A projection and contraction method for a class of linear complementarity problems and its application in convex quadratic programming

In this paper we propose a new iterative method for solving a class of linear complementarity problems:u 0,Mu + q 0, u^T(Mu + q)=0, where M is a givenl ×l positive semidefinite matrix (not necessarily symmetric) andq is a givenl-vector. The method makes two matrix-vector multiplications and a trivial projection onto the nonnegative orthant at each iteration, and the Euclidean distance of the iterates to the solution set monotonously converges to zero. The main advantages of the method presented are its simplicity, robustness, and ability to handle large problems with any start point. It is pointed out that the method may be used to solve general convex quadratic programming problems. Preliminary numerical experiments indicate that this method may be very efficient for large sparse problems.On leave from the Department of Mathematics, University of Nanjing, Nanjing, People's Republic of China.     

Identifying the optimal partition in convex quadratic programming

Given an optimal solution for a convex quadratic programming (QP) problem, the optimal partition of the QP can be computed by solving a pair of linear or QP problems for which nearly optimal solutions are known.     

Solving more linear complementarity problems with Murty's bard-type algorithm

The linear complementarity problem (M|q) is to findw andz inR ⁿ such thatw–Mz=q,w0,z0,w ^t z=0, givenM inR ^n×n andq in . Murty's Bard-type algorithm for solving LCP is modeled as a digraph.Murty's original convergence proof considered allq inR ⁿ andM inR ^n×n , aP-matrix. We show how to solve more LCP's by restricting the set ofq vectors and enlarging the class ofM matrices beyondP-matrices. The effect is that the graph contains an embedded graph of the type considered by Stickney and Watson wheneverM is a matrix containing a principal submatrix which is aP-matrix. Examples are presented which show what can happen when the hypotheses are further weakened.     

A note on a quadratic formulation for linear complementarity problems

In this note, we discuss some properties of a quadratic formulation for linear complementarity problems. Projected SOR methods proposed by Mangasarian apply to symmetric matrices only. The quadratic formulation discussed here makes it possible to use these SOR methods for solving nonsymmetric LCPs. SOR schemes based on this formulation preserve sparsity. For proper choice of a free parameter, this quadratic formulation also preserves convexity. The value of the quadratic function for the solution of original LCP is also known.     

Deriving an unconstrained convex program for linear programming

Consider a linear programming problem in Karmarkar's standard form. By perturbing its linear objective function with an entropic barrier function and applying generalized geometric programming theory to it, Fang recently proposed an unconstrained convex programming approach to finding an epsilon-optimal solution. In this paper, we show that Fang's derivation of an unconstrained convex dual program can be greatly simplified by using only one simple geometric inequality. In addition, a system of nonlinear equations, which leads to a pair of primal and dual epsilon-optimal solutions, is proposed for further investigation.This work was partially supported by the North Carolina Supercomputing Center and a 1990 Cray Research Grant. The authors are indebted to Professors E. L. Peterson and R. Saigal for stimulating discussions.     

An extension of Karmarkar's projective algorithm for convex quadratic programming

We present an extension of Karmarkar's linear programming algorithm for solving a more general group of optimization problems: convex quadratic programs. This extension is based on the iterated application of the objective augmentation and the projective transformation, followed by optimization over an inscribing ellipsoid centered at the current solution. It creates a sequence of interior feasible points that converge to the optimal feasible solution in O(Ln) iterations; each iteration can be computed in O(Ln ³) arithmetic operations, wheren is the number of variables andL is the number of bits in the input. In this paper, we emphasize its convergence property, practical efficiency, and relation to the ellipsoid method.     

On extremal behaviors of Murty's least index method

In this small note, we observe some extremal behaviors of Murty's least index method for solving linear complementarity problems. In particular, we show that the expected number of steps for solving Murty's exponential example with a random permutation of variable indices is exactly equal ton, wheren is the size of the input square matrix.Corresponding author.     

