Alper Atamturk and Vishnu Narayanan The submodular knapsack set is the discrete lower level set of a submodular function. The modular case reduces to the classical linear 0-1 knapsack set. One motivation for studying the submodular knapsack polytope is to address 0-1 programming problems with uncertain coefficients. Under various assumptions, a probabilistic constraint on 0-1 variables can be modeled as a submodular knapsack set. In this paper we describe cover inequalities for the submodular knapsack set and investigate their lifting problem. Each lifting problem is itself an optimization problem over a submodular knapsack set. We give sequence-independent upper and lower bounds on the valid lifting coefficients and show that whereas the upper bound can be computed in polynomial time, the lower bound problem is NP-hard. Furthermore, we present polynomial algorithms based on parametric linear programming and computational results for the conic quadratic 0-1 knapsack case. Research Report BCOL.08.03, IEOR, University of California-Berkeley. June 2008. subknap.pdf subknap.bib Shabbir Ahmed and Alper Atamturk Given a finite ground set N and a value vector a in R^N, we consider optimization problems involving maximization of a submodular utility function of the form h(S)= f(sum_{i in S} a_i), S subseteq N, where f is a strictly concave, increasing, differentiable function. Such problems arise, for instance, in the context of risk-averse capital budgeting under uncertainty and combinatorial auctions. Due to their combinatorial nature, these problems can be formulated as linear mixed 0-1 programs. However, the standard formulation of these problems based on existing submodular inequalities is ineffective for solving them except for very small instances. In this paper, we perform a polyhedral analysis of the 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 lifting of the inequalities. Computational experiments show the effectiveness of the new formulation. Research Report BCOL.08.02, IEOR, University of California-Berkeley. March 2008. submodular-utility.pdf submodular-utility.bib M. Selim Akturk, Alper Atamturk, and Sinan Gurel Many scheduling problems in practice involve rescheduling of disrupted schedules. In this study, we show that in contrast to fixed processing times, if we have the flexibility to control the processing times of the jobs, we can generate alternative reactive schedules considering the manufacturing cost implications in response to disruptions. We consider a non-identical parallel machining environment where processing times of the jobs are compressible at a certain manufacturing cost, which is a convex function of the compression on the processing time. In rescheduling it is highly desirable to catch up the original schedule as soon as possible by reassigning the jobs to the machines and compressing their processing times. On the other hand, one must also keep the manufacturing cost due to compression of the jobs low. Thus, one is faced with a tradeoff between match-up time and manufacturing cost criteria. We introduce alternative match-up scheduling problems for finding schedules on the efficient frontier of this time/cost tradeoff. We employ the recent advances in conic mixed-integer programming to model these problems effectively. We further provide a fast heuristic algorithm driven by dual prices of convex subproblems for generating approximate efficient schedules. Research Report BCOL.08.01, IEOR, University of California-Berkeley. January 2008. Revised June 2008. matchup.pdf matchup.bib Alper Atamturk and Vishnu Narayanan In financial markets high levels of risk are associated with large returns as well as large losses, whereas with lower levels of risk, the potential for either return or loss is small. Therefore, risk management is fundamentally concerned with finding an optimal tradeoff between risk and return matching an investor's risk tolerance. Managing risk is studied mostly in a financial context; nevertheless, it is certainly relevant in any area with a significant source of uncertainty. The mean-risk tradeoff is well-studied for problems with a convex feasible set. However, this is not the case in the discrete setting, even though, in practice, portfolios are often restricted to discrete choices. In this paper we study mean-risk minimization for problems with discrete decision variables. In particular, we consider discrete optimization problems with a submodular mean-risk minimization objective. We show the connection between extended polymatroids and the convex lower envelope of this mean-risk objective. For 0-1 problems a complete linear characterization of the convex lower envelope is given. For mixed 0-1 problems we derive an exponential class of conic quadratic inequalities. We report preliminary computational experiments on a risk-aware capital budgeting problem with uncertain returns on investments with discrete choices. The results show significant improvements in the solvability of the problem with the introduced convexification method. Forthcoming in Operations Research Letters. Research Report BCOL.07.05, IEOR, University of California-Berkeley. November 2007. Revised April 2008. conicobj.pdf conicobj.bib Alper Atamturk and Vishnu Narayanan Lifting is a procedure for deriving valid inequalities for mixed-integer sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to solve such problems with branch-and-cut algorithms. Here we generalize the theory of lifting to conic integer programming, i.e., integer programs with conic constraints. We show how to derive conic valid inequalities for a conic integer program from conic inequalities valid for its lower-dimensional restrictions. In order to simplify computations, we also discuss sequence-independent lifting for conic integer programs. When the cones are restricted to nonnegative orthants, conic lifting reduces to the lifting for linear integer programming. Research Report BCOL.07.04, IEOR, University of California-Berkeley. October 2007. coniclift.pdf coniclift.bib Alper Atamturk and Oktay Gunluk Mixed-integer rounding (MIR) is a simple, yet powerful procedure for generating valid inequalities for mixed-integer programs. When used as cutting planes, MIR inequalities are very effective for problems with unbounded integer variables. For problems with bounded integer variables, however, cutting planes based on lifting techniques appear to be more effective. This is not surprising as lifting techniques make explicit use of the bounds on variables, whereas the MIR procedure does not. In this paper we describe a simple procedure, which we call mingling, for incorporating variable bound information into mixed-integer rounding. By explicitly using the variable bounds, the mingling procedure leads to strong inequalities for mixed-integer sets with bounded variables. We show that facets of the mixed-integer knapsack sets derived earlier by superadditive lifting techniques are mingling inequalities. In particular, the mingling inequalities developed in this paper subsume the continuous cover and reverse continuous cover inequalities of Marchand and Wolsey as well as the continuous integer knapsack cover and pack inequalities of Atamturk. In addition, mingling inequalities give a generalization of the two-step MIR inequalities of Dash and Gunluk under some conditions. Research Report BCOL.07.03, IEOR, University of California-Berkeley. September 2007. Revised June 2008. mingle.pdf mingle.bib Alper Atamturk and Simge Kucukyavuz Lot-sizing problems with inventory bounds and fixed charges have not received much attention in the literature, even though there are many emerging applications in which highly specialized storage is the main activity of interest. The traditional infinite capacity and variable cost assumptions on inventory that have been prevalent in the literature are inappropriate in situations for which tight storage capacity and high fixed cost of specialized storage are critical. Here we present an O(n^2) dynamic programming algorithm for the lot-sizing problem with inventory bounds and fixed costs, where n is the number of time periods. The algorithm operates on a hierarchy of two layers of value functions to solve the problem efficiently. It improves the complexity bound of the classic O(n^3) algorithm of Love (1973) for lot sizing with concave cost and bounded inventory. Operations Research Letters 36, 297-299, 2008. lsbi-alg.pdf lsbi-alg.bib http://dx.doi.org/10.1016/j.orl.2007.08.004 M. Selim Akturk, Alper Atamturk, and Sinan Gurel We consider a machine-job assignment problem with separable convex cost. A major source of difficulty with solving the problem using relaxation methods is that optimal solutions to its continuous relaxations are highly fractional as they are typically found in the interior of the relaxation due to the convex cost function. Here we give a polynomial-size conic quadratic reformulation of the problem so as to strengthen the bounds from its continuous relaxation. The results in the paper are sufficiently general so that they can also be applied to other mixed 0-1 optimization problems with separable convex cost functions. Our computational results on the machine-job assignment problem with controllable processing times demonstrate that the proposed conic reformulation is very effective for solving the problem to optimality. Research Report BCOL.07.01, IEOR, University of California-Berkeley. April 2007. Revised December 2007. conicsch.pdf conicsch.bib Alper Atamturk and Vishnu Narayanan A conic integer program is an integer programming problem with conic constraints. Many important 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 continuous 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 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. Forthcoming in Mathematical Programming. Research Report BCOL.06.03, IEOR, University of California-Berkeley. December 2006. Revised March 2008. conicmip.pdf conicmip.bib Alper Atamturk and Simge Kucukyavuz We derive strong inequalities for capacitated fixed-charge network flow problems based on path-set relaxations. These inequalities are based on exact characterizations of submodular value functions for fixed-charge flow on simple paths and they generalize the well-known flow cover inequalities. Research Report BCOL.06.02, IEOR, University of California-Berkeley. March 2006. Alper Atamturk and Simge Kucukyavuz Flow cover inequalities are the most effective valid inequalities for solving capacitated fixed-charge network flow problems. These valid inequalities are implications on the flow quantity on the cut arcs of a two-partitioning of the network, depending on whether some of the cut arcs are open or closed. As the implications are only on the cut arcs, flow cover inequalities can be modeled by collapsing the subset of nodes defining the cut into a single node. In this paper we define new valid inequalities for the capacitated fixed-charge network flow problem by exploiting the internal structure of the subgraphs defining the cuts. In particular, the new inequalities are based on three-partitioning of the nodes and they can be modeled by collapsing the each partition into a single node. The three-partitioning inequalities include the flow cover inequalities as a special case. We discuss varying capacity and constant capacity cases and describe separation algorithms for the inequalities. Finally, we report our preliminary computational results with the new inequalities. Research Report BCOL.06.01, IEOR, University of California-Berkeley. February 2006. Alper Atamturk and Muhong Zhang The flow set with partial order is a mixed-integer set described by a budget on total flow and a partial order on the arcs that may carry positive flow. This set is a common substructure of resource allocation and scheduling problems with precedence constraints and robust network flow problems under demand/capacity uncertainty. We give a polyhedral analysis of the convex hull of the flow set with partial order. Unlike for the flow set without partial order, cover-type inequalities based on partial order structure are a function of a lifting sequence. We study the lifting sequences and describe structural results on the lifting coefficients for general and simpler special cases. We show that all lifting coefficients can be computed in polynomial time by solving maximum weight closure problems in general. For the special case of induced-minimal covers, we give a sequence-dependent characterization of the lifting coefficients. We prove, however, if the partial order is defined by an arborescence, then lifting is sequence-independent and all lifting coefficients can be computed in linear time. On the other hand, if the partial order is defined by a path (total order), then the coefficients can be expressed explicitly. We also give a complete polyhedral description of the flow set with partial order for the polynomially-solvable total order case. We show that finding an optimal lifting order for a given induced-minimal cover and a given fractional solution is a submodular optimization problem, which is solved greedily. Finally, we present preliminary computational results with a cutting-plane algorithm based on the lifting and separation results. Forthcoming in Mathematics of Operations Research. Research Report BCOL.05.03, IEOR, University of California-Berkeley. December 2005. Revised December 2007. pcflow.pdf pcflow.bib Alper Atamturk and Deepak Rajan We study the design of capacitated survivable networks using directed p-cycles. A p-cycle is a cycle with at least three arcs, used for rerouting disrupted flow during edge failures. Survivability of the network is accomplished by reserving sufficient slack on directed p-cycles so that if an edge fails, its flow can be rerouted along the p-cycles. We describe a model for designing capacitated survivable networks based on directed p-cycles. We motivate this model by comparing it with other means of ensuring survivability, and present a mixed-integer programming formulation for it. We derive valid inequalities for the model based on the minimum capacity requirement between partitions of the nodes and give facet conditions for them. We discuss the separation for these inequalities and present results of computational experiments for testing their effectiveness as cutting planes when incorporated in a branch-and-cut algorithm. Our experiments show that the proposed inequalities reduce the computational effort significantly. Discrete Optimization 5, 415-433, 2008. (Copyright Elsevier) _published/do5-2008.pdf sdp.bib http://dx.doi.org/10.1016/j.disopt.2007.08.002 Alper Atamturk and Oktay Gunluk In this paper we study the network design arc set with variable upper bounds. This set appears as a common substructure of many network design problems and is a relaxation of several fundamental mixed-integer sets studied earlier independently. In particular, the splittable flow arc set, the unsplittable flow arc set, the single node fixed-charge flow set, and the binary knapsack set are facial restrictions of the network design arc set with variable upper bounds. Here we describe families of strong valid inequalities that cut off all fractional extreme points of the continuous relaxation of the network design arc set with variable upper bounds. Interestingly, some of these inequalities are also new even for the aforementioned restrictions studied earlier. Networks 50, 17-28, 2007. _published/networks50-2007.pdf mixset.bib http:/dx.doi.org/10.1002/net.20162 Alper Atamturk and Muhong Zhang 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 single-stage optimization. However, this advantage often comes at a price: two-stage optimization is, in general, significantly harder than singe-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 NP-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 multi-commodity network flow and design, and give applications to lot-sizing and location-transportation 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 two-stage robust optimization approach with single-stage robust optimization as well as scenario-based two-stage stochastic optimization. Operations Research 55, 662-673, 2007. _published/or55-2007.pdf rn.bib Alper Atamturk and Deepak Rajan In this paper, we present new inequalities for the mixed-integer knapsack set. First, we analyze the mixed-integer knapsack set with two integer variables and one continuous variable, and develop a polynomial algorithm that enumerates all the facets in its convex hull. Then, we study the exact lifting function for the facets of the convex hull of this set, and describe super-additive lower bounds for it. These super-additive lower bounds are obtained from partial LP relaxations of the exact lifting function, and used in a sequence independent lifting framework to develop strong valid inequalities for the mixed-integer knapsack polyhedron. We present some sufficient conditions under which these lifted inequalities define facets of mixed-integer knapsack sets with at most one continuous variable. Finally, we summarize our computational experience with these inequalities. Research Report BCOL.04.02, IEOR, University of California-Berkeley. December 2004. 2var.pdf 2var.bib Alper Atamturk We introduce strong formulations for robust mixed 0-1 programming with uncertain objective coefficients. We focus on a polytopic uncertainty set described by as "a budget constraint" for allowed uncertainty in the objective. We show that for a robust 0-1 problem, there is an alpha-tight linear programming formulation with size polynomial in the size of an alpha-tight linear programming formulation for the nominal 0-1 problem. We give extensions to robust mixed 0-1 programming and present computational experiments with the proposed formulations. Mathematical Programming 108, 235-250, 2006. ./_published/mathprog108-2006.pdf robobj.bib http://dx.doi.org/10.1007/s10107-006-0709-5 Alper Atamturk and Simge Kucukyavuz We investigate the polyhedral structure of the lot-sizing problem with inventory bounds. We consider two models, one with linear costs on inventory, the other with linear and fixed costs on inventory. For both models, we identify facet-defining inequalities that make use of the inventory capacities explicitly and give exact separation algorithms. We also give a linear programming formulation of the problem when the order and inventory variable costs satisfy the Wagner-Whitin nonspeculative property. We present computational experiments that show the effectiveness of the results in tightening the linear programming relaxations of the lot-sizing problem with inventory bounds. Operations Research 53, 711-730, 2005. _published/or53-2005.pdf bif.bib Alper Atamturk and Martin Savelsbergh We review the recent developments in integer programming software systems that have improved tremendously our ability to solve large-scale instances. Our objective is to highlight the capabilities and advanced features of state-of-the-art optimization systems and discuss advances towards integrated modeling and solving environments. We conclude with perspectives on new features that we expect to see in integer programming software systems in the near future. Annals of Operations Research 140, 67-124, 2005. _published/aor140-2005.pdf ipsoftware.bib http://dx.doi.org/10.1007/s10479-005-3968-2 Alper Atamturk We review strong inequalities for fundamental knapsack relaxations of (mixed) integer programs. These relaxations are the 0-1 knapsack set, the mixed 0-1 knapsack set, the integer knapsack set, and the mixed integer knapsack set. Our aim is to give a common presentation of the inequalities based on covers and packs and highlight the connections among them. The focus of the paper is on recent research on the use of superadditive functions for the analysis of knapsack polyhedra. We also present some new results on integer knapsacks. In particular, we give integer version of cover inequalities and describe the necessary and sufficient facet condition for them. This condition generalizes the well-known facet condition of minimality of covers for 0-1 knapsacks. Annals of Operations Research 139, 21-38, 2005. _published/aor139-2005.pdf cover-pack.bib http://dx.doi.org/10.1007/s10479-005-3442-1 Alper Atamturk In the last decade our capability of solving integer programming problems has increased dramatically due to the effectiveness of cutting plane methods based on polyhedral investigations. Polyhedral cutting planes have become central features in optimization software packages for integer programming. Here we present some of the important polyhedral methods used in discrete optimization. We discuss applications to knapsack problems and robust combinatorial optimization. Trends in Optimization. S. Hosten, J. Lee, R. Thomas (eds.), Proc. of Symposia in Applied Mathematics 61, 21-37, 2004, American Mathematical Society. poly.pdf poly.bib Alper Atamturk and Juan C. Munoz The lot-sizing polytope is a fundamental structure contained in many practical production planning problems. Here we study this polytope and identify facet-defining inequalities that cut off all fractional extreme points of its linear programming relaxation, as well as liftings from those facets. We give a polynomial-time combinatorial separation algorithm for the inequalities when capacities are constant. We also report computational experiments on solving the lot-sizing problem with varying cost and capacity characteristics. Mathematical Programming 99, 443-465, 2004. _published/mathprog99-2004.pdf ls.bib http://dx.doi.org/10.1007/s10107-003-0465-8 Deepak Rajan and Alper Atamturk A network is said to be survivable if it has sufficient capacity for rerouting all of its flow under the failure of any one of its edges. Here we present a polyhedral approach for designing survivable networks. We describe a mixed-integer programming model, in which sufficient slack is explicitly introduced on the directed cycles of the network while flow routing decisions are made. In case of a failure, flow is rerouted along the slacks reserved on directed cycles. We give strong valid inequalities that use the survivability requirements. We present a computational study with a column-and-cut generation algorithm for designing capacitated survivable networks. Networks 43, 201-211, 2004. _published/networks43-2004.pdf surv-cyc.bib http://dx.doi.org/10.1002/net.20004 Alper Atamturk We show that superadditive lifting functions lead to sequence independent lifting of inequalities for general mixed integer programming. As an application, we note that mixed-integer rounding (MIR) may be viewed as sequence independent lifting. Consequently, we obtain facet conditions for MIR inequalities for mixed-integer knapsacks. Operations Research 52, 487-490, 2004. _published/or52-2004.pdf sil.bib http://dx.doi.org/10.1287/opre.1030.0099 Karen Smilowitz, Alper Atamturk, and Carlos Daganzo Rapid growth in the package delivery industry has led carriers to offer a wider range of transportation services defined by guaranteed delivery time. This paper studies the possible integration of long haul operations by transportation mode and service level. Specifically, we consider the allocation of deferred items to excess capacity on alternative modes in ways that allow all transportation modes to be utilized better. Model formulation and solution techniques are discussed. The solution techniques presented produce efficient solutions for large-scale problem instances. Allowing deferred items to travel by air reduces long haul transportation costs. These savings increase with the amount of excess air capacity. Transportation Research: Logistics and Transportation Review 39, 305-323, 2003. _published/tre39-2003.pdf divrp.bib http://dx.doi.org/10.1016/S1366-5545(02)00048-0 Alper Atamturk We study the mixed-integer knapsack polyhedron, that is, the convex hull of the mixed-integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet-defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable. These inequalities strengthen and/or generalize known inequalities for several special cases. We report computational results on using the inequalities as cutting planes for mixed-integer programming. Mathematical Programming 98, 145-175, 2003. _published/mathprog98-2003.pdf mip.bib http://dx.doi.org/10.1007/s10107-003-0400-z Alper Atamturk This paper provides an analysis of capacitated network design cut-set polyhedra. We give a complete linear description of the cut-set polyhedron of the single commodity - single facility capacitated network design problem. Then we extend the analysis to single commodity - multifacility and multicommodity - multifacility capacitated network design problems. The valid inequalities described here have coefficients for both inflow and outflow arcs of a cut-set and are applicable to network design problems with an arbitrary number of facility types and arbitrary capacities. We report a computational study to test the effectiveness of the new inequalities. Mathematical Programming 92, 425-437, 2002. _published/mathprog92-3-2002.pdf nd.bib http://dx.doi.org/10.1007/s101070100284 Alper Atamturk and Deepak Rajan We study the polyhedra of splittable and unsplittable arc sets of multicommodity flow capacitated network design problems. We investigate the optimization problems over these polyhedra 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 (Magnanti et al., 1993) and show that the separation problem of c-strong inequalities (Brockmuller et al., 1996) is NP-hard, but can be solved over the subspace of fractional variables only. We introduce two new classes of inequalities that generalize the c-strong inequalities and show that the lifting of one of them can be done in polynomial time. We present a summary of computational experiments with a branch-and-cut algorithm for multicommodity flow capacitated network design problems illustrating the effectiveness of the results presented here empirically. Mathematical Programming 92, 315-333, 2002. _published/mathprog92-2-2002.pdf arcset.bib http://dx.doi.org/10.1007/s101070100269 Deepak Rajan and Alper Atamturk A network is said to be survivable if all of the demands on the nodes can be met under the failure of any one of its links. In order to ensure that the flow on the network can be rerouted in the case of a failure, sufficient spare (excess) capacity must be available on the working links of the network. Since over-provisioning of capacity is a major concern due to the high investment costs required for installing capacity, designing capacity-efficient survivable networks is a highly critical problem in the telecommunication industry. In this paper we present a new mixed-integer programming model and a column generation method for the survivable design of telecommunication networks. In contrast with the failure scenario models, the new model has almost the same number of constraints as the regular network design problem, which makes it effective for large instances. Even though the complexity of pricing the exponentially many variables of the model is NP-hard, in our computational experiments, we are able to produce capacity-efficient survivable networks for dense graphs up to 70 nodes. Telecommunications Network Design and Management, G. Anandalingam and S. Raghavan (eds.), 65-82, Kluwer Academic Publishers, 2002. rajan-atamturk-kluwer.pdf surv-colgen.bib Alper Atamturk and Dorit Hochbaum The fundamental question encountered in acquiring capacity to meet nonstationary demand over a multi-period horizon is how to balance the tradeoff between having insufficient capacity in some periods and excess capacity in others. In the former situation part of the demand is subcontracted, while in the latter capacity that has been paid for is rendered idle. Capacity and subcontracting decisions arise in many economic activities ranging from production capacity planning in semiconductor fabs to leasing communication networks, from transportation contracts to staffing of call centers. In this paper, we investigate the tradeoffs between acquiring capacity, subcontracting, production, and holding inventory to satisfy nonstationary demand over a finite horizon. We present capacity acquisition models with holding and without holding inventory and identify forecast-robust properties of the models that restrict the dependence of optimal capacity decisions on the demand forecasts. We develop algorithms for numerous practical cost structures involving variable and fixed charges and prove that they all have polynomial time complexity. For models with inventory, we solve a sequence of constant capacity lot-sizing and subcontracting subproblems, which is also of independent interest. Management Science 47, 1081-1100, 2001. _published/ms47-2001.pdf cap.bib Alper Atamturk, George L. Nemhauser, and