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Showing papers in "Journal of Global Optimization in 1998"


Journal ArticleDOI
TL;DR: This paper introduces the reader to a response surface methodology that is especially good at modeling the nonlinear, multimodal functions that often occur in engineering and shows how these approximating functions can be used to construct an efficient global optimization algorithm with a credible stopping rule.
Abstract: In many engineering optimization problems, the number of function evaluations is severely limited by time or cost. These problems pose a special challenge to the field of global optimization, since existing methods often require more function evaluations than can be comfortably afforded. One way to address this challenge is to fit response surfaces to data collected by evaluating the objective and constraint functions at a few points. These surfaces can then be used for visualization, tradeoff analysis, and optimization. In this paper, we introduce the reader to a response surface methodology that is especially good at modeling the nonlinear, multimodal functions that often occur in engineering. We then show how these approximating functions can be used to construct an efficient global optimization algorithm with a credible stopping rule. The key to using response surfaces for global optimization lies in balancing the need to exploit the approximating surface (by sampling where it is minimized) with the need to improve the approximation (by sampling where prediction error may be high). Striking this balance requires solving certain auxiliary problems which have previously been considered intractable, but we show how these computational obstacles can be overcome.

6,914 citations


Journal ArticleDOI
TL;DR: An interactive approach that infers the parameters of an ELECTRE TRI model from assignment examples is proposed that lies in the possibility given to the DM to revise his/her assignment examples and/or to give additional information before the optimization phase restarts.
Abstract: Given a finite set of alternatives, the sorting problem consists in the assignment of each alternative to one of the pre-defined categories. In this paper, we are interested in multiple criteria sorting problems and, more precisely, in the existing method ELECTRE TRI. This method requires the elicitation of parameters (weights, thresholds, category limits,...) in order to construct the Decision Maker‘s (DM) preference model. A direct elicitation of these parameters being rather difficult, we proceed to solve this problem in a way that requires from the DM much less cognitive effort. We elicit these parameters indirectly using holistic information given by the DM through assignment examples. We propose an interactive approach that infers the parameters of an ELECTRE TRI model from assignment examples. The determination of an ELECTRE TRI model that best restitutes the assignment examples is formulated through an optimization problem. The interactive aspect of this approach lies in the possibility given to the DM to revise his/her assignment examples and/or to give additional information before the optimization phase restarts.

426 citations


Journal ArticleDOI
TL;DR: The classical 0–1 knapsack problem is considered with two objectives, and two methods of the “two–phases” type are developed to generate the set of efficient solutions.
Abstract: The classical 0–1 knapsack problem is considered with two objectives. Two methods of the “two–phases” type are developed to generate the set of efficient solutions. In the first phase, the set of supported efficient solutions is determined by optimizing a parameterized single-objective knapsack problem. Two versions are proposed for a second phase, determining the non-supported efficient solutions: both versions are Branch and Bound approaches, but one is “breadth first”, while the other is “depth first”. Extensive numerical experiments have been realized to compare the results of both methods.

220 citations


Journal ArticleDOI
TL;DR: The results tangibly demonstrate the usefulness of using the outcome set approach of the Outer Approximation Algorithm instead of a decision set-based approach.
Abstract: Various difficulties have been encountered in using decision set-based vector maximization methods to solve a multiple objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years have suggested that outcome set-based approaches should instead be developed and used to solve problem (MOLP). In this article, we present a finite algorithm, called the Outer Approximation Algorithm, for generating the set of all efficient extreme points in the outcome set of problem (MOLP). To our knowledge, the Outer Approximation Algorithm is the first algorithm capable of generating this set. As a by-product, the algorithm also generates the weakly efficient outcome set of problem (MOLP). Because it works in the outcome set rather than in the decision set of problem (MOLP), the Outer Approximation Algorithm has several advantages over decision set-based algorithms. It is also relatively easy to implement. Preliminary computational results for a set of randomly-generated problems are reported. These results tangibly demonstrate the usefulness of using the outcome set approach of the Outer Approximation Algorithm instead of a decision set-based approach.

