Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization

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.Comment: Full pape

Strongly Polynomial Primal-Dual Algorithms for Concave Cost Combinatorial Optimization Problems

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

A Linear Approximation Approach to Duality in Nonlinear Programming

Linear approximation and linear programming duality theory are used as unifying tools to develop saddlepoint, Fenchel and local duality theory. Among results presented is a new and elementary proof of the necessity and sufficiency of the stability condition for saddlepoint duality, an equivalence between the saddlepoint and Fenchel theories, and nasc for an optimal solution of an optimization problem to be a Kuhn-Tucker point. Several of the classic "constraint qualifications" are discussed with respect to this last condition. In addition, generalized versions of Fenchel and Rockafeller duals are introduced. Finally, a shortened proof is given of a result of Mangasarian and Fromowitz that under fairly general conditions an optimal point is also a Fritz John point.Supported in part by the US Army Research Office (Durham) under Contract DAHC04-70-C-005

Location Games and Bounds for Median Problems

We consider a two-person zero-sum game in which the maximizer selects a point in a given bounded planar region, the minimizer selects K points in that region,.and the payoff is the distance from the maximizer's location to the minimizer's location closest to it. In a variant of this game, the maximizer has the privilege of restricting the game to any subset of the given region. We evaluate/approximate the values (and the saddle point strategies) of these games for K = 1 as well as for K + , thus obtaining tight upper bounds (and worst possible demand distributions) for K-median problems

Network Flow Models for Designing Diameter-Constrained Minimum Spanning and Steiner Trees

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

Shortest Paths, Network Design and Associated Polyhedra

We study a specialized version of network design problems that arise in telecommunication, transportation and other industries. The problem, a generalization of the shortest path problem, is defined on an undirected network consisting of a set of arcs on which we can install (load), at a cost, a choice of up to three types of capacitated facilities. Our objective is to determine the configuration of facilities to load on each arc that will satisfy the demand of a single commodity at the lowest possible cost. Our results (i) demonstrate that the single-facility loading problem and certain "common breakeven point" versions of the two-facility and three-facility loading problems are polynomially solvable as a shortest path problem; (ii) show that versions of the twofacility loading problem are strongly NP-hard, but that a shortest path solution provides an asymptotically "good" heuristic; and (iii) characterize the optimal solution (that is, specify a linear programming formulation with integer solutions) of the common breakeven point versions of the two-facility and three-facility loading problems. In this development, we introduce two new families of facets, give geometric interpretations of our results, and demonstrate the usefulness of partitioning the space of the problem parameters to establish polyhedral integrality properties. Generalizations of our results apply to (i) multicommodity applications and (ii) situations with more than three facilities

Convergence Conditions for Variational Inequality Algorithms

Within the extensive variational inequality literature, researchers have developed many algorithms. Depending upon the problem setting, these algorithms ensure the convergence of (i) the entire sequence of iterates, (ii) a subsequence of the iterates, or (iii) averages of the iterates. To establish these convergence results, the literature repeatedly invokes several basic convergence theorems. In this paper, we review these theorems and a few convergence results they imply, and introduce a new result, called the orthogonality theorem, for establishing the convergence of several algorithms for solving a certain class of variational inequalities. Several of the convergence results impose a condition of strong-f-monotonicity on the problem function. We also provide a general overview of the properties of strong-f-monotonicity, including some new results (for example, the relationship between strong-f-monotonicity and convexity)

Strong Formulations for Network Design Problems with Connectivity Requirements

The network design problem with connectivity requirements (NDC) models a wide variety of celebrated combinatorial optimization problems including the minimum spanning tree, Steiner tree, and survivable network design problems. We develop strong formulations for two versions of the edge-connectivity NDC problem: unitary problems requiring 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 combinatorial 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 flow-based formulations for network design problems with connectivity requirements