Methods for Convex and General Quadratic Programmingв€—

Philip Electronic. GillвЂ At the WongвЂ

UCSD Department of Mathematics Specialized Report NA-10-01 September 2010

Abstract Computational methods are believed for п¬Ѓnding a point that satisп¬Ѓes the secondorder important conditions for the general (possibly nonconvex) quadratic program (QP). The п¬Ѓrst part of the paper deп¬Ѓnes a framework to get the formula and examination of feasible-point active-set techniques for QP. This kind of framework deп¬Ѓnes a class of methods where a primal-dual search pair is a solution of your equality-constrained subproblem involving a " functioning setвЂќ of linearly self-employed constraints. This framework can be discussed inside the context of two broad classes of active-set way for quadratic programming: binding-direction strategies and nonbinding-direction methods. We recast a binding-direction means for general QP п¬Ѓrst proposed by Fletcher, and consequently modiп¬Ѓed by Gould, being 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 gradient. It is shown that, under specific circumstances, the solution of this KKT-system may be current using a basic recurrence regards, thereby giving a signiп¬Ѓcant decrease in the number of KKT systems that need to be solved. Furthermore, the nonbinding-direction framework is applied to QP problems with constraints in common form, and the dual of a convex QP. The second part of the conventional paper focuses on execution issues. First, two strategies are considered pertaining to solving the constituent KKT systems. The п¬Ѓrst way uses a variable-reduction technique needing the computation of the Cholesky factor of the reduced Hessian. The second approach uses a symmetric indeп¬Ѓnite factorization of a п¬Ѓxed KKT matrix in conjunction with the factorization of a smaller matrix that is updated at each iteration. Finally, algorithms to get п¬Ѓnding a preliminary point intended for the method happen to be proposed. Particularly, single-phase strategies involving a linearly limited augmented Lagrangian are proposed that obviate the need for a unique procedure for п¬Ѓnding a feasible point. Key words. Large-scale quadratic programming, active-set methods, convex quadratic development, dual quadratic program, nonconvex quadratic encoding, KKT devices.

в€— Study supported partly by Countrywide Science Basis grants DMS-0511766 and DMS-0915220, and by Office of Energy grant DE-SC0002349. вЂ Department of Mathematics, College or university of California, San Diego, La Jolla, CALIFORNIA 92093-0112 ([email protected] edu, [email protected] edu).

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Convex and General Quadratic Coding

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Introduction

The quadratic programming (QP) problem is to minimize a quadratic objective function subject to thready constraints around the variables. Quadratic programs occur in many areas, including economics, applied scientific research and executive. Important applications of quadratic coding include stock portfolio analysis, support vector machines, structural evaluation and optimal control. Quadratic programming also forms a principal computational component of a large number of sequential quadratic programming (SQP) methods for non-linear programming (for a recent study, see Gill and Wong [16]). Inside the п¬Ѓrst part of the paper (comprising Sections two and 3), we review the optimality conditions for QP and deп¬Ѓnes a framework pertaining to the ingredients and evaluation of feasiblepoint active-set techniques for QP. This framework deп¬Ѓnes a class of methods in which a primal-dual search pair is definitely the solution of an equality-constrained subproblem involving a " operating setвЂќ of linearly 3rd party constraints. This framework can be discussed in the context of two wide-ranging classes of active-set way of quadratic programming: binding-direction methods and nonbinding-direction methods. Generally, the working arranged for a joining direction method consists of a subsection, subdivision, subgroup, subcategory, subclass of the effective constraints, although the working collection for a...

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