204 citations


Journal ArticleDOI
TL;DR: A new continuous reformulation of the maximum weight clique problem in undirected graphs is presented which considerably improves previous attacks both as numerical stability and interpretation of the results are concerned.
Abstract: A standard quadratic optimization problem (QP) consists of finding (global) maximizers of a quadratic form over the standard simplex. Standard QPs arise quite naturally in copositivity-based procedures which enable an escape from local solutions. Furthermore, several important applications yield optimization problems which can be cast into a standard QP in a straightforward way. As an example, a new continuous reformulation of the maximum weight clique problem in undirected graphs is presented which considerably improves previous attacks both as numerical stability and interpretation of the results are concerned. Apparently also for the first time, an equivalence between standard QPs and QPs on the positive orthant is established. Also, a recently presented global optimization procedure (GENF - genetical engineering via negative fitness) is shortly reviewed.

171 citations


Journal ArticleDOI
TL;DR: This paper derives new approaches for applying Lagrangian methods in discrete space, shows that an equilibrium is reached when a feasible assignment to the original problem is found and presents heuristic algorithms to look for equilibrium points, and proposes a new discrete Lagrange-multiplier-based global-search method (DLM) for solving satisfiability problems.
Abstract: Satisfiability is a class of NP-complete problems that model a wide range of real-world applications. These problems are difficult to solve because they have many local minima in their search space, often trapping greedy search methods that utilize some form of descent. In this paper, we propose a new discrete Lagrange-multiplier-based global-search method (DLM) for solving satisfiability problems. We derive new approaches for applying Lagrangian methods in discrete space, we show that an equilibrium is reached when a feasible assignment to the original problem is found and present heuristic algorithms to look for equilibrium points. Our method and analysis provides a theoretical foundation and generalization of local search schemes that optimize the objective alone and penalty-based schemes that optimize the constraints alone. In contrast to local search methods that restart from a new starting point when a search reaches a local trap, the Lagrange multipliers in DLM provide a force to lead the search out of a local minimum and move it in the direction provided by the Lagrange multipliers. In contrast to penalty-based schemes that rely only on the weights of violated constraints to escape from local minima, DLM also uses the value of an objective function (in this case the number of violated constraints) to provide further guidance. The dynamic shift in emphasis between the objective and the constraints, depending on their relative values, is the key of Lagrangian methods. One of the major advantages of DLM is that it has very few algorithmic parameters to be tuned by users. Besides the search procedure can be made deterministic and the results reproducible. We demonstrate our method by applying it to solve an extensive set of benchmark problems archived in DIMACS of Rutgers University. DLM often performs better than the best existing methods and can achieve an order-of-magnitude speed-up for some problems.

147 citations


Journal ArticleDOI
TL;DR: It can be proved that – for a wide class of problems – proximal regularization performed with appropriate regularization parameters ensures convexity of the auxiliary problems and each accumulation point of the method satisfies the necessary optimality conditions.
Abstract: The goal of this paper is to discover some possibilities for applying the proximal point method to nonconvex problems It can be proved that – for a wide class of problems – proximal regularization performed with appropriate regularization parameters ensures convexity of the auxiliary problems and each accumulation point of the method satisfies the necessary optimality conditions

110 citations


Journal ArticleDOI
TL;DR: A new model to assess customer satisfaction is developed based on the principles of multicriteria analysis, using ordinal regression techniques, which sufficiently describe customer behavior and they can be used in the strategic planning of an organization.
Abstract: A new model to assess customer satisfaction is developed through this paper. The proposed model is based on the principles of multicriteria analysis, using ordinal regression techniques. The procedure uses survey‘s data on customer satisfaction criteria and disaggregates simultaneously all the global satisfaction judgments via a linear programming disaggregation formulation. The model provides collective global and partial satisfaction functions as well as average satisfaction indices. These results sufficiently describe customer behavior and they can be used in the strategic planning of an organization. The implementation of the model in three real world applications is used for illustration and for testing the model‘s reliability. Finally, several extensions and future research in the area of customer satisfaction analysis are discussed.

