We describe a two-stage robust optimization approach for solving network flow and design problems with uncertain demand. In two-stage network optimization, one defers a subset of the flow decisions until after the realization of the uncertain demand. Availability of such a recourse action allows one to come up with less conservative solutions compared to singlestage optimization. However, this advantage often comes at a price: two-stage optimization is, in general, significantly harder than single-stage optimization.For network flow and design under demand uncertainty, we give a characterization of the first-stage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is -hard even for a network flow problem on a bipartite graph. We show, however, that if the second-stage network topology is totally ordered or an arborescence, then the separation problem is tractable.Unlike single-stage robust optimization under demand uncertainty, two-stage robust optimization allows one to control conservatism of the solutions by means of an allowed "budget for demand uncertainty." Using a budget of uncertainty, we provide an upper bound on the probability of infeasibility of a robust solution for a random demand vector.We generalize the approach to multicommodity network flow and design, and give applications to lot-sizing and locationtransportation problems. By projecting out second-stage flow variables, we define an upper bounding problem for the two-stage min-max-min optimization problem. Finally, we present computational results comparing the proposed twostage robust optimization approach with single-stage robust optimization as well as scenario-based two-stage stochastic optimization.

a b s t r a c tWe describe a polynomial-size conic quadratic reformulation for a machine-job assignment problem with separable convex cost. Because the conic strengthening is based only on the objective of the problem, it can also be applied to other problems with similar cost functions. Computational results demonstrate the effectiveness of the conic reformulation.

Given a finite ground set N and a value vector a ∈ R N , we consider optimization problems involving maximization of a submodular set utility function of the form h(S) = f i∈S a i , S ⊆ N , where f is a strictly concave, increasing, differentiable function. This utility function appears frequently in combinatorial optimization problems when modeling risk aversion and decreasing marginal preferences, for instance, in risk-averse capital budgeting under uncertainty, competitive facility location, and combinatorial auctions. These problems can be formulated as linear mixed 0-1 programs. However, the standard formulation of these problems using submodular inequalities is ineffective for their solution, except for very small instances. In this paper, we perform a polyhedral analysis of a relevant mixed-integer set and, by exploiting the structure of the utility function h, strengthen the standard submodular formulation significantly. We show the lifting problem of the submodular inequalities to be a submodular maximization problem with a special structure solvable by a greedy algorithm, which leads to an easily-computable strengthening by subadditive liftingThe research of the first author has been supported, in part, by Grant # FA9550-08-1-0117 from the Air Force Office of Scientific Research. The research of the second author has been supported, in part, by Grant # DMI0700203 from the National Science Foundation. The second author is grateful for the hospitality of the Georgia Institute of Technology, where part of this research was conducted.

Abstract. We study the polyhedra of splittable and unsplittable single arc-set relaxations of multicommodity flow capacitated network design problems. We investigate the optimization problems over these sets and the separation and lifting problems of valid inequalities for them. In particular, we give a linear-time separation algorithm for the residual capacity inequalities [19] and show that the separation problem of c-strong inequalities [7] is N P-hard, but can be solved over the subspace of fractional variables only. We introduce two classes of inequalities for the unsplittable flow problems. We present a summary of computational experiments with a branch-and-cut algorithm for multicommodity flow capacitated network design problems to test the effectiveness of the results presented here empirically.

Abstract. A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints.Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve either secondorder conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree.Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming.We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean-variance capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability.