Alper Atamtürk

Two-Stage Robust Network Flow and Design Under Demand Uncertainty

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 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 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 multicommodity 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.

INTEGER PROGRAMMING SOFTWARE SYSTEMS

Recent developments in integer-programming software systems have tremendously improved our ability to solve large-scale instances. We review the major algorithmic components of state-of-the-art solvers and discuss the options available to users for adjusting the behavior of these solvers when default settings do not achieve the desired performance level. Furthermore, we highlight advances towards integrated modeling and solution environments. We conclude with a discussion of model characteristics and substructures that pose challenges for integer-programming software systems and a perspective on features we may expect to see in these systems in the near future.

On capacitated network design cut–set polyhedra

Abstract.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. Valid inequalities described here are applicable to directed network design problems with any number of facility types and any level of capacities. We report results from a computational study done for testing the effectiveness of the new inequalities.

Capacity Acquisition, Subcontracting, and Lot Sizing

The fundamental question encountered in acquiring capacity to meet nonstationary demand over a multiperiod horizon is how to balance the trade-off 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 trade-offs 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.

Conflict graphs in solving integer programming problems

Abstract We report on the use of conflict graphs in solving integer programs. A conflict graph represents logical relations between binary variables. 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.

On the facets of the mixed–integer knapsack polyhedron

Abstract.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.

Conic mixed-integer rounding cuts

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 second-order 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.

On splittable and unsplittable flow capacitated network design arc–set polyhedra

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 ??–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.

A Conic Integer Programming Approach to Stochastic Joint Location-Inventory Problems

We study several joint facility location and inventory management problems with stochastic retailer demand. In particular, we consider cases with uncapacitated facilities, capacitated facilities, correlated retailer demand, stochastic lead times, and multicommodities. We show how to formulate these problems as conic quadratic mixed-integer problems. Valid inequalities, including extended polymatroid and extended cover cuts, are added to strengthen the formulations and improve the computational results. Compared to the existing modeling and solution methods, the new conic integer programming approach not only provides a more general modeling framework but also leads to fast solution times in general.

A study of the lot-sizing polytope

Abstract.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.