103 citations


Journal ArticleDOI
TL;DR: A new algorithm is proposed that finds the exact global minimum of this problem in a finite number of iterations and extends a guarantee of finiteness to all branch-and-bound algorithms for concave programming that (1) partition exhaustively using rectangular subdivisions and (2) branch on the incumbent solution when possible.
Abstract: Researchers first examined the problem of separable concave programming more than thirty years ago, making it one of the earliest branches of nonlinear programming to be explored. This paper proposes a new algorithm that finds the exact global minimum of this problem in a finite number of iterations. In addition to proving that our algorithm terminates finitely, the paper extends a guarantee of finiteness to all branch-and-bound algorithms for concave programming that (1) partition exhaustively using rectangular subdivisions and (2) branch on the incumbent solution when possible. The algorithm uses domain reduction techniques to accelerate convergence; it solves problems with as many as 100 nonlinear variables, 400 linear variables and 50 constraints in about five minutes on an IBM RS/6000 Power PC. An industrial application with 152 nonlinear variables, 593 linear variables, and 417 constraints is also solved in about ten minutes.

99 citations


Journal ArticleDOI
TL;DR: The goal of characterizing the global solutions of an optimization problem, i.e. getting at necessary and sufficient conditions for a feasible point to be a global minimizer (or maximizer) of the objective function, is pursued.
Abstract: In this paper bearing the same title as our earlier survey-paper [11] we pursue the goal of characterizing the global solutions of an optimization problem, i.e. getting at necessary and sufficient conditions for a feasible point to be a global minimizer (or maximizer) of the objective function. We emphasize nonconvex optimization problems presenting some specific structures like ’convex-anticonvex‘ ones or quadratic ones.

96 citations


Journal ArticleDOI
TL;DR: A new branch and bound algorithm using a rectangular partition and ellipsoidal technique for minimizing a nonconvex quadratic function with box constraints is proposed.
Abstract: In this paper we propose a new branch and bound algorithm using a rectangular partition and ellipsoidal technique for minimizing a nonconvex quadratic function with box constraints. The bounding procedures are investigated by d.c. (difference of convex functions) optimization algorithms, called DCA. This is based upon the fact that the application of the DCA to the problems of minimizing a quadratic form over an ellipsoid and/or over a box is efficient. Some details of computational aspects of the algorithm are reported. Finally, numerical experiments on a lot of test problems showing the efficiency of our algorithm are presented.

Journal ArticleDOI
TL;DR: This paper uses the concept of indifference thresholds for modelling the imprecision related to the goal values in the goal programming model.
Abstract: The goal programming (GP) model is probably the best known in mathematical programming with multiple objectives. Available in various versions, GP is one of the most powerful multiple objective methods which has been applied in much varied fields. It has also been the target of many criticisms among which are those related to the difficulty of determining precisely the goal values as well as those concerning the decision-maker‘s near absence in this modelling process. In this paper, we will use the concept of indifference thresholds for modelling the imprecision related to the goal values. Many classical imprecise and fuzzy GP model formulations can be considered as a particular case of the proposed formulation.

Journal ArticleDOI
Hanif D. Sherali1
TL;DR: An extension of the Reformulation-Linearization Technique (RLT) is developed to generate linear programming relaxations that are embedded within a branch-and-bound algorithm for nonconvex polynomial programming problems that arise in various engineering design, network distribution, and location-allocation contexts.
Abstract: This paper considers the solution of nonconvex polynomial programming problems that arise in various engineering design, network distribution, and location-allocation contexts. These problems generally have nonconvex polynomial objective functions and constraints, involving terms of mixed-sign coefficients (as in signomial geometric programs) that have rational exponents on variables. For such problems, we develop an extension of the Reformulation-Linearization Technique (RLT) to generate linear programming relaxations that are embedded within a branch-and-bound algorithm. Suitable branching or partitioning strategies are designed for which convergence to a global optimal solution is established. The procedure is illustrated using a numerical example, and several possible extensions and algorithmic enhancements are discussed.

