We report a computational study of two-stage SP models on a large set of benchmark problems and consider the following methods: (i) Solution of the deter- ministic equivalent problem by the simplex method and an interior point method, (ii) Benders decomposition (L-shaped method with aggregated cuts), (iii) Regular- ised decomposition of Ruszczyn ?ski (Math Program 35:309–333, 1986), (iv) Bend- ers decomposition with regularization of the expected recourse by the level method (Lemare?chal et al. in Math Program 69:111–147, 1995), (v) Trust region (regularisa- tion) method of Linderoth and Wright (Comput Optim Appl 24:207–250, 2003). In this study the three regularisation methods have been introduced within the computational structure of Benders decomposition. Thus second-stage infeasibility is controlled in the traditional manner, by imposing feasibility cuts. This approach allows extensions of the regularisation to feasibility issues, as in Fa?bia?n and Szo ?ke (Comput Manag Sci 4:313–353, 2007). We report computational results for a wide range of benchmark problems from the POSTS and SLPTESTSET collections and a collection of difficult test problems compiled by us. Finally the scale-up properties and the performance profiles of the methods are presented.

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Robustness is about reducing the feasible set of a given nominal opti- mization problem by cutting “risky” solutions away. To this end, the most popular approach in the literature is to extend the nominal model with a polynomial number of additional variables and constraints, so as to obtain its robust counterpart. Robustness can also be enforced by adding a possibly exponential family of cutting planes, which typically leads to an exponential formulation where cuts have to be generated at run time. Both approaches have pros and cons, and it is not clear which is the best one when approaching a specific problem. In this paper we computationally compare the two options on some prototype problems with different characteristics. We first address robust optimization a? la Bertsimas and Sim for linear programs, and show through computational experiments that a considerable speedup (up to 2 orders of magni- tude) can be achieved by exploiting a dynamic cut generation scheme. For integer linear problems, instead, the compact formulation exhibits a typically better perfor- mance. We then move to a probabilistic setting and introduce the uncertain set covering problem where each column has a certain probability of disappearing, and each row has to be covered with high probability. A related uncertain graph connectivity problem is also investigated, where edges have a certain probability of failure. For both problems, compact ILP models and cutting plane solution schemes are presented and compared through extensive computational tests. The outcome is that a compact ILP formulation (if available) can be preferable because it allows for a better use of the rich arsenal of preprocessing/cut generation tools available in modern ILP solvers. For the cases where such a compact ILP formulation is not available, as in the uncertain connectivity problem, we propose a restart solution strategy and computationally show its practical effectiveness.

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While semidefinite relaxations are known to deliver good approximations for combinatorial optimization problems like graph bisection, their practical scope is mostly associated with small dense instances. For large sparse instances, cutting plane techniques are considered the method of choice. These are also applicable for semi- definite relaxations via the spectral bundle method, which allows to exploit structural properties like sparsity. In order to evaluate the relative strengths of linear and semi- definite approaches for large sparse instances, we set up a common branch-and-cut framework for linear and semidefinite relaxations of the minimum graph bisection problem. It incorporates separation algorithms for valid inequalities of the bisection cut polytope described in a recent study by the authors. While the problem specific cuts help to strengthen the linear relaxation significantly, the semidefinite bound prof- its much more from separating the cycle inequalities of the cut polytope on a slightly enlarged support. Extensive numerical experiments show that this semidefinite branch- and-cut approach without problem specific cuts is a superior choice to the classical simplex approach exploiting bisection specific inequalities on a clear majority of our large sparse test instances from VLSI design and numerical optimization.

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