Martin W. P. Savelsbergh We study the facial structure of a polyhedron associated with the single node relaxation of network flow problems with additive variable upper bounds. This type of structure arises, for example, in production planning problems with setup times and in network certain expansion problems. We derive several classes of valid inequalities for this polyhedron and give conditions under which they are facet--defining. Our computational experience with large network expansion problems indicates that these inequalities are very effective in improving the quality of the linear programming relaxations. Mathematical Programming 91, 145-162, 2001. _published/mathprog91-2001.pdf avub.bib http://dx.doi.org/10.1007/s101070100235 Alper Atamturk We present a class of facet-defining valid inequalities for the single node fixed-charge flow polytope. We provide a comparison of the new inequalities with others from the literature. We also present computational results that show the effectiveness of these inequalities in solving fixed-charge network flow problems with a branch-and-cut algorithm. Operations Research Letters 29, 107-114, 2001. (Copyright: Elsevier) _published/orl29-2001.pdf fp.bib http://dx.doi.org/10.1016/S0167-6377(01)00100-6 Alper Atamturk, George L. Nemhauser, and Martin W. P. Savelsbergh We study a generalization of the vertex packing problem having both binary and bounded continuous variables, called the mixed vertex packing problem (MVPP). The well-known vertex packing model arises as a subproblem or relaxation of many 0-1 integer problems, whereas the mixed vertex packing model arises as a natural counterpart of vertex packing in the context of mixed 0-1 integer programming. We describe strong valid inequalities for the convex hull of solutions to the MVPP and separation algorithms for these inequalities. We give a summary of computational results with a branch-and-cut algorithm for solving the MVPP and using it to solve general mixed-integer problems. Mathematical Programming 89, 35-53, 2000. _published/mathprog89-2000.pdf mvp.bib http://dx.doi.org/10.1007/s101070000154 Alper Atamturk, George L. Nemhauser, and Martin W. P. Savelsbergh We report on our investigation of the use of conflict graphs in solving integer programs. A conflict graph represents logical relations between binary variables appearing in an integer program. We construct an extended conflict graph by using probing techniques based on feasibility as well as optimality considerations. Each node in the search tree of an LP based branch-and-bound algorithm has its own associated conflict graph. We develop algorithms and data structures that allow the effective and efficient construction, management, and use of dynamically changing conflict graphs. Our computational experiments show that the techniques presented work very well. European Journal of Operational Research 121, 40-55, 2000. (Copyright: Elsevier) _published/ejor121-2000.pdf cg.bib http://dx.doi.org/10.1016/S0377-2217(99)00015-6 Alper Atamturk, Ellis Johnson, Jeff Linderoth, and Martin W. P. Savelsbergh We discuss an integer linear programming modeling system based on relational algebra. In this system, all modeling related activities, such as model formulation, model instantiation, and model and instance management, are done using simple operations such as selection, projection, and predicated join. Operations Research 48, 846-857, 2000. _published/or48-2000.pdf armos.bib Alper Atamturk, George L. Nemhauser, and Martin W.P. Savelsbergh Given a finite ground set, a set of subsets and costs on the subsets, the set partitioning problem is to find a minimum cost partition of the ground set. Many combinatorial optimization problems can be formulated as set partitioning problems. We present an approximation algorithm that produces high quality solutions in an acceptable amount of computation time. The algorithm is iterative and combines problem size reduction techniques, such as logical implications derived from feasibility and optimality conditions and reduced cost fixing, with a primal heuristic based on cost perturbations embedded in a Lagrangian dual framework. Computational experiments illustrate the effectiveness of the approximation algorithm. Journal of Heuristics 1, 247-259, 1996. sp.pdf sp.bib http://dx.doi.org/10.1007/BF00127080 The documents distributed by this server have been provided by the contributing authors as a means to ensure timely dissemination of scholarly work on a noncommercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electronically. It is understood that all persons copying this information will adhere to the terms and constraints invoked by the authors and other copyright holders. These works may not be reposted without the explicit permission of the copyright holders.