A branch and bound algorithm for extreme point mathematical programming problems

This paper deals with a class of nonconvex mathematical programs called Extreme Point Mathematical Programs. This class is a generalization of zero-one integer programs and is a special case of the Generalized Lattice Point Problem, and finds applications in various areas such as production scheduling, load balancing, and concave programming. The current work existing on this class of problems is limited to certain dual types of extreme point ranking methods (which do not find a feasible solution until optimality) and some non-convergent cutting plane algorithms. No computational experience exists. This paper develops a finitely convergent branch and bound algorithm for solving the problem. The principles involved in the design of this algorithm are quite general and apply to a wider class of mathematical programs including the Generalized Lattice Point Problem. A random problem generator is described which is capable of generating problems of varying levels of difficulty. Computational experience on such problems is provided.     

A Global Optimization Approach to a Water Distribution Network Design Problem

In this paper, we address a global optimization approach to a waterdistribution network design problem. Traditionally, a variety of localoptimization schemes have been developed for such problems, each new methoddiscovering improved solutions for some standard test problems, with noknown lower bound to test the quality of the solutions obtained. A notableexception is a recent paper by Eiger et al. (1994) who present a firstglobal optimization approach for a loop and path-based formulation of thisproblem, using a semi-infinite linear program to derive lower bounds. Incontrast, we employ an arc-based formulation that is linear except forcertain complicating head-loss constraints and develop a first globaloptimization scheme for this model. Our lower bounds are derived through thedesign of a suitable Reformulation-Linearization Technique (RLT) thatconstructs a tight linear programming relaxation for the given problem, andthis is embedded within a branch-and-bound algorithm. Convergence to anoptimal solution is induced by coordinating this process with an appropriatepartitioning scheme. Some preliminary computational experience is providedon two versions of a particular standard test problem for the literature forwhich an even further improved solution is discovered, but one that isverified for the first time to be an optimum, without any assumed boundson the flows. Two other variants of this problem are also solved exactly forillustrative purposes and to provide researchers with additional test caseshaving known optimal solutions. Suggestions on a more elaborate study involving several algorithmic enhancements are presented for futureresearch.     

On the Solution of the Inverse Eigenvalue Complementarity Problem

In this paper, we discuss the solution of an Inverse Eigenvalue Complementarity Problem. Two nonlinear formulations are presented for this problem. A necessary and sufficient condition for a stationary point of the first of these formulations to be a solution of the problem is established. On the other hand, to assure global convergence to a solution of this problem when it exists, an enumerative algorithm is designed by exploiting the structure of the second formulation. The use of additional implied constraints for enhancing the efficiency of the algorithm is also discussed. Computational results are provided to highlight the performance of the algorithm.     

An Inverse Reliability-based Approach for Designing under Uncertainty with Application to Robust Piston Design

In this work, we propose an optimization framework for designing under uncertainty that considers both robustness and reliability issues. This approach is generic enough to be applicable to engineering design problems involving nonconvex objective and constraint functions defined in terms of random variables that follow any distribution. The problem formulation employs an Inverse Reliability Strategy that uses percentile performance to address both robustness objectives and reliability constraints. Robustness is achieved through a design objective that evaluates performance variation as a percentile difference between the right and left trails of the specified goals. Reliability requirements are formulated as Inverse Reliability constraints that are based on equivalent percentile performance levels. The general proposed approach first approximates the formulated problem via a Gaussian Kriging model. This is then used to evaluate the percentile performance characteristics of the different measures inherent in the problem formulation for various design variable settings via a Most Probable Point of Inverse Reliability search algorithm. By using these percentile evaluations in concert with the response surface methodology, a polynomial programming approximation is generated. The resulting problem formulation is finally solved to global optimality using the Reformulation–Linearization Technique (RLT) approach. We demonstrate this overall proposed approach by applying it to solve the problem of reducing piston slap, an undesirable engine noise due to the secondary motion of a piston within a cylinder.     

On an enumerative algorithm for solving eigenvalue complementarity problems

In this paper, we discuss the solution of linear and quadratic eigenvalue complementarity problems (EiCPs) using an enumerative algorithm of the type introduced by Júdice et al. (Optim. Methods Softw. 24:549–586, 2009). Procedures for computing the interval that contains all the eigenvalues of the linear EiCP are first presented. A nonlinear programming (NLP) model for the quadratic EiCP is formulated next, and a necessary and sufficient condition for a stationary point of the NLP to be a solution of the quadratic EiCP is established. An extension of the enumerative algorithm for the quadratic EiCP is also developed, which solves this problem by computing a global minimum for the NLP formulation. Some computational experience is presented to highlight the efficiency and efficacy of the proposed enumerative algorithm for solving linear and quadratic EiCPs.     

Models and algorithms for the scheduling of a doubles tennis training tournament

We address a doubles tennis scheduling problem in the context of a training tournament, and develop a 0–1 mixed-integer programming model that attempts to balance the partnership and the opponentship pairings among the players. We propose effective symmetry-defeating strategies that impose certain decision hierarchies within the model, which serve to significantly enhance algorithmic performance via their pruning effect. We also discuss the concept of symmetry compatible formulations, and highlight the importance of crafting formulations in discrete optimization in a fashion that enhances the interplay between the original model structure, branch-and-bound algorithms (as implemented in commercial packages such as CPLEX), and the structure of specific symmetry-defeating hierarchical constraints. Finally, various specialized heuristics are devised and are computationally evaluated along with the exact solution schemes using a set of realistic practical test problems.     

On the choice of step size in subgradient optimization

This paper recommends some procedures for the selection of step sizes in the context of subgradient optimization. The first of these procedures is developed in detail in this study and is a theoretically convergent scheme. This method has two phases, the first phase is designed to accelerate the solution procedure towards an optimal solution, while the second phase helps to close in on an optimal solution. A second technique recommended is a simple-minded scheme which, although not theoretically convergent, seems to be computationally very efficient. These two methods are shown to compare favorably with Held, Wolfe and Crowder's scheme for prescribing step sizes. We also suggest some modifications of the latter scheme to make it computationally more efficient.