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 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.     

A stable method for solving certain constrained least squares problems

This paper presents a feasible descent algorithm for solving certain constrained least squares problems. These problems are specially structured quadratic programming problems with positive semidefinite Hessian matrices that are allowed to be singular. The algorithm generates a finite sequence of subproblems that are solved using the numerically stable technique of orthogonal factorization with reorthogonalization and Given's transformation updating.This material is based upon work supported by the National Science Foundation under Grant No. MCS 78-06716 and by the International Institute for Applied Systems Analysis.     

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.     

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.     

一个关于二次规划问题的分段线性同伦算法

本文发展了一个关于二次规划问题的分段线性同伦算法。该算法可看作是外点罚函数法的一个变体。凡是符合外点罚函数法收敛条件的二次规划问题用该算法均可经有限次轮回运算得到稳定解。大量的关于随机的凸二次规划问题的数值实验结果表明它的计算效率是高的，在某些条件下可能是多项式时间算法。     

A dual differentiable exact penalty function

A new penalty function is associated with an inequality constrained nonlinear programming problem via its dual. This penalty function is globally differentiable if the functions defining the original problem are twice globally differentiable. In addition, the penalty parameter remains finite. This approach reduces the original problem to a simple problem of maximizing a globally differentiable function on the product space of a Euclidean space and the nonnegative orthant of another Euclidean space. Many efficient algorithms exist for solving this problem. For the case of quadratic programming, the penalty function problem can be solved effectively by successive overrelaxation (SOR) methods which can handle huge problems while preserving sparsity features. Sponsored by the United States Army under Contract No. DAAG 29-80-C-0041. This material is based upon work supported by the National Science Foundation under Grants No. MCS-790166 and ENG-7903881.     

On solving a primal geometric program by partial dual optimization

The several published methods for mapping a dual solution estimate to a primal solution estimate in posynomial geometric programming provide no criteria for deciding how much deviation from primal feasibility, or discrepancy between the primal and dual objective function values, should be permitted before the primal solution estimate is accepted by the designer. This paper presents a new and simple dual-to-primal conversion method that uses the cost coefficients to provide a sound economic criterion for determining when to accept a primal solution estimate. The primal solution estimate generated is the exact solution to a modified primal obtained from the given primal by modifying the cost coefficients, with the exponent matrix left unchanged. The method is shown to have desirable properties when coupled with a convergent dual algorithm.     

A convergence proof for an affine-scaling algorithm for convex quadratic programming without nondegeneracy assumptions

This paper presents a theoretical result on convergence of a primal affine-scaling method for convex quadratic programs. It is shown that, as long as the stepsize is less than a threshold value which depends on the input data only, Ye and Tse's interior ellipsoid algorithm for convex quadratic programming is globally convergent without nondegeneracy assumptions. In addition, its local convergence rate is at least linear and the dual iterates have an ergodically convergent property.Research supported in part by the NSF under grant DDM-8721709.     

Penalty for zero–one integer equivalent problem

The equivalence of zero–one integer programming and a concave quadratic penalty function problem has been shown by Raghavachari, for a sufficiently large value of the penalty. A lower bound for this penalty is obtained here, which in specific cases cannot be reduced.This research was supported in part by the Computer Science Section of the National Science Foundation under Research Grant MCS 8101214.     

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.     

Numerically stable methods for quadratic programming

Numerically stable algorithms for quadratic programming are discussed. A new algorithm is described for indefinite quadratic programming which utilizes methods for updating positivedefinite factorizations only. Consequently all the updating procedures required are common to algorithms for linearly-constrained optimization. The new algorithm can be used for the positive-definite case without loss of efficiency.     

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.     

Primal-dual proximal point algorithm for linearly constrained convex programming problems

A primal-dual version of the proximal point algorithm is developed for linearly constrained convex programming problems. The algorithm is an iterative method to find a saddle point of the Lagrangian of the problem. At each iteration of the algorithm, we compute an approximate saddle point of the Lagrangian function augmented by quadratic proximal terms of both primal and dual variables. Specifically, we first minimize the function with respect to the primal variables and then approximately maximize the resulting function of the dual variables. The merit of this approach exists in the fact that the latter function is differentiable and the maximization of this function is subject to no constraints. We discuss convergence properties of the algorithm and report some numerical results for network flow problems with separable quadratic costs.     

A dual perturbation view of linear programming

Solving standard-form linear prograrns via perturbation of the primal objective function has received much attention recently. In this paper, we investigate a new perturbation scheme which obtains a dual optimal solution by perturbing the dual feasible domain under different norms. A dual-to-primal conversion formula is also provided. We show that this new perturbation scheme actually generalizes the primal entropic perturbation approach to linear programming.Partially sponsored by the North Carolina Supercomputing Center 1994 Cray Research Grant and the National Textile Center Research Grant.     

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.     

On a Modified Subgradient Algorithm for Dual Problems via Sharp Augmented Lagrangian*

We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. *Partially Supported by 2003 UniSA ITEE Small Research Grant Ero2. Supported by CAPES, Brazil, Grant No. 0664-02/2, during her visit to the School of Mathematics and Statistics, UniSA.     

Dual Convergence for Penalty Algorithms in Convex Programming

Algorithms for convex programming, based on penalty methods, can be designed to have good primal convergence properties even without uniqueness of optimal solutions. Taking primal convergence for granted, in this paper we investigate the asymptotic behavior of an appropriate dual sequence obtained directly from primal iterates. First, under mild hypotheses, which include the standard Slater condition but neither strict complementarity nor second-order conditions, we show that this dual sequence is bounded and also, each cluster point belongs to the set of Karush–Kuhn–Tucker multipliers. Then we identify a general condition on the behavior of the generated primal objective values that ensures the full convergence of the dual sequence to a specific multiplier. This dual limit depends only on the particular penalty scheme used by the algorithm. Finally, we apply this approach to prove the first general dual convergence result of this kind for penalty-proximal algorithms in a nonlinear setting.     

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.