposed for proximal point subproblems associated with the latter. This merit function is based on Burachik-Iusem-Svaiter’s concept of ε-enlargement of a maximal monotone operator. For variational inequalities, we establish a precise relationship between the regularized gap function, which is a natural error measure in this context, and our new merit function. Some error bounds are derived using both merit functions for the corresponding formulations of the proximal subproblem. We further use the regularized gap function to devise a new inexact proximal point algorithm for solving monotone variational inequalities. This inexact proximal point method preserves all the desirable global and local convergence properties of the classical exact/inexact method, while providing a constructive error tolerance criterion, suitable for further practical applications. The use of other tolerance rules is also discussed. Received: April 28, 1999 / Accepted: March 24, 2000?Published online July 20, 2000     

Global s-type error bound for the extended linear complementarity problem and applications

For the extended linear complementarity problem over an affine subspace, we first study some characterizations of (strong) column/row monotonicity and (strong) R ₀-property. We then establish global s-type error bound for this problem with the column monotonicity or R ₀-property, especially for the one with the nondegeneracy and column monotonicity, and give several equivalent formulations of such error bound without the square root term for monotone affine variational inequality. Finally, we use this error bound to derive some properties of the iterative sequence produced by smoothing methods for solving such a problem under suitable assumptions. Received: May 2, 1999 / Accepted: February 21, 2000?Published online July 20, 2000     

Boundedness and Regularity Properties of Semismooth Reformulations of Variational Inequalities

The Karush-Kuhn-Tucker (KKT) system of the variational inequality problem over a set defined by inequality and equality constraints can be reformulated as a system of semismooth equations via an nonlinear complementarity problem (NCP) function. We give a sufficient condition for boundedness of the level sets of the norm function of this system of semismooth equations when the NCP function is metrically equivalent to the minimum function; and a sufficient and necessary condition when the NCP function is the minimum function. Nonsingularity properties identified by Facchinei, Fischer and Kanzow, 1998, SIAM J. Optim. 8, 850–869, for the semismooth reformulation of the variational inequality problem via the Fischer-Burmeister function, which is an irrational regular pseudo-smooth NCP function, hold for the reformulation based on other regular pseudo-smooth NCP functions. We propose a new regular pseudo-smooth NCP function, which is piecewise linear-rational and metrically equivalent to the minimum NCP function. When it is used to the generalized Newton method for solving the variational inequality problem, an auxiliary step can be added to each iteration to reduce the value of the merit function by adjusting the Lagrangian multipliers only. This work is supported by the Research Grant Council of Hong Kong This paper is dedicated to Alex Rubinov on the occasion of his 65th Birthday     

On existence and uniqueness of stationary points in semi-infinite optimization

The paper deals with semi-infinite optimization problems which are defined by finitely many equality constraints and infinitely many inequality constraints. We generalize the concept of strongly stable stationary points which was introduced by Kojima for finite problems; it refers to the local existence and uniqueness of a stationary point for each sufficiently small perturbed problem, where perturbations up to second order are allowed. Under the extended Mangasarian-Fromovitz constraint qualification we present equivalent conditions for the strong stability of a considered stationary point in terms of first and second derivatives of the involved functions. In particular, we discuss the case where the reduction approach is not satisfied. Received June 30, 1995 / Revised version received October 9, 1998? Published online June 11, 1999     

A non-interior continuation method for generalized linear complementarity problems

In this paper, we propose a non-interior continuation method for solving generalized linear complementarity problems (GLCP) introduced by Cottle and Dantzig. The method is based on a smoothing function derived from the exponential penalty function first introduced by Kort and Bertsekas for constrained minimization. This smoothing function can also be viewed as a natural extension of Chen-Mangasarian’s neural network smooth function. By using the smoothing function, we approximate GLCP as a family of parameterized smooth equations. An algorithm is presented to follow the smoothing path. Under suitable assumptions, it is shown that the algorithm is globally convergent and local Q-quadratically convergent. Few preliminary numerical results are also reported. Received September 3, 1997 / Revised version received April 27, 1999?Published online July 19, 1999     

