A new algorithm for minimizing convex functions over convex sets

Let be a convex set for which there is an oracle with the following property. Given any pointz∈ℝ ⁿ the oracle returns a “Yes” ifz∈S; whereas ifz∉S then the oracle returns a “No” together with a hyperplane that separatesz fromS. The feasibility problem is the problem of finding a point inS; the convex optimization problem is the problem of minimizing a convex function overS. We present a new algorithm for the feasibility problem. The notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm. Our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm. The algorithm for the feasibility problem easily adapts to the convex optimization problem.     

Benders's method revisited

The master problem in Benders's partitioning method is an integer program with a very large number of constraints, each of which is usually generated by solving the integer program with the constraints generated earlier. Computational experience shows that the subset B of those constraints of the master problem that are satisfied with equality at the linear programming optimum often play a crucial role in determining the integer optimum, in the sense that only a few of the remaining inequalities are needed. We characterize this subset B of inequalities. If an optimal basic solution to the linear program is nondegenerate in the continuous variables and has p integer-constrained basic variables, then the corresponding set B contains at most 2^p inequalities, none of which is implied by the others. We give an efficient procedure for generating an appropriate subset of the inequalities in B, which leads to a considerably improved version of Benders's method.     

Approximation Algorithm for MAX DICUT with Given Sizes of Parts

Given a directed graph G and an edge weight function w : A(G)→ R^ , the maximum directed cut problem (MAX DICUT) is that of finding a directed cut δ(S) with maximum total weight. We consider a version of MAX DICUT -- MAX DICUT with given sizes of parts or MAX DICUT WITH GSP -- whose instance is that of MAX DICUT plus a positive integer k, and it is required to find a directed cut δ(S) having maximum weight over all cuts δ(S) with |S| -- k. We present an approximation algorithm for this problem which is based on semidefinite programming (SDP) relaxation. The algorithm achieves the presently best performance guarantee for a range of k.     

非负约束条件下组合证券投资决策的分枝定界法

研究非负约束条件下 ,实现预期收益率的组合证券投资决策问题 ,将整数线性规划的分枝定界法用于该问题的求解 ,并应用于一个四元证券投资决策问题     

Polyhedral studies on the convex recoloring problem

A coloring of the vertices of a graph G is convex if, for each assigned color d, the vertices with color d induce a connected subgraph of G. We address the convex recoloring problem, defined as follows. Given a graph G and a coloring of its vertices, recolor a minimum number of vertices of G, so that the resulting coloring is convex. This problem is known to be NP-hard even when G is a path. We show an integer programming formulation for the weighted version of this problem on arbitrary graphs, and then specialize it for trees. We study the facial structure of the polytope defined as the convex hull of the integer points satisfying the restrictions of the proposed ILP formulation, present several classes of facet-defining inequalities and discuss separation algorithms.     

A Multi-Stage Stochastic Integer Programming Approach for Capacity Expansion under Uncertainty

This paper addresses a multi-period investment model for capacity expansion in an uncertain environment. Using a scenario tree approach to model the evolution of uncertain demand and cost parameters, and fixed-charge cost functions to model the economies of scale in expansion costs, we develop a multi-stage stochastic integer programming formulation for the problem. A reformulation of the problem is proposed using variable disaggregation to exploit the lot-sizing substructure of the problem. The reformulation significantly reduces the LP relaxation gap of this large scale integer program. A heuristic scheme is presented to perturb the LP relaxation solutions to produce good quality integer solutions. Finally, we outline a branch and bound algorithm that makes use of the reformulation strategy as a lower bounding scheme, and the heuristic as an upper bounding scheme, to solve the problem to global optimality. Our preliminary computational results indicate that the proposed strategy has significant advantages over straightforward use of commercial solvers.     

