Interior-point methods for nonconvex nonlinear programming: cubic regularization

Interior-point methods for nonconvex nonlinear programming: cubic regularization

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Article ID: iaor2014699
Volume: 58
Issue: 2
Start Page Number: 323
End Page Number: 346
Publication Date: Jun 2014
Journal: Computational Optimization and Applications
Authors: ,
Keywords: programming: nonlinear, heuristics, matrices
Abstract:

In this paper, we present a primal‐dual interior‐point method for solving nonlinear programming problems. It employs a Levenberg‐Marquardt (LM) perturbation to the Karush‐Kuhn‐Tucker (KKT) matrix to handle indefinite Hessians and a line search to obtain sufficient descent at each iteration. We show that the LM perturbation is equivalent to replacing the Newton step by a cubic regularization step with an appropriately chosen regularization parameter. This equivalence allows us to use the favorable theoretical results of Griewank (The modification of Newton’s method for unconstrained optimization by bounding cubic terms, 1981), Nesterov and Polyak (Math. Program., Ser. A 108:177–205, 2006), Cartis et al. (Math. Program., Ser. A 127:245–295, 2011; Math. Program., Ser. A 130:295–319, 2011), but its application at every iteration of the algorithm, as proposed by these papers, is computationally expensive. We propose a hybrid method: use a Newton direction with a line search on iterations with positive definite Hessians and a cubic step, found using a sufficiently large LM perturbation to guarantee a steplength of 1, otherwise. Numerical results are provided on a large library of problems to illustrate the robustness and efficiency of the proposed approach on both unconstrained and constrained problems.

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