In this paper, we introduce and study a two-stage distributionally robust mixed binary problem (TSDR-MBP) where the random parameters follow the worst-case distribution belonging to an uncertainty set of probability distributions. We present a decomposition algorithm, which utilizes distribution separation procedure and parametric cuts within Benders' algorithm or Lshaped method, to solve TSDR-MBPs with binary variables in the first stage and mixed binary programs in the second stage. We refer to this algorithm as distributionally robust integer (DRI) L-shaped algorithm. Using similar decomposition framework, we provide another algorithm to solve TSDR linear problem where both stages have only continuous variables. We investigate conditions and the families of ambiguity set for which our algorithms are finitely convergent. We present two examples of ambiguity set, defined using moment matching, or Kantorovich-Rubinstein distance (Wasserstein metric), which satisfy the foregoing conditions. We also present a cutting surface algorithm to solve TSDR-MBPs. We computationally evaluate the performance of the DRI Lshaped algorithm and the cutting surface algorithm in solving distributionally robust versions of a few instances from the Stochastic Integer Programming Library, in particular stochastic server location and stochastic multiple binary knapsack problem instances. We also discuss the usefulness of incorporating partial distribution information in two-stage stochastic optimization problems.

We study two-stage stochastic mixed integer programs (TSS-MIPs) with integer variables in the second stage. We show that under suitable conditions, the second stage MIPs can be convexified by adding parametric cuts a priori. As special cases, we extend the results of Miller and Wolsey (Math Program 98(1):73-88, 2003) to TSS-MIPs. Furthermore, we consider second stage programs that are generalizations of the well-known mixing (and continuous mixing) set, or certain piecewise-linear convex objective integer programs. These results allow us to relax the integrality restrictions on the second stage integer variables without effecting the integrality of the optimal solution of the TSS-MIP. We also use four variants of the two-stage stochastic capacitated lot-sizing problems as test problems for computational experiments, and present tight second-stage formulations for these problems. Our computational results show that adding parametric inequalities that a priori convexify the second stage formulation significantly reduces the total solution time taken to solve these problems.

In this paper, we derive (partial) convex hull for deterministic multi-constraint polyhedral conic mixed integer sets with multiple integer variables using conic mixed integer rounding (CMIR) cut-generation procedure of Atamtürk and Narayanan (Math Prog 122:1-20, 2008), thereby extending their result for a simple polyhedral conic mixed integer set with single constraint and one integer variable. We then introduce two-stage stochastic p-order conic mixed integer programs (denoted by TSS-CMIPs) in which the second stage problems have sum of l p -norms in the objective function along with integer variables. First, we present sufficient conditions under which the addition of scenario-based nonlinear cuts in the extensive formulation of TSS-CMIPs is sufficient to relax the integrality restrictions on the second stage integer variables without impacting the integrality of the optimal solution of the TSS-CMIP. We utilize scenario-based CMIR cuts for TSS-CMIPs and their distributionally robust generalizations with structured CMIPs in the second stage, and prove that these cuts provide conic/linear programming equivalent or approximation for the second stage CMIPs. We also perform extensive computational experiments by solving stochastic and distributionally robust capacitated facility location problem and randomly generated structured TSS-CMIPs with polyhedral CMIPs and second-order CMIPs in the second stage, i.e. p = 1 and p = 2, respectively. We observe that there is a significant reduction in the total time taken to solve these problems after adding the scenario-based cuts. KeywordsTwo-stage stochastic p-order conic mixed integer program • Scenario-based cutting planes • Two-stage distributionally robust program • (Partial) convex hull • Conic mixed integer rounding • Multi-module capacitated facility location B Manish Bansal

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We study the planar maximum coverage location problem (MCLP) with rectilinear distance and rectangular demand zones in the case where "partial coverage" is allowed in its true sense, i.e., when covering only part of a demand zone is allowed and the coverage accrued as a result of this is proportional to the demand of the covered part only. We pose the problem in a slightly more general form by allowing services zones to be rectangular instead of squares, thereby addressing applications in camera view-frame selection as well. More specifically, our problem, referred to as PMCLP-PCR, is to position a given number of rectangular service zones (SZs) on the two-dimensional plane to (partially) cover a set of existing (possibly overlapping) rectangular demand zones (DZs) such that the total covered demand is maximized. Previous studies on (planar) MCLP have assumed binary coverage, even when non-point objects such as lines or polygons have been used to represent demand. Under the binary coverage assumption, the problem can be readily formulated and solved as a binary linear program; whereas, partial coverage, although much more realistic, cannot be efficiently handled by binary linear programming making PMCLP-PCR much more challenging to solve. In this paper, we first prove that PMCLP-PCR is NP-hard if the number of SZs is part of the input. We then present an improved algorithm for the single-SZ PMCLP-PCR, which is at least two times faster than the existing exact Plateau Vertex Traversal algorithm. Next, we study multi-SZ PMCLP-PCR for the first time and prove theoretical properties which significantly reduce the search space for solving this problem and present a customized branchand-bound exact algorithm to solve it. Our computational experiments show that this algorithm can solve relatively large instances of multi-SZ PMCLP-PCR in a short time. We also propose a fast polynomial time heuristic algorithm. Having optimal solutions from our exact algorithm, we benchmark the quality of solutions obtained from our heuristic algorithm. Our results show that for all the random instances solved to optimality by our exact algorithm, our heuristic algorithm finds a solution in a fraction of a second, where its objective value is at least 91% of the optimal objective in 90% of the instances (and at least 69% of the optimal objective in all the instances).