Evrim Dalkiran

Reduced RLT representations for nonconvex polynomial programming problems

This paper explores equivalent, reduced size Reformulation-Linearization Technique (RLT)-based formulations for polynomial programming problems. Utilizing a basis partitioning scheme for an embedded linear equality subsystem, we show that a strict subset of RLT defining equalities imply the remaining ones. Applying this result, we derive significantly reduced RLT representations and develop certain coherent associated branching rules that assure convergence to a global optimum, along with static as well as dynamic basis selection strategies to implement the proposed procedure. In addition, we enhance the RLT relaxations with v-semidefinite cuts, which are empirically shown to further improve the relative performance of the reduced RLT method over the usual RLT approach. We present computational results for randomly generated instances to test the different proposed reduction strategies and to demonstrate the improvement in overall computational effort when such reduced RLT mechanisms are employed.

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.

Theoretical filtering of RLT bound-factor constraints for solving polynomial programming problems to global optimality

In this paper, we propose two sets of theoretically filtered bound-factor constraints for constructing reformulation-linearization technique (RLT)-based linear programming (LP) relaxations for solving polynomial programming problems. We establish related theoretical results for convergence to a global optimum for these reduced sized relaxations, and provide insights into their relative sizes and tightness. Extensive computational results are provided to demonstrate the relative effectiveness of the proposed theoretical filtering strategies in comparison to the standard RLT and a prior heuristic filtering technique using problems from the literature as well as randomly generated test cases.

Selecting Optimal Alternatives and Risk Reduction Strategies in Decision Trees

In this paper we conduct a quantitative analysis for a strategic risk management problem that involves allocating certain available failure-mitigating and consequence-alleviating resources to reduce the failure probabilities of system safety components and subsequent losses, respectively, together with selecting optimal strategic decision alternatives, to minimize the risk or expected loss in the event of a hazardous occurrence. Using a novel decision tree optimization approach to represent the cascading sequences of probabilistic events as controlled by key decisions and investment alternatives, the problem is modeled as a nonconvex mixed-integer 0-1 factorable program. We develop a specialized branch-and-bound algorithm in which lower bounds are computed via tight linear relaxations of the original problem that are constructed by utilizing a polyhedral outer-approximation mechanism in concert with two alternative linearization schemes having different levels of tightness and complexity. We also suggest three alternative branching schemes, each of which is proven to guarantee convergence to a global optimum for the underlying problem. Extensive computational results and sensitivity analyses are presented to provide insights and to demonstrate the efficacy of the proposed algorithm.

Combined bound-grid-factor constraints for enhancing RLT relaxations for polynomial programs

This paper studies the global optimization of polynomial programming problems using Reformulation-Linearization Technique (RLT)-based linear programming (LP) relaxations. We introduce a new class of bound-grid-factor constraints that can be judiciously used to augment the basic RLT relaxations in order to improve the quality of lower bounds and enhance the performance of global branch-and-bound algorithms. Certain theoretical properties are established that shed light on the effect of these valid inequalities in driving the discrepancies between RLT variables and their associated nonlinear products to zero. To preserve computational expediency while promoting efficiency, we propose certain concurrent and sequential cut generation routines and various grid-factor selection rules. The results indicate a significant tightening of lower bounds, which yields an overall reduction in computational effort for solving a test-bed of polynomial programming problems to global optimality in comparison with the basic RLT procedure as well as the commercial software BARON.

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.

Optimization of strategic planning processes for configurable products

Assortment planning aims to select the set of products that a retailer or manufacturer will offer to its customers to maximize profitability. While assortment planning research has been expanding in recent years, current models are inadequate for the needs of a configurable product manufacturer. In this paper, we develop models integrating assortment planning and supply chain management decisions for the strategic planning of a large automaker. Our model utilizes a multinomial logit choice model transformed into a mixed-integer linear program through the Charnes–Cooper transformation. It is able to scale to problems that contain thousands of configurations to possibly be offered, a necessity given the number of possible configurations an automaker can build. In addition, most research in assortment planning contains simplified costs associated with product complexity. We better account for design, integration, manufacturing, and supply chain complexities that stem from large product assortments. We believe that our model can significantly aid automotive manufacturers to balance their product complexity with supply chain complexity to improve profitability of vehicle programs. We also present results from a case study motivated by a large global automotive original equipment manufacturer.

On linear programming relaxations for solving polynomial programming problems

Abstract This paper studies linear programming (LP) relaxations for solving polynomial programming problems. A polynomial programming problem can be equivalently formulated as a quadratically constrained quadratic program (QCQP) by introducing new variables that represent nonlinear monomials and substituting them within the original formulation. Whereas Reformulation-Linearization Technique (RLT)-based LP relaxations constructed for the original formulation are tight, the relaxations generated using the equivalent lower degree formulations are smaller and require less computational effort to optimize in comparison to the effort the former requires. In this study, we analyze the strength and tractability of the standard RLT, the J-set, and recursive McCormick relaxations for polynomial programming problems, and identify the superior relaxation depending on the problem characteristics. Extensive computational results are provided to demonstrate the relative effectiveness of the standard RLT, J-set, and recursive McCormick algorithms using problems from the literature and randomly generated test instances.

Promoting sustainability of automotive products through strategic assortment planning

Abstract Assortment planning seeks to find an optimal set of products that the company should offer to its customers. Traditionally, it is a trade-off between offering larger assortments to maximize customer choice vs. smaller assortments to minimize costs associated with design, manufacturing, and distribution. Existing assortment planning models are quite lacking when it comes to configurable products such as automobiles and detail level of supply chain considerations. Further complications stem from increasingly strict environmental regulations and broader expectations for sustainable products and supply chains. We present a mixed-integer linear programming formulation for integrated assortment and supply chain network design models for automotive products to provide effective decision support and directional guidance to strategic product planners. Our models account for product use and supply chain emissions as well as fuel efficiency requirements. We also present an illustrative case study motivated by a global automaker to demonstrate the utility of the models and study the effects of sustainability requirements on the assortment and supply chain design.