A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities

In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions, we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent nonsmooth equation H(u,x)=0, where H:ℜ ²ⁿ →ℜ ²ⁿ , u∈ℜ ⁿ is a parameter variable and x∈ℜ ⁿ is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z ^k =(u ^k ,x ^k )} such that the mapping H(·) is continuously differentiable at each z ^k and may be non-differentiable at the limiting point of {z ^k }. We prove that three most often used Gabriel-Moré smoothing functions can generate strongly semismooth functions, which play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the proposed methods for particularly chosen smoothing functions are very promising. Received June 23, 1997 / Revised version received July 29, 1999?Published online December 15, 1999     

A hybrid Newton method for solving the variational inequality problem via the D-gap function

The variational inequality problem (VIP) can be reformulated as an unconstrained minimization problem through the D-gap function. It is proved that the D-gap function has bounded level sets for the strongly monotone VIP. A hybrid Newton-type method is proposed for minimizing the D-gap function. Under some conditions, it is shown that the algorithm is globally convergent and locally quadratically convergent. Received May 6, 1997 / Revised version received October 30, 1998?Published online June 11, 1999     

Properties of an interior embedding for solving nonlinear optimization problems

The paper presents an interior embedding of nonlinear optimization problems. This embedding satisfies a sufficient condition for the success of pathfollowing algorithms with jumps being applied to one-parametric optimization problems.?The one-parametric problem obtained by the embedding is supposed to be regular in the sense of Jongen, Jonker and Twilt. This asumption is analyzed, and its genericity is proved in the space of the original optimization problems. Received May 20, 1997 / Revised version received October 6, 1998?Published online May 12, 1999     

Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities

Let VIP(F,C) denote the variational inequality problem associated with the mapping F and the closed convex set C. In this paper we introduce weak conditions on the mapping F that allow the development of a convergent cutting-plane framework for solving VIP(F,C). In the process we introduce, in a natural way, new and useful notions of generalized monotonicity for which first order characterizations are presented. Received: September 25, 1997 / Accepted: March 2, 1999?Published online July 20, 2000     

Smooth methods of multipliers for complementarity problems

This paper describes several methods for solving nonlinear complementarity problems. A general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem, obtaining primal, dual, and primal-dual formulations. We derive Bregman-function-based generalized proximal algorithms for each of these formulations, generating three classes of complementarity algorithms. The primal class is well-known. The dual class is new and constitutes a general collection of methods of multipliers, or augmented Lagrangian methods, for complementarity problems. In a special case, it corresponds to a class of variational inequality algorithms proposed by Gabay. By appropriate choice of Bregman function, the augmented Lagrangian subproblem in these methods can be made continuously differentiable. The primal-dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods. Some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems. Received February 21, 1997 / Revised version received December 11, 1998? Published online May 12, 1999     

Forcing strong convergence of proximal point iterations in a Hilbert space

This paper concerns with convergence properties of the classical proximal point algorithm for finding zeroes of maximal monotone operators in an infinite-dimensional Hilbert space. It is well known that the proximal point algorithm converges weakly to a solution under very mild assumptions. However, it was shown by Güler [11] that the iterates may fail to converge strongly in the infinite-dimensional case. We propose a new proximal-type algorithm which does converge strongly, provided the problem has a solution. Moreover, our algorithm solves proximal point subproblems inexactly, with a constructive stopping criterion introduced in [31]. Strong convergence is forced by combining proximal point iterations with simple projection steps onto intersection of two halfspaces containing the solution set. Additional cost of this extra projection step is essentially negligible since it amounts, at most, to solving a linear system of two equations in two unknowns. Received January 6, 1998 / Revised version received August 9, 1999?Published online November 30, 1999     

Upper bounds for Gaussian stochastic programs

We present a construction which gives deterministic upper bounds for stochastic programs in which the randomness appears on the right–hand–side and has a multivariate Gaussian distribution. Computation of these bounds requires the solution of only as many linear programs as the problem has variables. Received December 2, 1997 / Revised version received January 5, 1999? Published online May 12, 1999     

New projection-type methods for monotone LCP with finite termination

Summary. In this paper we establish two new projection-type methods for the solution of monotone linear complementarity problem (LCP). The methods are a combination of the extragradient method and the Newton method, in which the active set strategy is used and only one linear system of equations with lower dimension is solved at each iteration. It is shown that under the assumption of monotonicity, these two methods are globally and linearly convergent. Furthermore, under a nondegeneracy condition they have a finite termination property. At last, the methods are extended to solving monotone affine variational inequality problem. Received October 10, 2000 / Revised version received May 22, 2001 / Published online October 17, 2001     

