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Anders Forsgren(andersfkth.se) Abstract: Computational methods are proposed for solving a convex quadratic program (QP). Activeset methods are defined for a particular primal and dual formulation of a QP with general equality constraints and simple lower bounds on the variables. In the first part of the paper, two methods are proposed, one primal and one dual. These methods generate a sequence of iterates that are feasible with respect to the equality constraints associated with the optimality conditions of the primaldual form. The primal method maintains feasibility of the primal inequalities while driving the infeasibilities of the dual inequalities to zero. In contrast, the dual method maintains feasibility of the dual inequalities while moving to satisfy the infeasibilities of the primal inequalities. In each of these methods, the search directions satisfy a KKT system of equations formed from Hessian and constraint components associated with an appropriate column basis. The composition of the basis is specified by an activeset strategy that guarantees the nonsingularity of each set of KKT equations. Each of the proposed methods is a conventional activeset method in the sense that an initial primal or dualfeasible point is required. In the second part of the paper, it is shown how the quadratic program may be solved as coupled pair of primal and dual quadratic programs created from the original by simultaneously shifting the simplebound constraints and adding a penalty term to the objective function. Any conventional column basis may be made optimal for such a primaldual pair of shiftedpenalized problems. The shifts are then updated using the solution of either the primal or the dual shifted problem. An obvious application of this approach is to solve a shifted dual QP to define an initial feasible point for the primal (or vice versa). The computational performance of each of the proposed methods is evaluated on a set of convex problems from the CUTEst test collection. Keywords: Quadratic programming, convex quadratic programming, activeset methods, primal activeset methods, dual activeset methods, primaldual methods, KKT systems Category 1: Nonlinear Optimization (Quadratic Programming ) Category 2: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: UCSD Center for Computational Mathematics Technical Report CCoM1502, March 2015 Download: [PDF] Entry Submitted: 03/28/2015 Modify/Update this entry  
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