Implementing proximal point methods for linear programming

We describe the application of proximal point methods to the linear programming problem. Two basic methods are discussed. The first, which has been investigated by Mangasarian and others, is essentially the well-known method of multipliers. This approach gives rise at each iteration to a weakly convex quadratic program which may be solved inexactly using a point-SOR technique. The second approach is based on the proximal method of multipliers, originally proposed by Rockafellar, for which the quadratic program at each iteration is strongly convex. A number of techniques are used to solve this subproblem, the most promising of which appears to be a two-metric gradient-projection approach. Convergence results are given, and some numerical experience is reported.This research was supported by National Science Foundation, Grant Nos. ASC-87-14009 and DMS-86-19903, and by Air Force Office of Scientific Research, Grant No. AFOSR-ISSA-87-0092. Part of this work was performed at Argonne National Laboratory. Computation was partly performed at the Pittsburgh Supercomputer Center under NSF Grant No. DMS-88-0033P and at the Argonne Advanced Computing Research Facility, whose support is gratefully acknowledged. The author thanks Olvi Mangasarian and Renato DeLeone for helpful discussions, and Jorge Moré for his valuable advice on the algorithms discussed in Section 3. The contributions of a referee, who provided helpful comments on an earlier version of this paper, are also acknowledged.     

Local minima of quadratic forms on convex cones

We study the local minima and the critical values of a quadratic form on the trace of a convex cone. This variational problem leads to the development of a spectral theory that combines matrix algebra and facial analysis of convex cones.      

基于递归型神经网络动力学求解时变凸二次规划

为了在线求解时变凸二次规划问题,实现误差精度更高、求解时间更短和收敛速度更快的目标。本文采用了求解问题更快的时变网络设计参数,选择了有限时间可以收敛的Sign-bi-power激活函数,构造了一种改进的归零神经网络动力学模型。其后,分析了模型的稳定性和收敛性,得到其解能够在有限时间内收敛。最后,在仿真算例中,与传统的梯度神经网络和归零神经网络模型相比,所提模型具有更高的误差精度、更短的求解时间和更快的收敛速度,优于前两种网络模型。     

Some properties of generalized proximal point methods for quadratic and linear programming

The proximal point method for convex optimization has been extended recently through the use of generalized distances (e.g., Bregman distances) instead of the Euclidean one. One advantage of these extensions is the possibility of eliminating certain constraints (mainly positivity) from the subproblems, transforming an inequality constrained problem into a sequence of unconstrained or equality constrained problems. We consider here methods obtained using a certain class of Bregman functions applied to convex quadratic (including linear) programming, which are of the interior-point type. We prove that the limit of the sequence generated by the method lies in the relative interior of the solution set, and furthermore is the closest optimal solution to the initial point, in the sense of the Bregman distance. These results do not hold for the standard proximal point method, i.e., when the square of the Euclidean norm is used as the Bregman distance.The research leading to this paper was partially supported by CNPq Grant 301280/86.We thank two anonymous referees whose comments and suggestions allowed us to improve our results in a significant way.     

A linear programming reformulation of the standard quadratic optimization problem

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note, we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LP’s whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.     

A new algorithm for solving the general quadratic programming problem

For the general quadratic programming problem (including an equivalent form of the linear complementarity problem) a new solution method of branch and bound type is proposed. The branching procedure uses a well-known simplicial subdivision and the bound estimation is performed by solving certain linear programs.     

A randomized scheme for speeding up algorithms for linear and convex programming problems with high constraints-to-variables ratio

We extend Clarkson's randomized algorithm for linear programming to a general scheme for solving convex optimization problems. The scheme can be used to speed up existing algorithms on problems which have many more constraints than variables. In particular, we give a randomized algorithm for solving convex quadratic and linear programs, which uses that scheme together with a variant of Karmarkar's interior point method. For problems withn constraints,d variables, and input lengthL, ifn = (d ²), the expected total number of major Karmarkar's iterations is O(d ²(logn)L), compared to the best known deterministic bound of O( L). We also present several other results which follow from the general scheme.     

