Reverse Convex Programming Approach in the Space of Extreme Criteria for Optimization over Efficient Sets

The problem of minimizing a convex function over the difference of two convex sets is called ‘reverse convex program’. This is a typical problem in global optimization, in which local optima are in general different from global optima. Another typical example in global optimization is the optimization problem over the efficient set of a multiple criteria programming problem. In this article, we investigate some special cases of optimization problems over the efficient set, which can be transformed equivalently into reverse convex programs in the space of so-called extreme criteria of multiple criteria programming problems under consideration. A suitable algorithm of branch and bound type is then established for globally solving resulting problems. Preliminary computational results with the proposed algorithm are reported.     

NONLINEAR INTEGER PROGRAMMING AND GLOBALOPTIMIZATION

1.IntroductionAlthoughthegenerallinearintegerprogrammingproblemisNP-hard,muchworkhasbeendevotedtoit(SeeNumhauserandWolsey[1988],Schrijver[1986]).Thesolutionmethodsincludethecuttingplane,theBranch-and-Bound,thedynamicprogrammingmethodsetc..However,thegeneralnonlinearintegerprogrammingproblemisdifficulttosolve.GareyandJohnson[1979]pointedoutthattheintegerprogrammingoverRewithalinearobjectivefunctionandquadraticconstraintsisundecidable.Soifanonlinearintegerprogrammingproblemishandled,itisalw…     

A finite,nonadjacent extreme-point search algorithm for optimization over the efficient set

The problem (P) of optimizing a linear function over the efficient set of a multiple-objective linear program serves many useful purposes in multiple-criteria decision making. Mathematically, problem (P) can be classified as a global optimization problem. Such problems are much more difficult to solve than convex programming problems. In this paper, a nonadjacent extreme-point search algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact extreme-point optimal solution for the problem after a finite number of iterations. It can be implemented using only linear programming methods. Convergence of the algorithm is proven, and a discussion is included of its main advantages and disadvantages.The author owes thanks to two anonymous referees for their helpful comments.     

Maximizing a concave function over the efficient or weakly-efficient set

An important approach in multiple criteria linear programming is the optimization of some function over the efficient or weakly-efficient set. This is a very difficult nonconvex optimization problem, even for the case that the function to be optimized is linear. In this article we consider the problem of maximizing a concave function over the efficient or weakly-efficient set. We show that this problem can essentially be formulated as a special global optimization problem in the space of the extreme criteria of the underlying multiple criteria linear program. An algorithm of branch and bound type is proposed for solving the resulting problem.     

An all-linear programming relaxation algorithm for optimizing over the efficient set

The problem (P) of optimizing a linear function over the efficient set of a multiple objective linear program has many important applications in multiple criteria decision making. Since the efficient set is in general a nonconvex set, problem (P) can be classified as a global optimization problem. Perhaps due to its inherent difficulty, it appears that no precisely-delineated implementable algorithm exists for solving problem (P) globally. In this paper a relaxation algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact optimal solution to the problem after a finite number of iterations. A detailed discussion is included of how to implement the algorithm using only linear programming methods. Convergence of the algorithm is proven, and a sample problem is solved.Research supported by a grant from the College of Business Administration, University of Florida, Gainesville, Florida, U.S.A.     

Optimization over the efficient set: Four special cases

Recently, researchers and practitioners have been increasingly interested in the problem (P) of maximizing a linear function over the efficient set of a multiple objective linear program. Problem (P) is generally a difficult global optimization problem which requires numerically intensive procedures for its solution. In this paper, simple linear programming procedures are described for detecting and solving four special cases of problem (P). When solving instances of problem (P), these procedures can be used as screening devices to detect and solve these four special cases.     

A Bilinear Algorithm for Optimizing a Linear Function over the Efficient Set of a Multiple Objective Linear Programming Problem

The problem Q of optimizing a linear function over the efficient set of a multiple objective linear program serves several useful purposes in multiple criteria decision making. However, Q is in itself a difficult global optimization problem, whose local optima, frequently large in number, need not be globally optimal. Indeed, this is due to the fact that the feasible region of Q is, in general, a nonconvex set. In this paper we present a monotonically increasing algorithm that finds an exact, globally-optimal solution for Q. Our approach does not require any hypothesis on the boundedness of neither the efficient set EP nor the optimal objective value. The proposed algorithm relies on a simplified disjoint bilinear program that can be solved through the use of well-known specifically designed methods within nonconvex optimization. The algorithm has been implemented in C and preliminary numerical results are reported.     

