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Primal-dual relationship between Levenberg-Marquardt and central trajectories for linearly constrained convex optimization

Roger Behling (rogerbehling***at***gmail.com)
Clovis Gonzaga (clovis***at***mtm.ufsc.br)
Gabriel Haeser (gabriel.haeser***at***unifesp.br)

Abstract: We consider the minimization of a convex function on a compact polyhedron defined by linear equality constraints and nonnegative variables. We define the Levenberg-Marquardt (L-M) and central trajectories starting at the analytic center and using the same parameter, and show that they satisfy a primal-dual relationship, being close to each other for large values of the parameter. Based on this we develop an algorithm that starts computing primal-dual feasible points on the L-M trajectory and eventually moves to the central path. Our main theorem is particularly relevant in quadratic programming, where points on the primal-dual L-M trajectory can be calculated by means of a system of linear equations. We present some computational tests related to box constrained trust region subproblems.

Keywords: central path, Levenberg-Marquardt, primal-dual, interior points, convex quadratic programming, trust region, initial point

Category 1: Nonlinear Optimization (Quadratic Programming )

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )


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Entry Submitted: 09/05/2012
Entry Accepted: 09/05/2012
Entry Last Modified: 03/21/2013

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