On the existence of efficient points

In this note we will study the existence ofefficient points of a set X with respect to a coneC by introducing the so-called quasi-C-bounded sets; relationships among C-compactness, C-boundedness and quasi-C-boundedness are studied in a systematic way, and new existence theorems are obtained which generalize some known results and which allow us to find existence conditions for vector optimization problem.     

A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem

In this article, we present and validate a simplicial branch and bound duality-bounds algorithm for globally solving the linear sum-of-ratios fractional program. The algorithm computes the lower bounds called for during the branch and bound search by solving ordinary linear programming problems. These problems are derived by using Lagrangian duality theory. The algorithm applies to a wide class of linear sum-of-ratios fractional programs. Two sample problems are solved, and the potential practical and computational advantages of the algorithm are indicated.     

Solving Large Quadratic Assignment Problems in Parallel

Quadratic Assignment problems are in practice among the mostdifficult to solve in the class of NP-complete problems. Theonly successful approach hitherto has been Branch-and-Bound-basedalgorithms, but such algorithms are crucially dependent on good boundfunctions to limit the size of the space searched. Much work hasbeen done to identify such functions for the QAP, but with limitedsuccess.Parallel processing has also been used in order to increase the sizeof problems solvable to optimality. The systems used have, however, oftenbeen systems with relatively few, but very powerful vector processors, andhave hence not been ideally suited for computations essentially involving non-vectorizable computations on integers.In this paper we investigate the combination of one of the best bound functions for a Branch-and-Bound algorithm (the Gilmore-Lawler bound) and various testing, variable binding and recalculation of bounds between branchings when used in aparallel Branch-and-Bound algorithm. The algorithm has been implemented on a 16-processor MEIKO Computing Surface with Intel i860processors. Computational results from the solution of a number of large QAPs, including the classical Nugent 20 are reported.     

Optimization over equilibrium sets ∗

We introduce a certain notion of equilibrium points, which constitute a generalization of Pareto efficient points, and we propose a branch-and-bound method to maximize a concave function over the set of all equilibrium points of a given set     

Theory and algorithms for linear multiple objective programs with zero–one variables

A new algorithm and theoretical results are presented for linear multiple objective programs with zero–one variables. A procedure to identify strong and weak efficient points as well as an extension of the main problem are analyzed. Extensive computational results are given and several topics for further research are discussed.     

Solving linear fractional bilevel programs

In this paper, we prove that an optimal solution to the linear fractional bilevel programming problem occurs at a boundary feasible extreme point. Hence, the Kth-best algorithm can be proposed to solve the problem. This property also applies to quasiconcave bilevel problems provided that the first level objective function is explicitly quasimonotonic.     

Global optimization for sum of geometric fractional functions

This paper presents an efficient branch and bound algorithm for globally solving sum of geometric fractional functions under geometric constraints, which arise in various practical problems. By using an equivalent transformation and a new linear relaxation technique, a linear relaxation programming problem of the equivalent problem is obtained. The proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems. Numerical results are reported to show the feasibility of our algorithm.     

A branch-reduce-cut algorithm for the global optimization of probabilistically constrained linear programs

We consider probabilistically constrained linear programs with general distributions for the uncertain parameters. These problems involve non-convex feasible sets. We develop a branch-and-bound algorithm that searches for a global optimal solution to this problem by successively partitioning the non-convex feasible region and by using bounds on the objective function to fathom inferior partition elements. This basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the partition elements and hence tighten bounds. The proposed branch-reduce-cut algorithm exploits the monotonicity properties inherent in the problem, and requires solving linear programming subproblems. We provide convergence proofs for the algorithm. Some illustrative numerical results involving problems with discrete distributions are presented.     

Finding all efficient extreme points for multiple objective linear programs

In this paper we develop a method for finding all efficient extreme points for multiple objective linear programs. Simple characterizations of the efficiency of an edge incident to a nondegenerate or a degenerate efficient vertex are given. These characterizations form the basis of an algorithm for enumerating all efficient vertices. The algorithm appears to have definite computational advantages over other methods. Some illustrative examples are included.     

Two-stage integer programs with stochastic right-hand sides: a superadditive dual approach

We consider two-stage pure integer programs with discretely distributed stochastic right-hand sides. We present an equivalent superadditive dual formulation that uses the value functions in both stages. We give two algorithms for finding the value functions. To solve the reformulation after obtaining the value functions, we develop a global branch-and-bound approach and a level-set approach to find an optimal tender. We show that our method can solve randomly generated instances whose extensive forms are several orders of magnitude larger than the extensive forms of those instances found in the literature. This work is supported by National Science Foundation grants DMI-0217190 and DMI-0355433.     

求线性比式和问题全局解的输出空间分枝定界算法

基于对p-1维输出空间进行剖分的思想,提出了一种求解线性比式和问题的分枝定界算法.通过一种两阶段转换方法得到原问题的一个等价问题,该问题的非凸性主要体现在新增加的p-1个非线性等式约束上.利用双线性函数的凹凸包络对这些非线性约束进行凸化,这就为等价问题构造了凸松弛子问题.将凸松弛子问题中的冗余约束去掉并进行等价转换,从而获得了一个比凸松弛子问题规模更小、约束更少的线性规划问题.证明了算法的理论收敛性和计算复杂性.数值实验表明该算法是有效可行的.     

