Enhancing RLT-based relaxations for polynomial programming problems via a new class of v-semidefinite cuts

In this paper, we propose to enhance Reformulation-Linearization Technique (RLT)-based linear programming (LP) relaxations for polynomial programming problems by developing cutting plane strategies using concepts derived from semidefinite programming. Given an RLT relaxation, we impose positive semidefiniteness on suitable dyadic variable-product matrices, and correspondingly derive implied semidefinite cuts. In the case of polynomial programs, there are several possible variants for selecting such particular variable-product matrices on which positive semidefiniteness restrictions can be imposed in order to derive implied valid inequalities. This leads to a new class of cutting planes that we call v-semidefinite cuts. We explore various strategies for generating such cuts, and exhibit their relative effectiveness towards tightening the RLT relaxations and solving the underlying polynomial programming problems in conjunction with an RLT-based branch-and-cut scheme, using a test-bed of problems from the literature as well as randomly generated instances. Our results demonstrate that these cutting planes achieve a significant tightening of the lower bound in contrast with using RLT as a stand-alone approach, thereby enabling a more robust algorithm with an appreciable reduction in the overall computational effort, even in comparison with the commercial software BARON and the polynomial programming problem solver GloptiPoly

Optimal Allocation of Risk-Reduction Resources in Event Trees

In this paper, we present a novel quantitative analysis for the strategic planning decision problem of allocating certain available prevention and protection resources to, respectively, reduce the failure probabilities of system safety measures and the total expected loss from a sequence of events. Using an event tree optimization approach, the resulting risk-reduction scenario problem is modeled and then reformulated as a specially structured nonconvex factorable program. We derive a tight linear programming relaxation along with related theoretical insights that serve to lay the foundation for designing a tailored branch-and-bound algorithm that is proven to converge to a global optimum. Computational experience is reported for a hypothetical case study, as well as for several realistic simulated test cases, based on different parameter settings. The results on the simulated test cases demonstrate that the proposed approach dominates the commercial software BARON v7.5 when the latter is applied to solve the original model by more robustly yielding provable optimal solutions that are at an average of 16.6% better in terms of objective function value; and it performs competitively when both models are used to solve the reformulated problem, particularly for larger test instances.risk management, risk reduction, event trees, system safety, global optimization, factorable programming, branch-and-bound

A discrete optimization approach for locating Automatic Vehicle Identification readers for the provision of roadway travel times

This paper develops an algorithm for optimally locating Automatic Vehicle Identification tag readers by maximizing the benefit that would accrue from measuring travel times on a transportation network. The problem is formulated as a quadratic 0-1 optimization problem where the objective function parameters represent benefit factors that capture the relevance of measuring travel times as reflected by the demand and travel time variability along specified trips. An optimization approach based on the Reformulation-Linearization Technique coupled with semidefinite programming concepts is designed to solve the formulated reader location problem. To illustrate the proposed methodology, we consider a transportation network that is comprised of freeway segments that might include merge, diverge, weaving, and bottleneck sections. In order to derive benefit factors for the various origin-destination pairs on this network, we employ a simulation package (INTEGRATION) in combination with a composite function, which estimates the travel time variability along a trip that is comprised of links that include any of the four identified sections. The simulation results are actually recorded as generic look-up tables that can be used for any such section for the purpose of computing the associated benefit factor coefficients. Computational results are presented using data pertaining to a freeway section in San Antonio, Texas, as well as synthetic test cases, to demonstrate the effectiveness of the proposed approach, and to study the sensitivity of the quality of the solution to variations in the number of available readers.