Rounding-based heuristics for nonconvex MINLPs

We propose two primal heuristics for nonconvex mixed-integer nonlinear programs. Both are based on the idea of rounding the solution of a continuous nonlinear program subject to linear constraints. Each rounding step is accomplished through the solution of a mixed-integer linear program. Our heuristics use the same algorithmic scheme, but they differ in the choice of the point to be rounded (which is feasible for nonlinear constraints but possibly fractional) and in the linear constraints. We propose a feasibility heuristic, that aims at finding an initial feasible solution, and an improvement heuristic, whose purpose is to search for an improved solution within the neighborhood of a given point. The neighborhood is defined through local branching cuts or box constraints. Computational results show the effectiveness in practice of these simple ideas, implemented within an open-source solver for nonconvex mixed-integer nonlinear programs

Decomposition algorithms for global solution of deterministic and stochastic pooling problems in natural gas value chains

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2009.Includes bibliographical references (leaves 153-158).In this thesis, a Benders decomposition algorithm is designed and implemented to solve both deterministic and stochastic pooling problems to global optimality. Convergence of the algorithm to a global optimum is proved and then it is implemented both in GAMS and C++ to get the best performance. A series of example problems are solved, both with the proposed Benders decomposition algorithm and commercially available global optimization software to determine the validity and the performance of the proposed algorithm. Moreover, a two stage stochastic pooling problem is formulated to model the optimal capacity expansion problem in pooling networks and the proposed algorithm is applied to this problem to obtain global optimum. A number of example stochastic pooling problems are solved, both with the proposed Benders decomposition algorithm and commercially available global optimization software to determine the validity and the performance of the proposed algorithm applied to stochastic problems.by Emre Armagan.S.M

Combinatorial optimization problems with concave costs

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2009.Includes bibliographical references (p. 83-89).In the first part, we study the problem of minimizing a separable concave function over a polyhedron. We assume the concave functions are nonnegative nondecreasing on R+, and the polyhedron is in RI' (these assumptions can be relaxed further under suitable technical conditions). We show how to approximate this problem to 1+ E precision in optimal value by a piecewise linear minimization problem so that the number of resulting pieces is polynomial in the input size of the original problem and linear in 1/c. For several concave cost problems, the resulting piecewise linear problem can be reformulated as a classical combinatorial optimization problem. As a result of our bound, a variety of polynomial-time heuristics, approximation algorithms, and exact algorithms for classical combinatorial optimization problems immediately yield polynomial-time heuristics, approximation algorithms, and fully polynomial-time approximation schemes for the corresponding concave cost problems. For example, we obtain a new approximation algorithm for concave cost facility location, and a new heuristic for concave cost multi commodity flow. In the second part, we study several concave cost problems and the corresponding combinatorial optimization problems. We develop an algorithm design technique that yields a strongly polynomial primal-dual algorithm for a concave cost problem whenever such an algorithm exists for the corresponding combinatorial optimization problem.(cont.) Our technique preserves constant-factor approximation ratios as well as ratios that depend only on certain problem parameters, and exact algorithms yield exact algorithms. For example, we obtain new approximation algorithms for concave cost facility location and concave cost joint replenishment, and a new exact algorithm for concave cost lot-sizing. In the third part, we study a real-time optimization problem arising in the operations of a leading internet retailer. The problem involves the assignment of orders that arrive via the retailer's website to the retailer's warehouses. We model it as a concave cost facility location problem, and employ existing primal-dual algorithms and approximations of concave cost functions to solve it. On past data, we obtain solutions on average within 1.5% of optimality, with running times of less than 100ms per problem.by Dan Stratila.Ph.D

Supply Chain Network Design with Concave Costs: Theory and Applications

Many practical decision models can be formulated as concave minimization problems. Supply chain network design problems (SCNDP) that explicitly account for economies-of-scale and/or risk pooling often lead to mathematical problems with a concave objective and linear constraints. In this thesis, we propose new solution approaches for this class of problems and use them to tackle new applications. In the first part of the thesis, we propose two new solution methods for an important class of mixed integer concave minimization problems over a polytope that appear frequently in SCNDP. The first is a Lagrangian decomposition approach that enables a tight bound and a high quality solution to be obtained in a single iteration by providing a closed-form expression for the best Lagrangian multipliers. The Lagrangian approach is then embedded within a branch-and-bound framework. Extensive numerical testing, including implementation on three SCNDP from the literature, demonstrates the validity and efficiency of the proposed approach. The second method is a Benders approach that is particularly effective when the number of concave terms is small. The concave terms are isolated in a low dimensional master problem that can be efficiently solved through enumeration. The subproblem is a linear program that is solved to provide a Benders cut. Branch-and-bound is then used to restore integrality if necessary. The Benders approach is tested and benchmarked against commercial solvers and is found to outperform them in many cases. In the second part, we formulate and solve the problem of designing a supply chain for chilled and frozen products. The cold supply chain design problem is formulated as a mixed-integer concave minimization problem with dual objectives of minimizing the total cost, including capacity, transportation, and inventory costs, and minimizing the global warming impact that includes, in addition to the carbon emissions from energy usage, the leakage of high global-warming-potential refrigerant gases. Demand is modeled as a general distribution, whereas inventory is assumed managed using a known policy but without explicit formulas for the inventory cost and maximum level functions. The Lagrangian approach proposed in the first part is combined with a simulation-optimization approach to tackle the problem. An important advantage of this approach is that it can be used with different demand distributions and inventory policies under mild conditions. The solution approach is verified through extensive numerical testing on two realistic case studies from different industries, and some managerial insights are drawn. In the third part, we propose a new mathematical model and a solution approach for the SCNDP faced by a medical sterilization service provider serving a network of hospitals. The sterilization network design problem is formulated as a mixed-integer concave minimization program that incorporates economies of scale and service level requirements under stochastic demand conditions, with the objective of minimizing long-run capacity, transportation, and inventory holding costs. To solve the problem, the resulting formulation is transformed into a mixed-integer second-order cone programming problem with a piecewise-linearized cost function. Based on a realistic case study, the proposed approach was found to reach high quality solutions efficiently. The results reveal that significant cost savings can be achieved by consolidating sterilization services as opposed to decentralization due to better utilization of resources, economies of scale, and risk pooling

Optimizing safety stock placement in general network supply chains

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2004.Includes bibliographical references (p. 207-214).The amount of safety stock to hold at each stage in a supply chain is an important problem for a manufacturing company that faces uncertain demand and needs to provide a high level of service to its customers. The amount of stock held should be small to minimize holding and storage costs while retaining the ability to serve customers on time and satisfy most, if not all, of the demand. This thesis analyzes this problem by utilizing the framework of deterministic service time models and provides an algorithm for safety stock placement in general-network supply chains. We first show that the general problem is NP-hard. Next, we develop several conditions that characterize an optimal solution of the general-network problem. We find that we can identify all possible candidates for the optimal service times for a stage by constructing paths from the stage to each other stage in the supply chain. We use this construct, namely these paths, as the basis for a branch and bound algorithm for the general-network problem. To generate the lower bounds, we create and solve a spanning-tree relaxation of the general-network problem. We provide a polynomial algorithm to solve these spanning tree problems. We perform a set of computational experiments to assess the performance of the general-network algorithm and to determine how to set various parameters for the algorithm. In addition to the general network case, we consider two-layer network problems. We develop a specialized branch and bound algorithm for these problems and show computationally that it is more efficient than the general-network algorithm applied to the two-layer networks.by Ekaterina Lesnaia.Ph.D