Global optimization of fractional programs

Dinkelbach's global optimization approach for finding the global maximum of the fractional programming problem is discussed. Based on this idea, a modified algorithm is presented which provides both upper and lower bounds at each iteration. The convergence of the lower and upper bounds to the global maximum function value is shown to be superlinear. In addition, the special case of fractional programming when the ratio involves only linear or quadratic terms is considered. In this case, the algorithm is guaranteed to find the global maximum to within any specified tolerance, regardless of the definiteness of the quadratic form.     

基于粒子群算法的非线性二层规划问题的求解算法

粒子群算法（Particle Swarm Optimization，PSO）是一种新兴的优化技术，其思想来源于人工生命和演化计算理论。PSO通过粒子追随自己找到的最好解和整个群的最好解来完成优化。该算法简单易实现，可调参数少，已得到了广泛研究和应用。本文根据该算法能够有效的求出非凸数学规划全局最优解的特点，对非线性二层规划的上下层问题求解，并根据二层规划的特点，给出了求解非线性二层规划问题全局最优解的有效算法。数值计算结果表明该算法有效。     

Global optimization of signomial geometric programming problems

This paper presents a global optimization approach for solving signomial geometric programming problems. In most cases nonconvex optimization problems with signomial parts are difficult, NP-hard problems to solve for global optimality. But some transformation and convexification strategies can be used to convert the original signomial geometric programming problem into a series of standard geometric programming problems that can be solved to reach a global solution. The tractability and effectiveness of the proposed successive convexification framework is demonstrated by seven numerical experiments. Some considerations are also presented to investigate the convergence properties of the algorithm and to give a performance comparison of our proposed approach and the current methods in terms of both computational efficiency and solution quality.     

一种改进的进化规划算法

本对于全局优化问题提出一个改进的进化规划算法，该算法以概率p接收基于电磁理论求出合力方向作为随机搜索方向，以概率1-p接收按正态分布产生的随机搜索方向。改进算法不仅克服了传统进化规划算法随机搜索的盲目性，而且保留了传统进化规划算法全局搜索性。本算法应用于几个典型例题，数值结果表明本算法是可行的，有效的。     

On Optimization over the Efficient Set in Linear Multicriteria Programming

The efficient set of a linear multicriteria programming problem can be represented by a reverse convex constraint of the form g(z)≤0, where g is a concave function. Consequently, the problem of optimizing some real function over the efficient set belongs to an important problem class of global optimization called reverse convex programming. Since the concave function used in the literature is only defined on some set containing the feasible set of the underlying multicriteria programming problem, most global optimization techniques for handling this kind of reverse convex constraint cannot be applied. The main purpose of our article is to present a method for overcoming this disadvantage. We construct a concave function which is finitely defined on the whole space and can be considered as an extension of the existing function. Different forms of the linear multicriteria programming problem are discussed, including the minimum maximal flow problem as an example. The research was partly done while the third author was visiting the Department of Mathematics, University of Trier with the support by the Alexander von Humboldt Foundation. He thanks the university as well as the foundation.     

New properties and computational improvement of the GOP algorithm for problems with quadratic objective functions and constraints

In Floudas and Visweswaran (1990, 1993), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the solution of the problem through a series ofprimal andrelaxed dual problems that provide valid upper and lower bounds respectively on the global solution. The algorithm was proved to have finite convergence to an -global optimum. In this paper, new theoretical properties are presented that help to enhance the computational performance of the GOP algorithm applied to problems of special structure. The effect of the new properties is illustrated through application of the GOP algorithm to a difficult indefinite quadratic problem, a multiperiod tankage quality problem that occurs frequently in the modeling of refinery processes, and a set of pooling/blending problems from the literature. In addition, extensive computational experience is reported for randomly generated concave and indefinite quadratic programming problems of different sizes. The results show that the properties help to make the algorithm computationally efficient for fairly large problems.     

