Lagrangian Smoothing Heuristics for Max-Cut

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), Humboldt-Universität zu Berlin (ISSN 0863-0976),

MSC 2000

90C22 Semidefinite programming

90C59 Approximation methods and heuristics

ZDM: N-60

CR: G.2

Abstract
This paper presents smoothing heuristics for an NP-hard combinatorial problem based on Lagrangian relaxation. We formulate the Lagrangian dual for this nonconvex quadratic problem and propose eigenvalue nonsmooth unconstrained optimization to solve the dual problem with bundle or subgradient methods. Derived heuristics are considered to obtain good primal solutions through pathfollowing methods using a projected gradient algorithm. Starting points are drawn using several sampling techniques that use randomization and eigenvectors. The proposed method turns out to be competitive with the most recent ones. The idea presented here is generic and can be generalized, to all problems where convex Lagrangian relaxation can be applied. Furthermore, to the best of our knowledge, this is the first time that a Lagrangian heuristic is combined with pathfollowing techniques.

Keywords: semidefinite programming, quadratic programming, combinatorial optimization, non-convex programming, approximation methods and heuristics, pathfollowing, homotopy