Multi-objective integer programming: A general approach for generating all non-dominated solutions

In this paper we develop a general approach to generate all non-dominated solutions of the multi-objective integer programming (MOIP) Problem. Our approach, which is based on the identification of objective efficiency ranges, is an improvement over classical ε-constraint method. Objective efficiency ranges are identified by solving simpler MOIP problems with fewer objectives. We first provide the classical ε-constraint method on the bi-objective integer programming problem for the sake of completeness and comment on its efficiency. Then present our method on tri-objective integer programming problem and then extend it to the general MOIP problem with k objectives. A numerical example considering tri-objective assignment problem is also provided.     

An approximation algorithm for the k-median warehouse-retailer network design problem

We study the generalizedk-median version of the warehouse-retailer network design problem(kWRND).We formulate the k-WRND as a binary integer program and propose a 6-approximation randomized algorithm based on Lagrangian relaxation.     

Integer programming formulation of combinatorial optimization problems

This paper considers in a somewhat general setting when a combinatorial optimization problem can be formulated as an (all-integer) integer programming (IP) problem. The main result is that any combinatorial optimization problem can be formulated as an IP problem if its feasible region S is finite but there are many rather sample problems that have no IP formulation if their S is infinite. The approach used for finite S usually gives a formulation with a relatively small number of additional variables for example, an integer polynomial of n 0?1 variables requires at most n + 1 additional variables by our approach, whereas 2ⁿ - (n + 1) additional variables at maximum are required by other existing methods. Finally, the decision problem of deciding whether an arbitrarily given combinatorial optimization problem has an IP formulation is considered and it is shown by an argument closely related to Hilbert's tenth problem (drophantine equations) that no such algorithm exists.     

On the complexity of a family of generalized matching problems

We consider a family of generalized matching problems called k-feasible matching (k-RM) problems, where k? {1,2,3,…} ∪ {∞}. We show each k-FM problem to be NP-complete even for very restricted cases. We develop a dynamic programming algorithm that solves in polynomial time the k-FM problem for graphs with width bounded by 2k. We also show that for any subset S of {1,2,…} ∪ {∞}, there is a set D of problem instances such that for k in S the k-FM problem is NP-complete on D, while for k not in S the k-FM problem is polynomially solvable on D.     

Solving a minimum-power covering problem with overlap constraint for cellular network design

We consider a type of covering problem in cellular networks. Given the locations of base stations, the problem amounts to determining cell coverage at minimum cost in terms of the power usage. Overlap between adjacent cells is required in order to support handover. The problem we consider is NP-hard. We present integer linear models and study the strengths of their continuous relaxations. Preprocessing is used to reduce problem size and tighten the models. Moreover, we design a tabu search algorithm for finding near-optimal solutions effectively and time-efficiently. We report computational results for both synthesized instances and networks originating from real planning scenarios. The results show that one of the integer models leads to tight bounds, and the tabu search algorithm generates high-quality solutions for large instances in short computing time.     

Discrete allocation of a linear resource according to a sum-of-products objective

The allocation of a linear resource according to the sum of the returns from independent activities is considered. The return from each activity is given by a product of concave and nondecreasing functions of a single allocation variable. The model can be used, for instance, to describe probabilities of success of several serial tasks, into which an activity is subdivided. An incremental algorithm is defined and conditions are given for the algorithm to generate an optimal solution; otherwise, the problem is solved by a two-step procedure involving the incremental maximization of the return corresponding to a single activity and the combination of the activities by dynamic programming. Examples are given of problems solvable and not solvable by the incremental algorithm.     

The Erd?s-Pósa property for vertex- and edge-disjoint odd cycles in graphs on orientable surfaces

We prove that for any orientable surface S and any non-negative integer k, there exists an integer f_S(k) such that every graph G embeddable in S has either k vertex-disjoint odd cycles or a vertex set A of cardinality at most f_S(k) such that G-A is bipartite. Such a property is called the Erd?s-Pósa property for odd cycles. We also show its edge version. As Reed [Mangoes and blueberries, Combinatorica 19 (1999) 267-296] pointed out, the Erd?s-Pósa property for odd cycles do not hold for all non-orientable surfaces.     

