A sequential quadratically constrained quadratic programming method for unconstrained minimax problems

In this paper, a sequential quadratically constrained quadratic programming (SQCQP) method for unconstrained minimax problems is presented. At each iteration the SQCQP method solves a subproblem that involves convex quadratic inequality constraints and a convex quadratic objective function. The global convergence of the method is obtained under much weaker conditions without any constraint qualification. Under reasonable assumptions, we prove the strong convergence, superlinearly and quadratic convergence rate.     

A parameterized hessian quadratic programming problem

We present a general active set algorithm for the solution of a convex quadratic programming problem having a parametrized Hessian matrix. The parametric Hessian matrix is a positive semidefinite Hessian matrix plus a real parameter multiplying a symmetric matrix of rank one or two. The algorithm solves the problem for all parameter values in the open interval upon which the parametric Hessian is positive semidefinite. The algorithm is general in that any of several existing quadratic programming algorithms can be extended in a straightforward manner for the solution of the parametric Hessian problem.This research was supported by the Natural Sciences and Engineering Research Council under Grant No. A8189 and under a Postgraduate Scholarship, by an Ontario Graduate Scholarship, and by the University of Windsor Research Board under Grant No. 9432.     

A parameterized hessian quadratic programming problem

We present a general active set algorithm for the solution of a convex quadratic programming problem having a parametrized Hessian matrix. The parametric Hessian matrix is a positive semidefinite Hessian matrix plus a real parameter multiplying a symmetric matrix of rank one or two. The algorithm solves the problem for all parameter values in the open interval upon which the parametric Hessian is positive semidefinite. The algorithm is general in that any of several existing quadratic programming algorithms can be extended in a straightforward manner for the solution of the parametric Hessian problem. This research was supported by the Natural Sciences and Engineering Research Council under Grant No. A8189 and under a Postgraduate Scholarship, by an Ontario Graduate Scholarship, and by the University of Windsor Research Board under Grant No. 9432.     

A two-step matrix splitting method for the mixed linear complementarity problem

In this paper, on the base of the methodology of the new modulus-based matrix splitting method in [Optim. Lett., (2022) 16:1427-1443], we establish a two-step matrix splitting (TMS) method for solving the mixed linear complementarity problem (MLCP). Two sufficient conditions to ensure the convergence of the proposed method are presented. Numerical examples are provided to illustrate the feasibility and efficiency of the proposed method.     

A quadratic programming approach to the multi-product newsvendor problem with side constraints

A quadratic programming approach is proposed for solving the newsvendor problem with side constraints. Among its salient features are the facts that it: utilizes familiar packages to solve the problem such as Excel Solver and Lingo, can accommodate lower bounds of product’s demands that are larger than zero, and facilitates the performance of sensitivity analysis tasks.     

Bounds for the quadratic assignment problem using the bundle method

Semidefinite programming (SDP) has recently turned out to be a very powerful tool for approximating some NP-hard problems. The nature of the quadratic assignment problem (QAP) suggests SDP as a way to derive tractable relaxations. We recall some SDP relaxations of QAP and solve them approximately using a dynamic version of the bundle method. The computational results demonstrate the efficiency of the approach. Our bounds are currently among the strongest ones available for QAP. We investigate their potential for branch and bound settings by looking also at the bounds in the first levels of the branching tree.      

Relaxed inertial proximal Peaceman-Rachford splitting method for separable convex programming

The strictly contractive Peaceman-Rachford splitting method is one of effective methods for solving separable convex optimization problem, and the inertial proximal Peaceman-Rachford splitting method is one of its important variants. It is known that the convergence of the inertial proximal Peaceman-Rachford splitting method can be ensured if the relaxation factor in Lagrangian multiplier updates is underdetermined, which means that the steps for the Lagrangian multiplier updates are shrunk conservatively. Although small steps play an important role in ensuring convergence, they should be strongly avoided in practice. In this article, we propose a relaxed inertial proximal Peaceman-Rachford splitting method, which has a larger feasible set for the relaxation factor. Thus, our method provides the possibility to admit larger steps in the Lagrangian multiplier updates. We establish the global convergence of the proposed algorithm under the same conditions as the inertial proximal Peaceman-Rachford splitting method. Numerical experimental results on a sparse signal recovery problem in compressive sensing and a total variation based image denoising problem demonstrate the effectiveness of our method.     

A primal interior point method for the linear semidefinite programming problem

The linear semidefinite programming problem is examined. A primal interior point method is proposed to solve this problem. It extends the barrier-projection method used for linear programs. The basic properties of the proposed method are discussed, and its local convergence is proved.     

A column generation approach for the unconstrained binary quadratic programming problem

This paper proposes a column generation approach based on the Lagrangean relaxation with clusters to solve the unconstrained binary quadratic programming problem that consists of maximizing a quadratic objective function by the choice of suitable values for binary decision variables. The proposed method treats a mixed binary linear model for the quadratic problem with constraints represented by a graph. This graph is partitioned in clusters of vertices forming sub-problems whose solutions use the dual variables obtained by a coordinator problem. The column generation process presents alternative ways to find upper and lower bounds for the quadratic problem. Computational experiments were performed using hard instances and the proposed method was compared against other methods presenting improved results for most of these instances.     

