New approaches for optimizing over the semimetric polytope

Abstract

The semimetric polytope is an important polyhedral structure lying at the heart of hard combinatorial problems. Therefore, linear optimization over the semimetric polytope is crucial for a number of relevant applications. Building on some recent polyhedral and algorithmic results about a related polyhedron, the rooted semimetric polytope, we develop and test several approaches, mainly based over Lagrangian relaxation and application of Non Differentiable Optimization algorithms, for linear optimization over the semimetric polytope. We show that some of these approaches can obtain very accurate primal and dual solutions in a small fraction of the time required for the same task by state-of-the-art general purpose linear programming technology. In some cases, good estimates of the dual optimal solution (but not of the primal solution) can be obtained even quicker.

A stabilized structured Dantzig–Wolfe decomposition method

“…A weaker version (that is sufficient in practice) showing that v(29) → 0 even if infinitely many aggregations are performed is also possible: one could then resort to employing the "poorman cutting-plane model" fx (cf. §4.1) at all steps, which basically makes the algorithm a subgradient-type approach with deflection [4,13] and results in much slower convergence [10,23]. Thus, this kind of development seems to be of little interest in our case.…”

“…Implementing them may require little more than access to a general-purpose LP solver, such as in the case of 5 where the dynamic generation of rows can be handled by the standard callback routines that are provided for the purpose. Other cases, such as 2, can be solved by general-purpose bundle codes such as that already used with success in several other applications [10,22,23]. Yet, other cases, such as 3 and 4, require development of entirely ad-hoc approaches.…”

Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

“…When in the above cutting-plane algorithm a simplex or a barrier algorithm is used to solve each LP, computation times can really blow up: for example, [20] reports computation times of more than one hour for complete graphs of 150 nodes just to solve the LP relaxation. A different way to compute the relaxation is to keep, as explicit constraints, only a small subset of the inequalities (5) and to dualize all the others that would be used by the cutting-plane algorithm.…”

“…Yet another approach is proposed in [20]. It is assumed that the graph has a node r adjacent to all other nodes.…”

“…The Lagrangian dual is then solved with a bundle method pretty much in the same way as described in Section 4.1. In [20] time savings of up to two orders of magnitude with respect to simplex or barrier based cutting-plane algorithms are reported for graphs of up to 150 nodes. The primal and dual infeasibilities of the solutions are comparable with those obtained with the simplex or the barrier algorithm.…”