A branch-and-cut algorithm for the capacitated open vehicle routing problem

In open vehicle routing problems, the vehicles are not required to return to the depot after completing service. In this paper, we present the first exact optimization algorithm for the open version of the well-known capacitated vehicle routing problem (CVRP). The algorithm is based on branch-and-cut. We show that, even though the open CVRP initially looks like a minor variation of the standard CVRP, the integer programming formulation and cutting planes need to be modified in subtle ways. Computational results are given for several standard test instances, which enables us for the first time to assess the quality of existing heuristic methods, and to compare the relative difficulty of open and closed versions of the same problem.     

Reducing the number of variables in integer quadratic programming problem

In this paper, a new variable reduction technique is presented for general integer quadratic programming problem (GP), under which some variables of (GP) can be fixed at zero without sacrificing optimality. A sufficient condition and a necessary condition for the identification of dominated terms are provided. By comparing the given data of the problem and the upper bound of the variables, if they meet certain conditions, some variables can be fixed at zero. We report a computational study to demonstrate the efficacy of the proposed technique in solving general integer quadratic programming problems. Furthermore, we discuss separable integer quadratic programming problems in a simpler and clearer form.     

A new lower bound for the linear knapsack problem with general integer variables

It is well known that the linear knapsack problem with general integer variables (LKP) is NP-hard. In this paper we first introduce a special case of this problem and develop an O(n) algorithm to solve it. We then show how this algorithm can be used efficiently to obtain a lower bound for a general instance of LKP and prove that it is at least as good as the linear programming lower bound. We also present the results of a computational study that show that for certain classes of problems the proposed bound on average is tighter than other bounds proposed in the literature.     

Resource assignment with preference conditions

This paper deals with a modification of the standard assignment problem, where subsets of resources express preferences in being, or not being, assigned together to the same activity. The problem arises in several real settings, among which the job assignment of the crew personnel of an airline company. We provide an integer programming formulation for both the Split Preference Problem, where couples of assignees do not want to work together, and for the Join Preference Problem, where, oppositely, couples of assignees want to work together. The mathematical nature of the two problems is indeed different, as for the first one it is possible to determine a minimum cost flow formulation on a suitable graph, and thus a polynomial time algorithm, while for the second one we face a NP-hard problem and device some heuristic solution approaches. Experimental tests conducted on instances of variable size confirm the effectiveness of the models and of the algorithms proposed.     

A primal-dual interior point method whose running time depends only on the constraint matrix

We propose a primal-dual “layered-step” interior point (LIP) algorithm for linear programming with data given by real numbers. This algorithm follows the central path, either with short steps or with a new type of step called a “layered least squares” (LLS) step. The algorithm returns an exact optimum after a finite number of steps—in particular, after O(n ^3.5 c(A)) iterations, wherec(A) is a function of the coefficient matrix. The LLS steps can be thought of as accelerating a classical path-following interior point method. One consequence of the new method is a new characterization of the central path: we show that it composed of at mostn ² alternating straight and curved segments. If the LIP algorithm is applied to integer data, we get as another corollary a new proof of a well-known theorem by Tardos that linear programming can be solved in strongly polynomial time provided thatA contains small-integer entries.     

The time constrained maximal covering salesman problem

We introduce the time constrained maximal covering salesman problem (TCMCSP) which is the generalization of the covering salesman and orienting problems. In this problem, we are given a set of vertices including a central depot, customer and facility vertices where each facility can supply the demand of some customers within its pre-determined coverage distance. Starting from the depot, the goal is to maximize the total number of covered customers by constructing a length constrained Hamiltonian cycle over a subset of facilities. We propose several mathematical programming models for the studied problem followed by a heuristic algorithm. The developed algorithm takes advantage of different procedures including swap, deletion, extraction-insertion and perturbation. Finally, an integer linear programming based improvement technique is designed to try to improve the quality of the solutions. Extensive computational experiments on a set of randomly generated instances indicate the effectiveness of the algorithm.