Journal ArticleDOI
TL;DR: IHR is a sequential random search method that has been successfully used in several engineering design applications, such as the optimal design of composite structures, and several variations have been applied to the composites design problem.
Abstract: Engineering design problems often involve global optimization of functions that are supplied as ’black box‘ functions. These functions may be nonconvex, nondifferentiable and even discontinuous. In addition, the decision variables may be a combination of discrete and continuous variables. The functions are usually computationally expensive, and may involve finite element methods. An engineering example of this type of problem is to minimize the weight of a structure, while limiting strain to be below a certain threshold. This type of global optimization problem is very difficult to solve, yet design engineers must find some solution to their problem – even if it is a suboptimal one. Sometimes the most difficult part of the problem is finding any feasible solution. Stochastic methods, including sequential random search and simulated annealing, are finding many applications to this type of practical global optimization problem. Improving Hit-and-Run (IHR) is a sequential random search method that has been successfully used in several engineering design applications, such as the optimal design of composite structures. A motivation to IHR is discussed as well as several enhancements. The enhancements include allowing both continuous and discrete variables in the problem formulation. This has many practical advantages, because design variables often involve a mixture of continuous and discrete values. IHR and several variations have been applied to the composites design problem. Some of this practical experience is discussed.

Journal ArticleDOI
TL;DR: An analytical equivalent for the inclusion of a set to the Lebesque set of a convex function is given and global optimality conditions related to classical optimization theory for convex maximization and reverse-convex optimization are obtained.
Abstract: In this paper we give an analytical equivalent for the inclusion of a set to the Lebesque set of a convex function. Using this results, we obtain global optimality conditions (GOC) related to classical optimization theory for convex maximization and reverse-convex optimization. Several examples illustrate the effectiveness of these optimality conditions allowing to escape from stationary points and local extremums.

Journal ArticleDOI
Ulrich Raber1
TL;DR: The presented algorithm often outperforms a comparable rectangular branch-and-bound method and under the assumption that a feasible point of the all-quadratic program is known, the algorithm guarantees an ε-approximate optimal solution in a finite number of iterations.
Abstract: In this paper we present an algorithm for solving nonconvex quadratically constrained quadratic programs (all-quadratic programs). The method is based on a simplicial branch-and-bound scheme involving mainly linear programming subproblems. Under the assumption that a feasible point of the all-quadratic program is known, the algorithm guarantees an e-approximate optimal solution in a finite number of iterations. Computational experiments with an implementation of the procedure are reported on randomly generated test problems. The presented algorithm often outperforms a comparable rectangular branch-and-bound method.

Journal ArticleDOI
TL;DR: It is shown that cutting plane algorithms can be designed to solve the equivalent problems which minimize a linear function over a convex region and several classes of valid inequalities of the conveX region are proposed.
Abstract: We apply a linearization technique for nonconvex quadratic problems with box constraints. We show that cutting plane algorithms can be designed to solve the equivalent problems which minimize a linear function over a convex region. We propose several classes of valid inequalities of the convex region which are closely related to the Boolean quadric polytope. We also describe heuristic procedures for generating cutting planes. Results of preliminary computational experiments show that our inequalities generate a polytope which is a fairly tight approximation of the convex region.

Journal ArticleDOI
TL;DR: An estimation model is presented to describe the behavior of real estate markets using mathematical programming tools within a multiple criteria analysis context and the usefulness and applicability is empirically shown through an implementation using the data of the City of Edmonton, Alberta, Canada.
Abstract: Real estate evaluation is of great importance and interest to many socio-economic agents, especially to property buyers and sellers for personal benefits, municipalities for tax purposes, financial institutions for loan policies, and to real estate brokerage firms for marketing activities. Although these agents are motivated in their actions by different objectives, even conflicting at times, they all desire to have a realistic description of the real estate market behavior in order to make right and timely decisions. This article presents an estimation model to describe the behavior of real estate markets. The model is based on certain observable real estate market data as well as on the perceptions of real estate agents who are active in the market. The parameters that describe the behavior of the real estate market are estimated, through the estimation model, using mathematical programming tools within a multiple criteria analysis context. The usefulness and applicability of the approach is empirically shown through an implementation using the data of the City of Edmonton, Alberta, Canada.

