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Interior-Point Methods for Nonconvex Nonlinear Programming: Primal-Dual Methods and Cubic Regularization

Hande Y. Benson(benson***at***drexel.edu)
David F. Shanno(shannod***at***comcast.net)

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 (1981), Nesterov and Polyak (2006), and Cartis et.al. (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. This paper extends our previous work (Benson, Shanno, 2012) from purely primal barrier methods to the primal-dual interior-point method framework.

Keywords: interior-point methods, nonlinear programming, cubic regularization, Newton's method

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )


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Entry Submitted: 05/08/2012
Entry Accepted: 05/08/2012
Entry Last Modified: 05/08/2012

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