[Show abstract][Hide abstract] ABSTRACT: We introduce an algorithm design technique for a class of combinatorial
optimization problems with concave costs. This technique yields a strongly
polynomial primal-dual algorithm for a concave cost problem whenever such an
algorithm exists for the fixed-charge counterpart of the problem. For many
practical concave cost problems, the fixed-charge counterpart is a well-studied
combinatorial optimization problem. Our technique preserves constant factor
approximation ratios, as well as ratios that depend only on certain problem
parameters, and exact algorithms yield exact algorithms.
Using our technique, we obtain a new 1.61-approximation algorithm for the
concave cost facility location problem. For inventory problems, we obtain a new
exact algorithm for the economic lot-sizing problem with general concave
ordering costs, and a 4-approximation algorithm for the joint replenishment
problem with general concave individual ordering costs.
[Show abstract][Hide abstract] ABSTRACT: We study the problem of minimizing a nonnegative separable concave function
over a compact feasible set. We approximate this problem to within a factor of
1+epsilon by a piecewise-linear minimization problem over the same feasible
set. Our main result is that when the feasible set is a polyhedron, the number
of resulting pieces is polynomial in the input size of the polyhedron and
linear in 1/epsilon. For many practical concave cost problems, the resulting
piecewise-linear cost problem can be formulated as a well-studied discrete
optimization problem. As a result, a variety of polynomial-time exact
algorithms, approximation algorithms, and polynomial-time heuristics for
discrete optimization problems immediately yield fully polynomial-time
approximation schemes, approximation algorithms, and polynomial-time heuristics
for the corresponding concave cost problems.
We illustrate our approach on two problems. For the concave cost
multicommodity flow problem, we devise a new heuristic and study its
performance using computational experiments. We are able to approximately solve
significantly larger test instances than previously possible, and obtain
solutions on average within 4.27% of optimality. For the concave cost facility
location problem, we obtain a new 1.4991+epsilon approximation algorithm.
[Show abstract][Hide abstract] ABSTRACT: Because of its imbedded network flow structure, the generic network design problem is an attractive candidate for integer
programming decomposition. This paper studies the application and acceleration of Benders decomposition for uncapacitated
models from this problem class and illustrates the potential flexibility of the Benders solution strategy. In particular,
it (i) shows that several lower bounding inequalities from the literature can be derived as Benders cuts; and (ii) introduces
new Benders cuts for the network design problem.
The paper also reports on computational experience in using Benders decomposition with a dual ascent and variable elimination
preprocessing procedure to solve uncapacitated network design problems with up to 90 binary variables and 15 080 continuous
variables, or 45 binary variables and 105 600 continuous variables.
[Show abstract][Hide abstract] ABSTRACT: We consider two types of hop-indexed models for the unit-demand asymmetric Capacitated Vehicle Routing Problem (CVRP): (a) capacitated models guaranteeing that the number of commodities (paths) traversing any given arc does not exceed a specified capacity; and (b) hop-constrained models guaranteeing that any route length (number of nodes) does not exceed a given value. The latter might, in turn, be divided into two classes: (b1) those restricting the length of the path from the depot to any node k, and (b2) those restricting the length of the circuit passing through any node k. Our results indicate that formulations based upon circuit lengths (b2) lead to models with a linear programming relaxation that is tighter than the linear programming relaxation of models based upon path lengths (b1), and that combining features from capacitated models with those of circuit lengths can lead to formulations for the CVRP with a tight linear programming bound. Computational results on a small number of problem instances with up to 41 nodes and 440 edges show that the combined model with capacities and circuit lengths produce average gaps of less than one percent. We also briefly examine the asymmetric travelling salesman problem (ATSP), showing the potential use of the ideas developed for the vehicle routing problem to derive models for the ATSP with a linear programming relaxation bound that is tighter than the linear programming relaxation bound of the standard Dantzig, Fulkerson and Johnson [G. Dantzig, D. Fulkerson, D. Johnson, Solution of large-scale travelling salesman problem, Operations Research 2 (1954) 393–410] formulation.