A numerically stable least squares solution to the quadratic programming problem

The strictly convex quadratic programming problem is transformed to the least distance problem — finding the solution of minimum norm to the system of linear inequalities. This problem is equivalent to the linear least squares problem on the positive orthant. It is solved using orthogonal transformations, which are memorized as products. Like in the revised simplex method, an auxiliary matrix is used for computations. Compared to the modified-simplex type methods, the presented dual algorithm QPLS requires less storage and solves ill-conditioned problems more precisely. The algorithm is illustrated by some difficult problems.      

一类二次规划逆问题的交替方向数值方法

考虑求解一类二次规划逆问题的交替方向数值算法．首先给出矩阵变量子问题解的显示表达式,而后构造了两个求解向量变量子问题近似解的数值算法,其中一个算法基于不动点原理,另一算法则应用半光滑牛顿法．数值实验表明,所提出的算法能够快速高效地求解二次规划逆问题．     

On the convergence properties of Hildreth's quadratic programming algorithm

We prove the linear convergence rate of Hildreth's method for quadratic programming, in both its sequential and simulateneous versions. We give bounds on the asymptotic error constant and compare these bounds to those given by Mandel for the cyclic relaxation method for solving linear inequalities.Research of this author was partially supported by CNPq grant No. 301280/86.On leave from the Universidade Federal do Rio de Janeiro, Instituto de Matemática, Rio de Janeiro, R.J. 21.910, Brazil. Research of this author was partially supported by NIH grant HL28438.     

A dual-active-set algorithm for positive semi-definite quadratic programming

Because of the many important applications of quadratic programming, fast and efficient methods for solving quadratic programming problems are valued. Goldfarb and Idnani (1983) describe one such method. Well known to be efficient and numerically stable, the Goldfarb and Idnani method suffers only from the restriction that in its original form it cannot be applied to problems which are positive semi-definite rather than positive definite. In this paper, we present a generalization of the Goldfarb and Idnani method to the positive semi-definite case and prove finite termination of the generalized algorithm. In our generalization, we preserve the spirit of the Goldfarb and Idnani method, and extend their numerically stable implementation in a natural way. Supported in part by ATERB, NSERC and the ARC. Much of this work was done in the Department of Mathematics at the University of Western Australia and in the Department of Combinatorics and Optimization at the University of Waterloo.     

A numerically efficient procedure for the Theil-Van de Panne quadratic programming method

A procedure is given for implementing the Theil-Van de Panne algorithm, which utilizes the Cholesky decomposition to reduce the computations involved in matrix inversions.     

A second-order method for the general nonlinear programming problem

This paper presents a multiplier-type method for nonlinear programming problems with both equality and inequality constraints. Slack variables are used for the inequalities. The penalty coefficient is adjusted automatically, and the method converges quadratically to points satisfying second-order conditions.The work of the first author was supported by NSF RANN and JSEP Contract No. F44620-71-C-0087; the work of the second author was supported by the National Science Foundation Grant No. ENG73-08214A01 and US Army Research Office Durham Contract No. DAHC04-73-C-0025.     

A quadratic programming algorithm

By using conjugate directions a method for solving convex quadratic programming problems is developed. The algorithm generates a sequence of feasible solutions and terminates after a finite number of iterations. Extensions of the algorithm for nonconvex and large structured quadratic programming problems are discussed.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 and in part by the Natural Sciences and Engineering Research Council of Canada under Grant Nos. A 8189 and E 5582.     

Parallel orthogonal factorization null-space method for dynamic quadratic programming

An algorithm has been developed to solve quadratic programs that have a dynamic programming structure. It has been developed for use as part of a parallel trajectory optimization algorithm and aims to achieve significant speed without sacrificing numerical stability. the algorithm makes use of the dynamic programming problem structure and the domain decomposition approach. It parallelizes the orthogonal factorization null-space method of quadratic programming by developing a parallel orthogonal factorization and a parallel Cholesky factorization. Tests of the algorithm on a 32-node INTEL iPSC/2 hypercube demonstrate speedup factors as large as 10 in comparison to the fastest known equivalent serial algorithm.This research was supported in part by the National Aeronautics and Space Administration under Grant No. NAG-1-1009.     

Optimal objective function approximation for separable convex quadratic programming

We present an optimal piecewise-linear approximation method for the objective function of separable convex quadratic programs. The method provides guidelines on how many grid points to use and how to position them for a piecewise-linear approximation if the error induced by the approximation is to be bounded a priori.Corresponding author.     

Partial Lagrangian relaxation for general quadratic programming

We give a complete characterization of constant quadratic functions over an affine variety. This result is used to convexify the objective function of a general quadratic programming problem (Pb) which contains linear equality constraints. Thanks to this convexification, we show that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear constraints are not relaxed. We apply these results by comparing two semidefinite relaxations made from two sets of null quadratic functions over an affine variety.      

Quadratic convex reformulations for quadratic 0–1 programming

This is a summary of the author’s PhD thesis supervised by A. Billionnet and S. Elloumi and defended on November 2006 at the CNAM, Paris (Conservatoire National des Arts et Métiers). The thesis is written in French and is available from http://www.cedric.cnam.fr/PUBLIS/RC1115. This work deals with exact solution methods based on reformulations for quadratic 0–1 programs under linear constraints. These problems are generally not convex; more precisely, the associated continuous relaxation is not a convex problem. We developed approaches with the aim of making the initial problem convex and of obtaining a good lower bound by continuous relaxation. The main contribution is a general method (called QCR) that we implemented and applied to classical combinatorial optimization problems.