Journal ArticleDOI
TL;DR: This work systematically inserts Steiner points between edges of the minimal spanning tree meeting at angles less than 120 degrees, performing a local optimization at the end, and minimizes over all connections.
Abstract: The Euclidean Steiner tree problem is to find the tree with minimal Euclidean length spanning a set of fixed points in the plane, allowing the addition of auxiliary points to the set (Steiner points). The problem is NP-hard, so polynomial-time heuristics are desired. We present two such heuristics, both of which utilize an efficient method for computing a locally optimal tree with a given topology. The first systematically inserts Steiner points between edges of the minimal spanning tree meeting at angles less than 120 degrees, performing a local optimization at the end. The second begins by finding the Steiner tree for three of the fixed points. Then, at each iteration, it introduces a new fixed point to the tree, connecting it to each possible edge by inserting a Steiner point, and minimizes over all connections, performing a local optimization for each. We present a variety of test cases that demonstrate the strengths and weaknesses of both algorithms.

Journal ArticleDOI
TL;DR: This paper addresses the problem of finding the number, K, of phases present at equilibrium and their composition, in a chemical mixture of ns substances, and presents an algorithmic approach that reduces this global optimization problem to a finite sequence of local optimization steps in K(ns-1) -space, K ≤ mb, and global optimization Steps in (ns- 1)-space.
Abstract: This paper addresses the problem of finding the number, K, of phases present at equilibrium and their composition, in a chemical mixture of n_s substances. This corresponds to the global minimum of the Gibbs free energy of the system, subject to constraints representing m_b independent conserved quantities, wherem_b=n_s when no reaction is possible and m_b ≤ n_e+1 when reaction is possible and n_e is the number of elements present. After surveying previous work in the field and pointing out the main issues, we extend the necessary and sufficient condition for global optimality based on the ’’reaction tangent-plane criterion‘‘, to the case involving different thermodynamical models (multiple phase classes). We then present an algorithmic approach that reduces this global optimization problem (involving a search space of m_b(n_s-1) dimensions) to a finite sequence of local optimization steps inK(n_s-1) -space, K ≤ m_b, and global optimization steps in (n_s-1)-space. The global step uses the tangent-plane criterion to determine whether the current solution is optimal, and, if it is not, it finds an improved feasible solution either with the same number of phases or with one added phase. The global step also determines what class of phase (e.g. liquid or vapour) is to be added, if any phase is to be added. Given a local minimization procedure returning a Kuhn–Tucker point and a global optimization procedure (for a lower-dimensional search space) returning a global minimum, the algorithm is proved to converge to a global minimum in a finite number of the above local and global steps. The theory is supported by encouraging computational results.

Journal ArticleDOI
TL;DR: A branch-and-bound approach is suggested transforming the problem of finding an approximate global minimum of the original nonconvex optimization problem into the solution of a finite number of convex problems.
Abstract: A nonconvex generalized semi-infinite programming problem is considered, involving parametric max-functions in both the objective and the constraints. For a fixed vector of parameters, the values of these parametric max-functions are given as optimal values of convex quadratic programming problems. Assuming that for each parameter the parametric quadratic problems satisfy the strong duality relation, conditions are described ensuring the uniform boundedness of the optimal sets of the dual problems w.r.t. the parameter. Finally a branch-and-bound approach is suggested transforming the problem of finding an approximate global minimum of the original nonconvex optimization problem into the solution of a finite number of convex problems.

Journal ArticleDOI
TL;DR: In this paper, several trials in order to overcome the difficulties of the multi-surface method are suggested and it will be shown that using the suggested methods, the additional learning can be easily made.
Abstract: Pattern classification is one of the main themes in pattern recognition, and has been tackled by several methods such as the statistic one, artificial neural networks, mathematical programming and so on. Among them, the multi-surface method proposed by Mangasarian is very attractive, because it can provide an exact discrimination function even for highly nonlinear problems without any assumption on the data distribution. However, the method often causes many slits on the discrimination curve. In other words, the piecewise linear discrimination curve is sometimes too complex resulting in a poor generalization ability. In this paper, several trials in order to overcome the difficulties of the multi-surface method are suggested. One of them is the utilization of goal programming in which the auxiliary linear programming problem is formulated as a goal programming in order to get as simple discrimination curves as possible. Another one is to apply fuzzy programming by which we can get fuzzy discrimination curves with gray zones. In addition, it will be shown that using the suggested methods, the additional learning can be easily made. These features of the methods make the discrimination more realistic. The effectiveness of the methods is shown on the basis of some applications.

