A numerically stable dual method for solving strictly convex quadratic programs

An efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point. Its implementation utilizes the Cholesky and QR factorizations and procedures for updating them. The performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code (the principal motivation for the development of the algorithm). These computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available. The algorithm is also compared theoretically to the modified-simplex type dual methods of Lemke and Van de Panne and Whinston and it is illustrated by a numerical example. This research was supported in part by the Army Research Office under Grant No. DAAG 29-77-G-0114 and in part by the National Science Foundation under Grant No. MCS-6006065.     

A weighted gram-schmidt method for convex quadratic programming

Range-space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases. In this paper we describe a range-space method based upon updating a weighted Gram-Schmidt factorization of the constraints in the active set. The updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active-set strategy. Many quadratic programming problems include simple bounds on all the variables as well as general linear constraints. A feature of the proposed method is that it is able to exploit the structure of simple bound constraints. This allows the method to retain efficiency when the number ofgeneral constraints active at the solution is small. Furthermore, the efficiency of the method improves as the number of active bound constraints increases. This research was supported by the U.S. Department of Energy Contract DE-AC03-76SF00326, PA No. DE-AT03-76ER72018; National Science Foundation Grants MCS-7926009 and ECS-8012974; the Office of Naval Research Contract N00014-75-C-0267; and the U.S. Army Research Office Contract DAAG29-79-C-0110. The work of Nicholas Gould was supported by the Science and Engineering Research Council of Great Britain.     

Mixed-integer quadratic programming

This paper considers mixed-integer quadratic programs in which the objective function is quadratic in the integer and in the continuous variables, and the constraints are linear in the variables of both types. The generalized Benders' decomposition is a suitable approach for solving such programs. However, the program does not become more tractable if this method is used, since Benders' cuts are quadratic in the integer variables. A new equivalent formulation that renders the program tractable is developed, under which the dual objective function is linear in the integer variables and the dual constraint set is independent of these variables. Benders' cuts that are derived from the new formulation are linear in the integer variables, and the original problem is decomposed into a series of integer linear master problems and standard quadratic subproblems. The new formulation does not introduce new primary variables or new constraints into the computational steps of the decomposition algorithm.The author wishes to thank two anonymous referees for their helpful comments and suggestions for revising the paper.     

A decomposition algorithm for quadratic programming

A decomposition algorithm using Lemke's method is proposed for the solution of quadratic programming problems having possibly unbounded feasible regions. The feasible region for each master program is a generalized simplex of minimal size. This property is maintained by a dropping procedure which does not affect the finiteness of the convergence. The details of the matrix transformations associated with an efficient implementation of the algorithm are given. Encouraging preliminary computational experience is presented.     

Equivalence of some quadratic programming algorithms

We formulate a general algorithm for the solution of a convex (but not strictly convex) quadratic programming problem. Conditions are given under which the iterates of the algorithm are uniquely determined. The quadratic programming algorithms of Fletcher, Gill and Murray, Best and Ritter, and van de Panne and Whinston/Dantzig are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points. The various methods are shown to differ only in the manner in which they solve the linear equations expressing the Kuhn-Tucker system for the associated equality constrained subproblems. Equivalence results have been established by Goldfarb and Djang for the positive definite Hessian case. Our analysis extends these results to the positive semi-definite case. This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A8189.     

A unified approach to one-parametric general quadratic programming

A method is proposed for finding local minima to the parametric general quadratic programming problem where all the coefficients are linear or polynomial functions of a scalar parameter. The local minimum vector and the local minimum value are determined explicitly as rational functions of the parameter. A numerical example is given.     

Newton-KKT interior-point methods for indefinite quadratic programming

Two interior-point algorithms are proposed and analyzed, for the (local) solution of (possibly) indefinite quadratic programming problems. They are of the Newton-KKT variety in that (much like in the case of primal-dual algorithms for linear programming) search directions for the “primal” variables and the Karush-Kuhn-Tucker (KKT) multiplier estimates are components of the Newton (or quasi-Newton) direction for the solution of the equalities in the first-order KKT conditions of optimality or a perturbed version of these conditions. Our algorithms are adapted from previously proposed algorithms for convex quadratic programming and general nonlinear programming. First, inspired by recent work by P. Tseng based on a “primal” affine-scaling algorithm (à la Dikin) [J. of Global Optimization, 30 (2004), no. 2, 285–300], we consider a simple Newton-KKT affine-scaling algorithm. Then, a “barrier” version of the same algorithm is considered, which reduces to the affine-scaling version when the barrier parameter is set to zero at every iteration, rather than to the prescribed value. Global and local quadratic convergence are proved under nondegeneracy assumptions for both algorithms. Numerical results on randomly generated problems suggest that the proposed algorithms may be of great practical interest. The work of the first author was supported in part by the School of Computational Science of Florida State University through a postdoctoral fellowship. Part of this work was done while this author was a Research Fellow with the Belgian National Fund for Scientific Research (Aspirant du F.N.R.S.) at the University of Liège. The work of the second author was supported in part by the National Science Foundation under Grants DMI9813057 and DMI-0422931 and by the US Department of Energy under Grant DEFG0204ER25655. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation or those of the US Department of Energy.     

On solving convex quadratic semi-infinite programming probelms

In this paper we study a from of convex quadratic semi-infinite programming problems with finitely many variables and infinitely many constraints over a compact metric space. An entropic path-following algorithum is introduced with a convergence proof. Some practical implementations and numerical experiments are also included     

An equivalence between two algorithms for quadratic programming

In this paper, we demonstrate that the Van de Panne—Whinston symmetric simplex method when applied to a certain implicit formulation of a quadratic program generates the same sequence of primal feasible vectors as does the Von Hohenbalken simplicial decomposition algorithm specialized to the same program. Such an equivalence of the two algorithms extends earlier results for a least-distance program due to Cottle—Djang.This research was prepared as part of the activities of the Management Sciences Research Group, Carnegie-Mellon University, under Contract N00014-75-C-0621 NR 047-048 with the Office of Naval Research.     

