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

Author(s)

Behling, Roger ^[1] ; Gonzaga, Clovis ^[2] ; Haeser, Gabriel ^[3]

Affiliation

^[1] Catolica SC, Jaragua Do Sul, SC - Brazil

^[2] Univ Fed Santa Catarina, Florianopolis, SC - Brazil

^[3] Univ Fed Sao Paulo, Inst Sci & Technol, Sao Jose Dos Campos, SP - Brazil

• Document type: Journal article
• Source: JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS; v. 162, n. 3, p. 705-717, SEP 2014.
• Web of Science Citations: 2

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. (AU)

• FAPESP's process: 10/19720-5 - Optimality conditions and inexact restoration
• Grantee: Gabriel Haeser
• Support Opportunities: Research Grants - Young Investigators Grants