The prize collecting Steiner tree problem: models and Lagrangian dual optimization approaches

We propose a generalized version of the Prize Collecting Steiner Tree Problem (PCSTP), which offers a fundamental unifying model for several well-known -hard tree optimization problems. The PCSTP also arises naturally in a variety of network design applications including cable television and local access networks. We reformulate the PCSTP as a minimum spanning tree problem with additional packing and knapsack constraints and we explore various nondifferentiable optimization algorithms for solving its Lagrangian dual. We report computational results for nine variants of deflected subgradient strategies, the volume algorithm (VA), and the variable target value method used in conjunction with the VA and with a generalized Polyak–Kelley cutting plane technique. The performance of these approaches is also compared with an exact stabilized constraint generation procedure.     

Network interdiction to minimize the maximum probability of evasion with synergy between applied resources

In this paper, we model and solve the network interdiction problem of minimizing the maximum probability of evasion by an entity traversing a network from a given source to a designated terminus, while incorporating novel forms of superadditive synergy between resources applied to arcs in the network. Inspired primarily by operations to coordinate Iraqi and U.S. security forces seeking to interdict an evader attempting to avoid detection while transiting part of the nearly rectilinear street network in East Baghdad, this study motivates and examines either linear or concave (nonlinear) synergy relationships between the applied resources within our formulations. We also propose an alternative model for sequential overt and covert deployment of subsets of interdiction resources, and conduct theoretical as well as empirical comparative analyses between models for purely overt (with or without synergy) and composite overt-covert strategies to provide insights into absolute and relative threshold criteria for recommended resource utilization. Our empirical results confirm the value of tactical patience regarding decisions on the covert utilization of resources for network interdiction. Furthermore, considering non-integral and integral resource allocations, we identify (theoretically and empirically) parametric characteristics of instances that exhibit the relative worth of employing partially covert operations. Under the relatively more practical scenario involving integral resource allocations, we demonstrate that the composite overt-covert strategy of deploying resources has a greater potential to improve over a purely overt resource deployment strategy, both with and without synergy, particularly when costs are positively correlated, resources are plentiful, and a sufficiently high ratio of covert to overt resources exists. Moreover, should an interdictor be able to ascertain an optimal evader path, the potential and magnitude of this relative improvement for the overt-covert resource allocation strategy is significantly greater.     

Optimal base station selection for anycast routing in wireless sensor networks

Energy constraints have a significant impact on the design and operation of wireless sensor networks. This paper investigates the base station (BS) selection (or anycast) problem in wireless sensor networks. A wireless sensor network having multiple BSs (data sink nodes) is considered. Each source node must send all its locally generated data to only one of the BSs. To maximize network lifetime, it is essential to optimally match each source node to a particular BS and find an optimal routing solution. A polynomial time heuristic is proposed for optimal BS selection and anycast via a sequential fixing procedure. Through extensive simulation results, it is shown that this algorithm has excellent performance behavior and provides a near-optimal solution.     

RLT-POS: Reformulation-Linearization Technique-based optimization software for solving polynomial programming problems

In this paper, we introduce a Reformulation-Linearization Technique-based open-source optimization software for solving polynomial programming problems (RLT-POS). We present algorithms and mechanisms that form the backbone of RLT-POS, including constraint filtering techniques, reduced RLT representations, and semidefinite cuts. When implemented individually, each model enhancement has been shown in previous papers to significantly improve the performance of the standard RLT procedure. However, the coordination between different model enhancement techniques becomes critical for an improved overall performance since special structures in the original formulation that work in favor of a particular technique might be lost after implementing some other model enhancement. More specifically, we discuss the coordination between (1) constraint elimination via filtering techniques and reduced RLT representations, and (2) semidefinite cuts for sparse problems. We present computational results using instances from the literature as well as randomly generated problems to demonstrate the improvement over a standard RLT implementation and to compare the performances of the software packages BARON, COUENNE, and SparsePOP with RLT-POS.     

A Pseudo-Global Optimization Approach with Application to the Design of Containerships

In this paper, we present a new class of pseudo-global optimization procedures for solving formidable optimization problems in which the objective and/or constraints might be analytically complex and expensive to evaluate, or available only as black-box functions. The proposed approach employs a sequence of polynomial programming approximations that are constructed using the Response Surface Methodology (RSM), and embeds these within a branch-and-bound framework in concert with a suitable global optimization technique. The lower bounds constructed in this process might only be heuristic in nature, and hence, this is called a pseudo-global optimization approach. We develop two such procedures, each employing two alternative branching techniques, and apply these methods to the problem of designing containerships. The model involves five design variables given by the design draft, the depth at side, the speed, the overall length, and the maximum beam. The constraints imposed enforce the balance between the weight and the displacement, a required acceptable length to depth ratio, a restriction on the metacentric height to ensure that the design satisfies the Coast Guard wind heel criterion, a minimum freeboard level as governed by the code of federal regulations (46 CFR 42), and a lower bound on the rolling period to ensure sea-worthiness. The objective function seeks to minimize the required freight rate that is induced by the design in order to recover capital and operating costs, expressed in dollars per metric ton per nautical mile. The model formulation also accommodates various practical issues in improving the representation of the foregoing considerations, and turns out to be highly nonlinear and nonconvex. A practical test case is solved using the proposed methodology, and the results obtained are compared with those derived using a contemporary commercialized design optimization tool. The prescribed solution yields an improved design that translates to an estimated increase in profits of about $18.45 million, and an estimated 27% increase in the return on investment, over the life of the ship.     

Enumeration Approach for Linear Complementarity Problems Based on a Reformulation-Linearization Technique

In this paper, we consider the linear complementarity problem (LCP) and present a global optimization algorithm based on an application of the reformulation-linearization technique (RLT). The matrix M associated with the LCP is not assumed to possess any special structure. In this approach, the LCP is formulated first as a mixed-integer 0–1 bilinear programming problem. The RLT scheme is then used to derive a new equivalent mixed-integer linear programming formulation of the LCP. An implicit enumeration scheme is developed that uses Lagrangian relaxation, strongest surrogate and strengthened cutting planes, and a heuristic, designed to exploit the strength of the resulting linearization. Computational experience on various test problems is presented.     

Enhanced intersection cutting-plane approach for linear complementarity problems

In this paper, we develop an enhanced intersection cutting-plane algorithm for solving a mixed integer 0–1 bilinear programming formulation of the linear complementarity problem (LCP). The matrixM associated with the LCP is not assumed to possess any special structure, except that the corresponding feasible region is assumed to be bounded. A procedure is described to generate cuts that are deeper versions of the Tuy intersection cuts, based on a relaxation of the usual polar set. The proposed algorithm then attempts to find an LCP solution in the process of generating either a single or a pair of such strengthened intersection cuts. The process of generating these cuts involves a vertexranking scheme that either finds an LCP solution, or else these cuts eliminate the entire feasible region leading to the conclusion that no LCP solution exists. Computational experience on various test problems is provided.This material is based upon work supported by the National Science Foundation under Grant No. DMII-9121419 to the first author and Grant No. DMII-9114489 to the third author. The authors gratefully acknowledge the constructive suggestions of a referee that helped focus the approach and its presentation.