NDLI: A Solver for Nonconvex Bound-Constrained Quadratic Optimization

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Robinson, Daniel P. Mohy-ud-Din, Hassan
Copyright Year	2015
Abstract	We present a new algorithm for nonconvex bound-constrained quadratic optimization. In the strictly convex case, our method is equivalent to the state-of-the-art algorithm by Dostál and Schöberl [Comput. Optim. Appl., 30 (2005), pp. 23--43]. Unlike their method, however, we establish a convergence theory for our algorithm that holds even when the problems are nonconvex. This is achieved by carefully addressing the challenges associated with directions of negative curvature, in particular, those that may naturally arise when applying the conjugate gradient algorithm to an indefinite system of equations. Our presentation and analysis deal explicitly with both lower and upper bounds on the optimization variables, whereas the work by Dostál and Schöberl considers only strictly convex problems with lower bounds. To handle this generality, we introduce the reduced chopped gradient that is analogous to the reduced free gradient previously used. The reduced chopped gradient leads to a new condition that is used to determine when optimization over a given subspace should be terminated. This condition, although not equivalent, is motivated by a similar condition used by Dostál and Schöberl. Numerical results illustrate the superior performance of our method over commonly used solvers that employ gradient projection steps and subspace acceleration.
Starting Page	2385
Ending Page	2407
Page Count	23
File Format	PDF
ISSN	10526234
DOI	10.1137/15M1022100
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	4
Volume Number	25
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2015-11-19
Access Restriction	Subscribed
Subject Keyword	nonconvex conjugate gradient quadratic optimization Analysis of algorithms and problem complexity Systems of equations active set Mathematical programming methods Newton-type methods bound constraint Nonlinear programming Implicit function theorems; global Newton methods Abstract computational complexity for mathematical programming problems Methods of nonlinear programming type
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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