Journal ArticleDOI
TL;DR: Optimal shape design problems for systems governed by an elliptic hemivariational inequality are considered and a general existence result is established by the mapping method.
Abstract: Optimal shape design problems for systems governed by an elliptic hemivariational inequality are considered. A general existence result for this problem is established by the mapping method.

Journal ArticleDOI
TL;DR: A brief survey of recent developments in the field of stochastic global optimization methods will be presented, and it is argued that for moderately sized problems this approach might prove more efficient than those based upon uniform random samples.
Abstract: In this paper a brief survey of recent developments in the field of stochastic global optimization methods will be presented. Most methods discussed fall in the category of two-phase algorithms, consisting in a global or exploration phase, obtained through sampling in the feasible domain, and a second or local phase, consisting of refinement of local knowledge, obtained through classical descent routines. A new class of methods is also introduced, characterized by the fact that sampling is performed through deterministic, well distributed, sample points. It is argued that for moderately sized problems this approach might prove more efficient than those based upon uniform random samples.

Journal ArticleDOI
TL;DR: Simple necessary optimality conditions are formulated for a function f of the form f–h, where g and h are nonsmooth functions and related sufficient conditions are given for local minimization and global minimization.
Abstract: Simple necessary optimality conditions are formulated for a functionf of the form f=g-h, where g andh are nonsmooth functions. Related sufficient conditions are given for local minimization and global minimization.

Journal ArticleDOI
TL;DR: The aim here is to show that recent developments in Nonsmooth Analysis (especially in Exact Penalization Theory) allow one to treat successfully even some quite ‘smooth’ problems by tools of NonsMooth Analysis and Nondifferentiable Optimization.
Abstract: The nonsmoothness is viewed by many people as at least an undesirable (if not unavoidable) property. Our aim here is to show that recent developments in Nonsmooth Analysis (especially in Exact Penalization Theory) allow one to treat successfully even some quite ’’smooth‘‘ problems by tools of Nonsmooth Analysis and Nondifferentiable Optimization. Our approach is illustrated by one Classical Control Problem of finding optimal parameters in a system described by ordinary differential equations.

Journal ArticleDOI
TL;DR: A simultaneous axiomatic extension of these two classical models conjugate duality with arbitrary coupling functions is obtained, which requires only the mentioned support property.
Abstract: We study conjugate duality with arbitrary coupling functions. Our only tool is a certain support property, which is automatically fulfilled in the two most widely used special cases, namely the case where the underlying space is a topological vector space and the coupling functions are the continuous linear ones, and the case where the underlying space is a metric space and the coupling functions are the continuous ones. We obtain thereby a simultaneous axiomatic extension of these two classical models. Also included is a condition for global optimality, which requires only the mentioned support property.

Journal ArticleDOI
TL;DR: This paper investigates a technique for constructing test functions for global optimization problems for which the problem dimension, the number of local minima, the local minata points, and the function values of theLocal minima are fixed a priori.
Abstract: Functions with local minima and size of their ’region of attraction‘ known a priori, are often needed for testing the performance of algorithms that solve global optimization problems. In this paper we investigate a technique for constructing test functions for global optimization problems for which we fix a priori: (i) the problem dimension, (ii) the number of local minima, (iii) the local minima points, (iv) the function values of the local minima. Further, the size of the region of attraction of each local minimum may be made large or small. The technique consists of first constructing a convex quadratic function and then systematically distorting selected parts of this function so as to introduce local minima.


Journal ArticleDOI
TL;DR: The aim of the present paper is to discuss the influence which certain perturbations have on the solution of the eigenvalue problem for hemivariational inequalities on a sphere of given radius.
Abstract: The aim of the present paper is to discuss the influence which certain perturbations have on the solution of the eigenvalue problem for hemivariational inequalities on a sphere of given radius. The perturbation results in adding a term of the type g^0(x, u(x); v(x)) to the hemivariational inequality, where g is a locally Lipschitz nonsmooth and nonconvex energy functional. Applications illustrate the theory.