MPC

MPC 2015, ISSUE 2



Mathematical Programming Computation, Volume 7, Issue 2, June 2015

The strength of multi-row models

Quentin Louveaux, Laurent Poirrier, Domenico Salvagnin

We develop a method for computing facet-defining valid inequalities for any mixed-integer set PJ . While our practical implementation does not return only facet-defining inequalities, it is able to find a separating cut whenever one exists. The separator is not comparable in speed with the specific cutting-plane generators used in branch-and-cut solvers, but it is general-purpose. We can thus use it to compute cuts derived from any reasonably small relaxation PJ of a general mixed-integer problem, even when there exists no specific implementation for computing cuts with PJ . Exploiting this, we evaluate, from a computational perspective, the usefulness of cuts derived from several types of multi-row relaxations. In particular, we presentresults with four different strengthenings of the two-row intersection cut model, and multi-row models with up to fifteen rows. We conclude that only fully-strengthened two-row cuts seem to offer a significant advantage over two-row intersection cuts. Our results also indicate that the improvement obtained by going frommodelswith very few rows to models with up to fifteen rows may not be worth the increased computing cost.

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Mathematical Programming Computation, Volume 7, Issue 2, June 2015

Generalized alternating direction method of multipliers: new theoretical insights and applications

Ethan X. Fang, Bingsheng He, Han Liu, Xiaoming Yuan

Recently, the alternating direction method of multipliers (ADMM) has received intensive attention from a broad spectrum of areas. The generalized ADMM (GADMM) proposed by Eckstein and Bertsekas is an efficient and simple acceleration scheme of ADMM. In this paper, we take a deeper look at the linearized version of GADMM where one of its subproblems is approximated by a linearization strategy. This linearized version is particularly efficient for a number of applications arising from different areas. Theoretically, we show the worstcase O(1/k) convergence rate measured by the iteration complexity (k represents the iteration counter) in both the ergodic and a nonergodic senses for the linearized version of GADMM. Numerically, we demonstrate the efficiency of this linearized version of GADMM by some rather new and core applications in statistical learning. Code packages in Matlab for these applications are also developed.

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Mathematical Programming Computation, Volume 7, Issue 2, June 2015

Solving network design problems via iterative aggregation

Andreas Bärmann, Frauke Liers, Alexander Martin, Maximilian Merkert, Christoph Thurner, Dieter Weninger

In this work, we present an exact approach for solving network design problems that is based on an iterative graph aggregation procedure. The scheme allows existing preinstalled capacities. Starting with an initial aggregation, we solve a sequence of network design master problems over increasingly fine-grained representations of the original network. In each step, a subproblem is solved that either proves optimality of the solution or gives a directive where to refine the representation of the network in the subsequent iteration. The algorithm terminates with a globally optimal solution to the original problem. Our implementation uses a standard integer programming solver for solving the master problems as well as the subproblems. The computational results on random and realistic instances confirm the profitable use of the iterative aggregation technique. The computing time often reduces drastically when our method is compared to solving the original problem from scratch.

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Mathematical Programming Computation, Volume 7, Issue 2, June 2015

On solving a hard quadratic 3-dimensional assignment problem

Hans D. Mittelmann, Domenico Salvagnin

We address the solution of a very challenging (and previously unsolved) instance of the quadratic 3-dimensional assignment problem, arising in digitalwireless communications.The paper describes the techniques developed to solve this instance to optimality, from the choice of an appropriate mixed-integer programming formulation, to cutting planes and symmetry handling. Using these techniques we were able to solve the target instance with moderate computational effort (2.5 million nodes and 1 week of computations on a standard PC).

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