A randomized algorithm for fixed-dimensional linear programming

We give a (Las Vegas) randomized algorithm for linear programming in a fixed dimensiond for which the expected computation time is , where lim _{d d} = 0. This improves the corresponding worst-case complexity, . The method is based on a recent idea of Clarkson. Two variations on the algorithm are examined briefly.     

A sparse sequential quadratic programming algorithm

Described here is the structure and theory for a sequential quadratic programming algorithm for solving sparse nonlinear optimization problems. Also provided are the details of a computer implementation of the algorithm along with test results. The algorithm maintains a sparse approximation to the Cholesky factor of the Hessian of the Lagrangian. The solution to the quadratic program generated at each step is obtained by solving a dual quadratic program using a projected conjugate gradient algorithm. An updating procedure is employed that does not destroy sparsity.     

On the convergence of the method of analytic centers when applied to convex quadratic programs

This work examines the method of analytic centers of Sonnevend when applied to solve generalized convex quadratic programs — where also the constraints are given by convex quadratic functions. We establish the existence of a two-sided ellipsoidal approximation for the set of feasible points around its center and show, that a simple (zero order) algorithm starting from an initial center of the feasible set generates a sequence of strictly feasible points whose objective function values converge to the optimal value. Concerning the speed of convergence it is shown that an upper bound for the gap in between the objective function value and the optimal value is reduced by a factor of with iterations wherem is the number of inequality constraints. Here, each iteration involves the computation of one Newton step. The bound of Newton iterations to guarantee an error reduction by a factor of in the objective function is as good as the one currently given forlinear programs. However, the algorithm considered here is of theoretical interest only, full efficiency of the method can only be obtained when accelerating it by some (higher order) extrapolation scheme, see e.g. the work of Jarre, Sonnevend and Stoer.This work was supported by the Deutsche Forschungsgemeinschaft, Schwerpunktprogramm für anwendungsbezogene Optimierung und Steuerung.     

Reformulation of mathematical programming problems as linear complementarity problems and investigation of their solution methods

A family of complementarity problems is defined as extensions of the well-known linear complementarity problem (LCP). These are:

Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming

A classical method for solving the variational inequality problem is the projection algorithm. We show that existing convergence results for this algorithm follow from one given by Gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Moreover, we extend the projection algorithm to solveany monotone affine variational inequality problem. When applied to linear complementarity problems, we obtain a matrix splitting algorithm that is simple and, for linear/quadratic programs, massively parallelizable. Unlike existing matrix splitting algorithms, this algorithm converges under no additional assumption on the problem. When applied to generalized linear/quadratic programs, we obtain a decomposition method that, unlike existing decomposition methods, can simultaneously dualize the linear constraints and diagonalize the cost function. This method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets.This research is partially supported by the U.S. Army Research Office, contract DAAL03-86-K-0171 (Center for Intelligent Control Systems), and by the National Science Foundation under grant NSF-ECS-8519058.Thanks are due to Professor J.-S. Pang for his helpful comments.     

Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited

Recently, Zhang, Tapia, and Dennis (Ref. 1) produced a superlinear and quadratic convergence theory for the duality gap sequence in primal-dual interior-point methods for linear programming. In this theory, a basic assumption for superlinear convergence is the convergence of the iteration sequence; and a basic assumption for quadratic convergence is nondegeneracy. Several recent research projects have either used or built on this theory under one or both of the above-mentioned assumptions. In this paper, we remove both assumptions from the Zhang-Tapia-Dennis theory.Dedicated to the Memory of Magnus R. Hestenes, 1906–1991This research was supported in part by NSF Cooperative Agreement CCR-88-09615 and was initiated while the first author was at Rice University as a Visiting Member of the Center for Research in Parallel Computation.The authors thank Yinyu Ye for constructive comments and discussions concerning this material.This author was supported in part by NSF Grant DMS-91-02761 and DOE Grant DE-FG05-91-ER25100.This author was supported in part by AFOSR Grant 89-0363, DOE Grant DE-FG05-86-ER25017, and ARO Grant 9DAAL03-90-G-0093.