Entropic proximal decomposition methods for convex programs and variational inequalities

We consider convex optimization and variational inequality problems with a given separable structure. We propose a new decomposition method for these problems which combines the recent logarithmic-quadratic proximal theory introduced by the authors with a decomposition method given by Chen-Teboulle for convex problems with particular structure. The resulting method allows to produce for the first time provably convergent decomposition schemes based on C ^∞ Lagrangians for solving convex structured problems. Under the only assumption that the primal-dual problems have nonempty solution sets, global convergence of the primal-dual sequences produced by the algorithm is established. Received: October 6, 1999 / Accepted: February 2001?Published online September 17, 2001     

Approximating the complexity measure of Vavasis-Ye algorithm is NP-hard

Given an m×n integer matrix A of full row rank, we consider the problem of computing the maximum of ∥B ^-1 A∥₂ where B varies over all bases of A. This quantity appears in various places in the mathematical programming literature. More recently, logarithm of this number was the determining factor in the complexity bound of Vavasis and Ye’s primal-dual interior-point algorithm. We prove that the problem of approximating this maximum norm, even within an exponential (in the dimension of A) factor, is NP-hard. Our proof is based on a closely related result of L. Khachiyan [1]. Received November 13, 1998 / Revised version received January 20, 1999? Published online May 12, 1999     

A new inexact alternating directions method for monotone variational inequalities

The alternating directions method (ADM) is an effective method for solving a class of variational inequalities (VI) when the proximal and penalty parameters in sub-VI problems are properly selected. In this paper, we propose a new ADM method which needs to solve two strongly monotone sub-VI problems in each iteration approximately and allows the parameters to vary from iteration to iteration. The convergence of the proposed ADM method is proved under quite mild assumptions and flexible parameter conditions. Received: January 4, 2000 / Accepted: October 2001?Published online February 14, 2002     

A branch-and-cut method for 0-1 mixed convex programming

We generalize the disjunctive approach of Balas, Ceria, and Cornuéjols [2] and devevlop a branch-and-cut method for solving 0-1 convex programming problems. We show that cuts can be generated by solving a single convex program. We show how to construct regions similar to those of Sherali and Adams [20] and Lovász and Schrijver [12] for the convex case. Finally, we give some preliminary computational results for our method. Received January 16, 1996 / Revised version received April 23, 1999?Published online June 28, 1999     

Filled function method for nonlinear equations

Systems of nonlinear equations are ubiquitous in engineering, physics and mechanics, and have myriad applications. Generally, they are very difficult to solve. In this paper, we will present a filled function method to solve nonlinear systems. We will first convert the nonlinear systems into equivalent global optimization problems with the property: x^∗ is a global minimizer if and only if its function value is zero. A filled function method is proposed to solve the converted global optimization problem. Numerical examples are presented to illustrate our new techniques.     

An interior Newton method for quadratic programming

We propose a new (interior) approach for the general quadratic programming problem. We establish that the new method has strong convergence properties: the generated sequence converges globally to a point satisfying the second-order necessary optimality conditions, and the rate of convergence is 2-step quadratic if the limit point is a strong local minimizer. Published alternative interior approaches do not share such strong convergence properties for the nonconvex case. We also report on the results of preliminary numerical experiments: the results indicate that the proposed method has considerable practical potential. Received October 11, 1993 / Revised version received February 20, 1996 Published online July 19, 1999     

Applying a Power Penalty Method to Numerically Pricing American Bond Options

In this paper, we aim to develop a numerical scheme to price American options on a zero-coupon bond based on a power penalty approach. This pricing problem is formulated as a variational inequality problem (VI) or a complementarity problem (CP). We apply a fitted finite volume discretization in space along with an implicit scheme in time, to the variational inequality problem, and obtain a discretized linear complementarity problem (LCP). We then develop a power penalty approach to solve the LCP by solving a system of nonlinear equations. The unique solvability and convergence of the penalized problem are established. Finally, we carry out numerical experiments to examine the convergence of the power penalty method and to testify the efficiency and effectiveness of our numerical scheme.