Qwertyui
Methods for Convex and General Quadratic Programming∗
Philip E. Gill† Elizabeth Wong†
UCSD Department of Mathematics Technical Report NA-10-01 September 2010
Abstract Computational methods are considered for finding a point that satisfies the secondorder necessary conditions for a general (possibly nonconvex) quadratic program (QP). The first part of the paper defines a framework for the formulation and analysis of feasible-point active-set methods for QP. This framework defines a class of methods in which a primal-dual search pair is the solution of an equality-constrained subproblem involving a “working set” of linearly independent constraints. This framework is discussed in the context of two broad classes of active-set method for quadratic programming: binding-direction methods and nonbinding-direction methods. We recast a binding-direction method for general QP first proposed by Fletcher, and subsequently modified by Gould, as a nonbinding-direction method. This reformulation gives the primal-dual search pair as the solution of a KKT-system formed from the QP Hessian and the working-set constraint gradients. It is shown that, under certain circumstances, the solution of this KKT-system may be updated using a simple recurrence relation, thereby giving a significant reduction in the number of KKT systems that need to be solved. Furthermore, the nonbinding-direction framework is applied to QP problems with constraints in standard form, and to the dual of a convex QP. The second part of the paper focuses on implementation issues. First, two approaches are considered for solving the constituent KKT systems. The first approach uses a variable-reduction technique requiring the calculation of the Cholesky factor of the reduced Hessian. The second approach uses a symmetric indefinite factorization of a fixed KKT matrix in conjunction with the factorization of a smaller matrix that is updated at each iteration. Finally, algorithms for finding an initial point for the method
Philip E. Gill† Elizabeth Wong†
UCSD Department of Mathematics Technical Report NA-10-01 September 2010
Abstract Computational methods are considered for finding a point that satisfies the secondorder necessary conditions for a general (possibly nonconvex) quadratic program (QP). The first part of the paper defines a framework for the formulation and analysis of feasible-point active-set methods for QP. This framework defines a class of methods in which a primal-dual search pair is the solution of an equality-constrained subproblem involving a “working set” of linearly independent constraints. This framework is discussed in the context of two broad classes of active-set method for quadratic programming: binding-direction methods and nonbinding-direction methods. We recast a binding-direction method for general QP first proposed by Fletcher, and subsequently modified by Gould, as a nonbinding-direction method. This reformulation gives the primal-dual search pair as the solution of a KKT-system formed from the QP Hessian and the working-set constraint gradients. It is shown that, under certain circumstances, the solution of this KKT-system may be updated using a simple recurrence relation, thereby giving a significant reduction in the number of KKT systems that need to be solved. Furthermore, the nonbinding-direction framework is applied to QP problems with constraints in standard form, and to the dual of a convex QP. The second part of the paper focuses on implementation issues. First, two approaches are considered for solving the constituent KKT systems. The first approach uses a variable-reduction technique requiring the calculation of the Cholesky factor of the reduced Hessian. The second approach uses a symmetric indefinite factorization of a fixed KKT matrix in conjunction with the factorization of a smaller matrix that is updated at each iteration. Finally, algorithms for finding an initial point for the method
References: [18] N. I. M. Gould. On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem. Math. Program., 32:90–99, 1985. [19] H. M. Huynh. A Large-Scale Quadratic Programming Solver Based on Block-LU Updates of the KKT System. PhD thesis, Program in Scientific Computing and Computational Mathematics, Stanford University, Stanford, CA, 2008. [20] A. Majthay. Optimality conditions for quadratic programming. Math. Programming, 1:359–365, 1971. [21] J. Nocedal and S. J. Wright. Numerical Optimization. Springer-Verlag, New York, 1999. [22] P. M. Pardalos and G. Schnitger. Checking local optimality in constrained quadratic programming is NP-hard. Oper. Res. Lett., 7(1):33–35, 1988. [23] P. M. Pardalos and S. A. Vavasis. Quadratic programming with one negative eigenvalue is NP-hard. J. Global Optim., 1(1):15–22, 1991. [24] M. J. D. Powell. On the quadratic programming algorithm of Goldfarb and Idnani. Math. Programming Stud., (25):46–61, 1985. [25] J. A. Tomlin. On pricing and backward transformation in linear programming. Math. Programming, 6:42–47, 1974.