Solving staircase linear programs by the simplex method, 2: Pricing

This and a companion paper consider how current implementations of the simplex method may be adapted to better solve linear programs that have a staged, or ‘staircase’, structure. The preceding paper considered ‘inversion’ routines that factorize the basis and solve linear systems. The present paper examines ‘pricing’ routines that compute reduced costs for nonbasic variables and that select a variable to enter the basis at each iteration. Both papers describe extensive (although preliminary) computer experiments, and can point to some quite promising results. For pricing in particular, staircase computation strategies appear to offer modest but consistent savings; staircase selection strategies, properly chosen, may offer substantial savings in number of iterations, time per iteration, or both.     

Solving staircase linear programs by the simplex method, 1: Inversion

This and a companion paper consider how current implementations of the simplex method may be adapted to better solve linear programs that have a staged, or staircase, structure. The present paper looks at inversion routines within the simplex method, particularly those for sparse triangular factorization of a basis by Gaussian elimination and for solution of triangular linear systems. The succeeding paper examines pricing routines. Both papers describe extensive (though preliminary) computational experience, and can point to some quite promising results.     

A statistical model for global optimization by means of select and clone

A statistical model tor giobai optimization is constructed generalizing some properties ofthe Wiener process to the multidimensional case. A new approach, which is similar to the Branch and Bound approach, is proposed to the construction of algorithms based on statistical models. A two dimensional version of the algorithm is implemented, and test results are presented     

Solving sum of quadratic ratios fractional programs via monotonic function

In this paper, a branch-reduce-bound algorithm is proposed for globally solving a sum of quadratic ratios fractional programming with nonconvex quadratic constraints. Due to its intrinsic difficulty, less work has been devoted to globally solving this problem. The proposed algorithm is based on reformulating the problem as a monotonic optimization problem, and it turns out that the optimal solution which is provided by the algorithm is adequately guaranteed to be feasible and to be close to the actual optimal solution. Convergence of the algorithm is shown and the numerical experiments are given to show the feasibility of the proposed algorithm.     

Global optimization for the sum of generalized polynomial fractional functions

In this paper, a branch and bound approach is proposed for global optimization problem (P) of the sum of generalized polynomial fractional functions under generalized polynomial constraints, which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solving this problem. By utilizing an equivalent problem and some linear underestimating approximations, a linear relaxation programming problem of the equivalent form is obtained. Consequently, the initial non-convex nonlinear problem (P) is reduced to a sequence of linear programming problems through successively refining the feasible region of linear relaxation problem. The proposed algorithm is convergent to the global minimum of the primal problem by means of the solutions to a series of linear programming problems. Numerical results show that the proposed algorithm is feasible and can successfully be used to solve the present problem (P).     

Cost optimal allocation of rail passenger lines

We consider the problem of cost optimal railway line allocation for passenger trains for the Dutch railway system. At present, the allocation of passenger lines by Dutch Railways is based on maximizing the number of direct travelers. This paper develops an alternative approach that takes operating costs into account. A mathematical programming model is developed which minimizes the operating costs subject to service constraints and capacity requirements. The model optimizes on lines, line types, routes, frequencies and train lengths. First, the line allocation model is formulated as an integer nonlinear programming model. This model is transformed into an integer linear programming model with binary decision variables. An algorithm is presented which solves the problem to optimality. The algorithm is based upon constraint satisfaction and a Branch and Bound procedure. The algorithm is applied to a subnetwork of the Dutch railway system for which it shows a substantial cost reduction. Further application and extension seem promising.     

Optimization problems with algebraic solutions: Quadratic fractional programs and ratio games

A mathematical program with a rational objective function may have irrational algebraic solutions even when the data are integral. We suggest that for such problems the optimal solution will be represented as follows: If λ* denotes the optimal value there will be given an intervalI and a polynomialP(λ) such thatI contains λ* and λ* is the unique root ofP(λ) inI. It is shown that with this representation the solutions to convex quadratic fractional programs and ratio games can be obtained in polynomial time.     

Solving fractional problems with dynamic multistart improving hit-and-run

Fractional programming has numerous applications in economy and engineering. While some fractional problems are easy in the sense that they are equivalent to an ordinary linear program, other problems like maximizing a sum or product of several ratios are known to be hard, as these functions are highly nonconvex and multimodal. In contrast to the standard Branch-and-Bound type algorithms proposed for specific types of fractional problems, we treat general fractional problems with stochastic algorithms developed for multimodal global optimization. Specifically, we propose Improving Hit-and-Run with restarts, based on a theoretical analysis of Multistart Pure Adaptive Search (cf. the dissertation of Khompatraporn (2004)) which prescribes a way to utilize problem specific information to sample until a certain level α of confidence is achieved. For this purpose, we analyze the Lipschitz properties of fractional functions, and then utilize a unified method to solve general fractional problems. The paper ends with a report on numerical experiments. This work was initiated while Mirjam Dür was spending a three-month research visit at the University of Washington. She would like to thank the Fulbright Commission for financial support and the optimization group at UW for their warm hospitality. The work of C. Khompatraporn and Z.B. Zabinsky was partially supported by the NSF grant DMI-0244286.