[Show abstract][Hide abstract] ABSTRACT: The function need not be continuous; it can have positive or negative jumps, though we do assume that the function is lower semi-continuous, that is, g a (x a ) lim inf x # a #xa g a (x # a ) for 1 any sequence x # a that approaches x a . Without loss of generality, we also assume, through a simple translation of the costs if necessary, that g a (0) = 0. Such a piecewise linear function can be fully characterized by its segments. On each arc a, each segment s of the function has a non-negative variable cost, c a (the slope), a non-negative fixed cost, f a (the intercept), and upper and lower bounds, b a and b a , on the flow of that segment. Since the total flow on each arc can always be bounded from above by either the arc capacity or the total demand flowing through the network, we assume that there is a finite number of segments on each arc a, which we represent by the set S a . We further introduce the following notation: K denotes the set of commodities, N is the |V
[Show abstract][Hide abstract] ABSTRACT: By adding a set of redundant constraints, and by iteratively refining the approximation, we show that a commercial solver is able to routinely solve moderate-size strategic safety stock placement problems to optimality. The speed-up arises because the solver automatically generates strong flow cover cuts using the redundant constraints.
[Show abstract][Hide abstract] ABSTRACT: In a previous paper, Gouveia and Magnanti (2003) found diameter-constrained minimal spanning and Steiner tree problems to
be more difficult to solve when the tree diameter D is odd. In this paper, we provide an alternate modeling approach that views problems with odd diameters as the superposition
of two problems with even diameters. We show how to tighten the resulting formulation to develop a model with a stronger linear
programming relaxation. The linear programming gaps for the tightened model are very small, typically less than 0.5–, and
are usually one third to one tenth of the gaps of the best previous model described in Gouveia and Magnanti (2003). Moreover,
the new model permits us to solve large Euclidean problem instances that are not solvable by prior approaches.
Annals of Operations Research 01/2006; 146:19-39. · 1.03 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The network design problem with connectivity requirements (NDC) models a wide variety of celebrated combinatorial optimization problems including the minimum span- ning tree, Steiner tree, and survivable network design problems. We develop strong for- mulations for two versions of the edge-connectivity NDC problem: unitary problems re- quiring connected network designs, and nonunitary problems permitting non-connected networks as solutions. We (i) present a new directed formulation for the unitary NDC problem that is stronger than a natural undirected formulation, (ii) project out several classes of valid inequalities—partition inequalities, odd-hole inequalities, and combi- natorial design inequalities—that generalize known classes of valid inequalities for the Steiner tree problem to the unitary NDC problem, and (iii) show how to strengthen and direct nonunitary problems. Our results provide a unifying framework for strengthening formulations for NDC problems, and demonstrate the strength and power of ßow-based formulations for net- work design problems with connectivity requirements.
[Show abstract][Hide abstract] ABSTRACT: We introduce a general adaptive line search framework for solving fixed point and variational inequality problems. Our goals are to develop iterative schemes that (i) compute solutions when the underlying map satisfies properties weaker than contractiveness, for example, weaker forms of nonexpansiveness, (ii) are more efficient than the classical methods even when the underlying map is contractive, and (iii) unify and extend several convergence results from the fixed point and variational inequality literatures. To achieve these goals, we introduce and study joint compatibility conditions imposed upon the underlying map and the iterative step sizes at each iteration and consider line searches that optimize certain potential functions. As a special case, we introduce a modified steepest descent method for solving systems of equations that does not require a previous condition from the literature (the square of the Jacobian matrix is positive definite). Since the line searches we propose might be difficult to perform exactly, we also consider inexact line searches.
[Show abstract][Hide abstract] ABSTRACT: We develop integer programming formulations and solution methods for addressing operational issues in merge-in-transit distribution systems. The models account for various complex problem features, including the integration of inventory and transportation decisions, the dynamic and multimodal components of the application, and the nonconvex piecewise linear structure of the cost functions. To accurately model the cost functions, we introduce disaggregation techniques that allow us to derive a hierarchy of linear programming relaxations. To solve these relaxations, we propose a cutting-plane procedure that combines constraint and variable generation with rounding and branch-and-bound heuristics. We demonstrate the effectiveness of this approach on a large set of test problems with instances derived from actual data from the computer industry that contain almost 500,000 integer variables.
[Show abstract][Hide abstract] ABSTRACT: The Diameter-Constrained Minimum Spanning Tree Problem seeks a least cost spanning tree subject to a (diameter) bound imposed on the number of edges in the tree between any node pair. A traditional multicommodity flow model with a commodity for every pair of nodes was unable to solve a 20-node and 100-edge problem after one week of computation. We formulate the problem as a directed tree from a selected central node or a selected central edge. Our model simultaneously finds a central node or a central edge and uses it as the source for the commodities in a directed multicommodity flow model with hop constraints. The new model has been able to solve the 20-node, 100-edge instance to optimality after less than four seconds. We also present model enhancements when the diameter bound is odd (these situations are more difficult). We show that the linear programming relaxation of the best formulations discussed in this paper always give an optimal integer solution for two special, polynomially-solvable cases of the problem. We also examine the Diameter Constrained Minimum Steiner Tree problem. We present computational experience in solving problem instances with up to 100 nodes and 1000 edges. The largest model contains more than 250,000 integer variables and more than 125,000 constraints.