Implementations of special ordered sets in MP software

Special ordered sets (SOS) have been introduced as a practical device for efficiently handling special classes of nonconvex optimization problems. They are now implemented in most commercial codes for mathematical programming (MP software). The paper gives a survey of possible applications as multiple choice restrictions, conditional multiple choice restrictions, discrete variables, discontinuous variables and piecewise linear functions, global optimization of separable programming problems, alternative right-hand sides, overlapping special ordered sets and the solution of quadratic programming problems. Alternative problem formulations are discussed. Since special ordered sets are not defined uniquely modelling facilities depend on the definition of a special orderedset in a code. The paper demonstrates the superiority of SOS to the application of binary variables if they are treated judiciously.     

Zero-one integer programs with few constraints —Lower bounding theory

The zero-one integer programming problem and its special case, the multiconstraint knapsack problem frequently appear as subproblems in many combinatorial optimization problems. We present several methods for computing lower bounds on the optimal solution of the zero-one integer programming problem. They include Lagrangean, surrogate and composite relaxations. New heuristic procedures are suggested for determining good surrogate multipliers. Based on theoretical results and extensive computational testing, it is shown that for zero-one integer problems with few constraints surrogate relaxation is a viable alternative to the commonly used Lagrangean and linear programming relaxations. These results are used in a follow up paper to develop an efficient branch and bound algorithm for solving zero-one integer programming problems.     

A mathematical programming approach to the computation of the omega invariant of a numerical semigroup

In this paper we present a mathematical programming formulation for the ω-invariant of a numerical semigroup for each of its minimal generators which is an useful index in commutative algebra (in particular in factorization theory) to analyze the primality of the elements in the semigroup. The model consists of solving a problem of optimizing a linear function over the efficient set of a multiobjective linear integer program. We offer a methodology to solve this problem and we provide some computational experiments to show the efficiency of the proposed algorithm.     

Global Optimization Versus Integer Programming in Portfolio Optimization under Nonconvex Transaction Costs

This paper is concerned with a portfolio optimization problem under concave and piecewise constant transaction cost. We formulate the problem as nonconcave maximization problem under linear constraints using absolute deviation as a measure of risk and solve it by a branch and bound algorithm developed in the field of global optimization. Also, we compare it with a more standard 0–1 integer programming approach. We will show that a branch and bound method elaborating the special structure of the problem can solve the problem much faster than the state-of-the integer programming code.     

Optimizing a Particular Real Root of a Polynomial by a Special Cylindrical Algebraic Decomposition

We study the problem of optimizing over parameters a particular real root of a polynomial with parametric coefficients. We propose an efficient symbolic method for solving the optimization problem based on a special cylindrical algebraic decomposition algorithm, which asks for a semi-algebraic decomposition into cells in terms of number-of-roots-invariance.     

一种改进的进化规划算法及其收敛性

1　引　言进化算法是一类借鉴生物界自然选择和自然遗传机制的随机搜索算法 ,主要包括遗传算法 ( genetic algorithms,GA)、进化规划 ( evolutionary programming,EP)和进化策略( evolution strategies,ES) [1 ] .遗传算法 ,进化策略和进化规划分别由 J.H.Holland,I.Rechenberg和 L.J.Fogel提出[2 ,3,4] .这些算法有许多相似性 ,且近年来都被应用于实值连续函数全局优化问题 ,并取得了较好的效果 .一些学者对进化算法的收敛性进行了分析[5,6,7,8] ,但总的说来 ,有关进化算法的理论研究成果目前还不是很多 ,尤其是进化规划和进化策略几…     

Maximization of the Ratio of Two Convex Quadratic Functions over a Polytope

In this paper, we will develop an algorithm for solving a quadratic fractional programming problem which was recently introduced by Lo and MacKinlay to construct a maximal predictability portfolio, a new approach in portfolio analysis. The objective function of this problem is defined by the ratio of two convex quadratic functions, which is a typical global optimization problem with multiple local optima. We will show that a well-designed branch-and-bound algorithm using (i) Dinkelbach's parametric strategy, (ii) linear overestimating function and (iii) -subdivision strategy can solve problems of practical size in an efficient way. This algorithm is particularly efficient for Lo-MacKinlay's problem where the associated nonconvex quadratic programming problem has low rank nonconcave property.