ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems

Abstract

This paper proposes an implementation of a constrained analytic center cutting plane method to solve nonlinear multicommodity flow problems. The new approach exploits the property that the objective of the Lagrangian dual problem has a smooth component with second order derivatives readily available in closed form. The cutting planes issued from the nonsmooth component and the epigraph set of the smooth component form a localization set that is endowed with a self-concordant augmented barrier. Our implementation uses an approximate analytic center associated with that barrier to query the oracle of the nonsmooth component. The paper also proposes an approximation scheme for the original objective. An active set strategy can be applied to the transfomed problem: it reduces the dimension of the dual space and accelerates computations. The new approach solves huge instances with high accuracy. The method is compared to alternative approaches proposed in the literature.

A bundle-type algorithm for routing in telecommunication data networks

“…-Each polyhedral function Π k is approximated by cutting planes, as in Kelley's method [9,2]. In a way, the above method can be viewed as a bundle variant [14], see also [1], in which -the cutting-plane paradigm is applied to a part of the (dual) objective function, namely Π, -stabilization aroundû is obtained by the Newtonian term u ∇ 2 Φ(û)u, instead of an artificial |u| 2 weighted by a hardto-tune penalty coefficient. Finally, since Lagrangian relaxation is column generation, our algorithm can be viewed as a Dantzig-Wolfe variant where the masters are suitably stabilized.…”

“…If such a u is aû, thenΦ j will degenerate and (13) will perhaps be unbounded from above 1 . 1 This difficulty can be eliminated, though.…”

“…Problem (1), often called (convex) multicommodity flow, occurs in data communication networks and plays an important role in the optimization of network performances. Various methods have been proposed in the literature to solve it, we refer to [15] for a review.…”

“…There we find proximal, ACCPM or subgradient methods. Our present proposal belongs to this latter class and solves the dual by a convex-optimization algorithm, comparable to the recent approach of [1]. We fully exploit the structure of the dual objective function, which is a sum of two pieces: the convex conjugate of the delay function f , and a polyhedral function which can be approximated through cutting planes.…”

“…We propose in §2 a (classical) Lagrangian relaxation of (1) and in §3 our model of the Lagrangian dual function using a second-order oracle. Our adaptation of the bundling technique to cope with this special model is the subject of §4, while §5 recalls the aggregation mechanism, important for primal recovery.…”