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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*…show more content…*

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. The linear constraints of a QP may include an arbitrary mixture of equality and inequality constraints, where the inequality constraints may be subject to lower and/or upper bounds. Many mathematically equivalent formulations of the constraints are possible, and the choice of formulation often depends on the context. We consider the generic quadratic program minimize n x∈R 1 ϕ(x) = cTx + 2 xTHx subject to Ax = b, Dx ≥ f, (1.1) where A, b, c, D, f and H are constant, H is symmetric, A is m × n, and D is mD × n. (In order to simplify the notation, it is assumed that the inequalities involve only lower bounds.) However, the methods to be described can be generalized to treat all forms of linear constraints. No assumptions are made about H (other than symmetry), which implies that the objective ϕ(x) need not be convex. In the nonconvex case, however, convergence will be to local minimizers only. In Section 4, the nonbinding direction method is extended to problems

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. The linear constraints of a QP may include an arbitrary mixture of equality and inequality constraints, where the inequality constraints may be subject to lower and/or upper bounds. Many mathematically equivalent formulations of the constraints are possible, and the choice of formulation often depends on the context. We consider the generic quadratic program minimize n x∈R 1 ϕ(x) = cTx + 2 xTHx subject to Ax = b, Dx ≥ f, (1.1) where A, b, c, D, f and H are constant, H is symmetric, A is m × n, and D is mD × n. (In order to simplify the notation, it is assumed that the inequalities involve only lower bounds.) However, the methods to be described can be generalized to treat all forms of linear constraints. No assumptions are made about H (other than symmetry), which implies that the objective ϕ(x) need not be convex. In the nonconvex case, however, convergence will be to local minimizers only. In Section 4, the nonbinding direction method is extended to problems

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