Primal-Dual Relationship Between Levenberg–Marquardt and Central Trajectories for Linearly Constrained Convex Optimization

Roger Behling (), Clovis Gonzaga () and Gabriel Haeser ()
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Roger Behling: Católica SC
Clovis Gonzaga: Federal University of Santa Catarina
Gabriel Haeser: Federal University of São Paulo

Journal of Optimization Theory and Applications, 2014, vol. 162, issue 3, No 2, 705-717

Abstract: Abstract We consider the minimization of a convex function on a bounded polyhedron (polytope) represented by linear equality constraints and non-negative variables. We define the Levenberg–Marquardt and central trajectories starting at the analytic center 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 Levenberg–Marquardt trajectory and eventually moves to the central path. Our main theorem is particularly relevant in quadratic programming, where points on the primal-dual Levenberg–Marquardt 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 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10957-013-0492-4

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