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A Limited-Memory Quasi-Newton Algorithm for Bound-Constrained Nonsmooth Optimization

Nitish Shirish Keskar(keskar.nitish***at***u.northwestern.edu)
Andreas Waechter(waechter***at***iems.northwestern.edu)

Abstract: We consider the problem of minimizing a continuous function that may be nonsmooth and nonconvex, subject to bound constraints. We propose an algorithm that uses the L-BFGS quasi-Newton approximation of the problem's curvature together with a variant of the weak Wolfe line search. The key ingredient of the method is an active-set selection strategy that defines the subspace in which search directions are computed. To overcome the inherent shortsightedness of the gradient for a nonsmooth function, we propose two strategies. The first relies on an approximation of the $\epsilon$-minimum norm subgradient, and the second uses an iterative corrective loop that augments the active set based on the resulting search directions. We describe a Python implementation of the proposed algorithm and present numerical results on a set of standard test problems to illustrate the efficacy of our approach.

Keywords: nonsmooth optimization; bound constraints; quasi-Newton; L-BFGS; active-set method; active-set correction

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Nonlinear Optimization (Bound-constrained Optimization )


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Entry Submitted: 12/21/2016
Entry Accepted: 12/21/2016
Entry Last Modified: 12/21/2016

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