[Show abstract][Hide abstract] ABSTRACT: We study a generic minimization problem with separable nonconvex piecewise linear costs, showing that the linear programming (LP) relaxation of three textbook mixed-integer programming formulations each approximates the cost function by its lower convex envelope. We also show a relationship between this result and classical Lagrangian duality theory.
[Show abstract][Hide abstract] ABSTRACT: gically ordered". page 80, line-15. "set of path and cycle flow" should be "set of path and cycle flows". page 80, line-15. "cycle fow" should be "set of path and cycle flows". page 80, line-7. "path and cycle flow" should be "a path and cycle flow". page 83, line 22. "cx = cx 0 + cx " should be "x = x 0 + x,,. page 85, line 1. "used" should be "use". page 85, line 13. "correspond" should be "corresponds". page 86, Exercise 3.9. line 4. "every pairs of elements" should be "every pair of elements". page 86, Exercise 3.9. line 5. "k = 1,..., n" should be "k = 1,..., (n-l)". page 86, Exercise 3.9. line 5. "k = 1,..., (n4)" should be "k = 1,..., (n-l)". page 90, Exercise 3.34. "the longest path in any depth-first search tree, no matter which node is selected as the source node" should be "the longest path in some depth-first search tree". page 92, Figure 3.15. (b) Arrows should be added: 1-->4, 2-->1, 2-->3, 3-->1, 3-->4, 4-->5, 5-->1, 5-->2. page 92, Exercise 3.51. line 1. "nodej ( i
[Show abstract][Hide abstract] ABSTRACT: The network restoration problem is a specialized capacitated network design problem requiring the installation of spare capacity to fully restore disrupted network flows if any edge in a telecommunications network fails. We present a new mixed-integer programming formulation for a line restoration version of the problem using a single type of capacitated facility. We examine two different models, for distinct and integrated spare-capacity systems, reflecting technologies used in synchronous transfer mode (STM) and asynchronous transfer mode (ATM) networks. The problem is NP-complete in the strong sense. We study the problem's polyhedral structure to identify strong valid inequalities that tighten the problem formulation. Our computational results on several real and randomly generated problems show that these inequalities considerably reduce the integrality gap from an average of 10% to an average of under 1%. These results indicate that strong cutting planes combined with branch-and-bound can provide efficient algorithms for solving a class of real-world problems in the telecommunications industry.
Operations Research 01/2002; 50:617-635. · 1.79 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: An intuitive solution-doubling argument establishes well known results concerning the worst-case performance of spanning tree-based heuristics for the Steiner network problem and the traveling salesman problem. This note shows that the solution-doubling argument and its implications apply to certain more general Low Connectivity Steiner (LCS) problems that are important in the design of survivable telecommunication networks. We use the doubling strategy to establish worst-case upper bounds on the value of tree-based heuristics relative to the optimal value for some versions of the LCS problem, and also provide a tight lower bound based on solutions to matching problems.
Operations Research Letters 01/2001; 29:99-106. · 0.52 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: To ensure uninterrupted service, telecommunication networks contain excess (spare) capacity for rerouting (restoring) traffic in the event of a link failure. We study the NP-hard capacity planning problem of economically installing spare capacity on a network to permit link restoration of steady-state traffic. We present a planning model that incorporates multiple facility types, and develop optimization-based heuristic solution methods based on solving a linear programming relaxation and minimum cost network flow subproblems. We establish bounds on the performance of the algorithms, and discuss problem instances that nearly achieve these worst-case bounds. In tests on three real-world problems and numerous randomly-generated problems containing up to 50 nodes and 150 edges, the heuristics provide good solutions (often within 0.5% of optimality) to problems with single facility type, in equivalent or less time than methods from the literature. For multi-facility problems, the gap between our heuristic solution values and the linear programming bounds are larger. However, for small graphs, we show that the optimal linear programming value does not provide a tight bound on the optimal integer value, and our heuristic solutions are closer to optimality than implied by the gaps.
Annals of Operations Research 01/2001; 106:127-154. · 1.03 Impact Factor