SEQUENTIAL QUADRATIC PROGRAMMING METHODS FOR OPTIMAL CONTROL PROBLEMS WITH STATE CONSTRAINTS

A Kind of direct methods is presented for the solution of optimal control problems with state constraints.These methods are sequential quadratic programming methods.At every iteration a quadratic programming which is obtained by quadratic approximation to Lagrangian function and Linear approximations to constraints is solved to get a search direction for a merit function.The merit function is formulated by augmenting the Lagrangian funetion with a penalty term.A line search is carried out along the search direction to determine a step length such that the merit function is decreased.The methods presented in this paper include continuous sequential quadratic programming methods and discreate sequential quadrade programming methods.     

Global minimization of large-scale constrained concave quadratic problems by separable programming

The global minimization of a large-scale linearly constrained concave quadratic problem is considered. The concave quadratic part of the objective function is given in terms of the nonlinear variablesx R ⁿ , while the linear part is in terms ofy R ^k. For large-scale problems we may havek much larger thann. The original problem is reduced to an equivalent separable problem by solving a multiple-cost-row linear program with 2n cost rows. The solution of one additional linear program gives an incumbent vertex which is a candidate for the global minimum, and also gives a bound on the relative error in the function value of this incumbent. Ana priori bound on this relative error is obtained, which is shown to be 0.25, in important cases. If the incumbent is not a satisfactory approximation to the global minimum, a guaranteed-approximate solution is obtained by solving a single liner zero–one mixed integer programming problem. This integer problem is formulated by a simple piecewise-linear underestimation of the separable problem.This research was supported by the Division of Computer Research, National Science Foundation under Research Grant DCR8405489.Dedicated to Professor George Dantzig in honor of his 70th Birthday.     

An unconstrained convex programming approach to solving convex quadratic programming problems

In this paper, we derive an unconstrained convex programming approach to solving convex quadratic programming problems in standard form. Related duality theory is established by using two simple inequalities. An ?-optimal solution is obtained by solving an unconstrained dual convex program. A dual-to-primal conversion formula is also provided. Some preliminary computational results of using a curved search method is included     

Remarks on a difficult test problem for quadratic boolean programming

We show that a “difficult” example is only difficult for special kinds of algorithms     

Least-index resolution of degeneracy in quadratic programming

In this study, we combine least-index pivot selection rules with Keller's algorithm for quadratic programming to obtain a finite method for processing degenerate problems.Research and reproduction of this report were partially supported by National Science Foundation Grant MCS76-81259; and the Office of Naval Research Contract N00014-75-C-0267.     

Some generalizations of the criss-cross method for quadratic programming

Three generalizations of the criss-cross method for quadratic programming are presented here. Tucker’s, Cottle’s and Dantzig’s principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program

A finite criss-cross method, based on least-index resolution, is constructed for solving the LCP. In proving finiteness, orthogonality properties of pivot tableaus and positive semidefiniteness of quadratic matrices are used

In the last section some special cases and two further variants of the quadratic criss-cross method are discussed. If the matrix of the LCP has full rank, then a surprisingly simple algorithm follows, which coincides with Murty’s ‘Bard type schema’ in the P matrix case     

A polynomially bounded algorithm for a singly constrained quadratic program

This paper presents a characterization of the solutions of a singly constrained quadratic program. This characterization is then used in the development of a polynomially bounded algorithm for this class of problems.     

One way to solve the parametric quadratic programming problem

A method is presented for the solution of the parametric quadratic programming problem by the use of conjugate directions. It is based on the method for quadratic programming proposed by the author in [1].While engaged in this research the author had a part-time post with the Manpower Services Commission.     

An alternative method for a global analysis of quadratic programs in a finite number of steps

This paper considers the global analysis of general quadratic programs in a finite number of steps. A procedure is presented for recursively finding either the global minimum or a halfline of the constraint set along which the minimand is unbounded below.Research was partially supported by the U.S. Energy Research and Development Administration Contract EY-76-S-03-0326 PA #18; the Office of Naval Research Contracts N00014-75-C-0267 and N00014-75-C-0865; and the National Science Foundation Grants MCS76-20019 and MCS76-81259.     

A quadratic programming algorithm using conjugate search directions

A quadratic programming algorithm is presented, resembling Beale's 1955 quadratic programming algorithm and Wolfe's Reduced Gradient method. It uses conjugate search directions. The algorithm is conceived as being particularly appropriate for problems with a large Hessian matrix. An experimental computer program has been written to validate the concepts, and has performed adequately, although it has not been used on very large problems. An outline of the solution to the quadratic capacity-constrained transportation problem using the above method is also presented.While engaged in this research the author had a part-time post with the Manpower Services Commission.     

A clique algorithm for standard quadratic programming

A standard Quadratic Programming problem (StQP) consists in minimizing a (nonconvex) quadratic form over the standard simplex. For solving a StQP we present an exact and a heuristic algorithm, that are based on new theoretical results for quadratic and convex optimization problems. With these results a StQP is reduced to a constrained nonlinear minimum weight clique problem in an associated graph. Such a clique problem, which does not seem to have been studied before, is then solved with an exact and a heuristic algorithm. Some computational experience shows that our algorithms are able to solve StQP problems of at least one order of magnitude larger than those reported in the literature.