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Showing papers in "Journal of Global Optimization in 1994"


Book ChapterDOI
TL;DR: A survey of results concerning algorithms, complexity, and applications of the maximum clique problem is presented and enumerative and exact algorithms, heuristics, and a variety of other proposed methods are discussed.
Abstract: The maximum clique problem is a classical problem in combinatorial optimization which finds important applications in different domains. In this paper we try to give a survey of results concerning algorithms, complexity, and applications of this problem, and also provide an updated bibliography. Of course, we build upon precursory works with similar goals [39, 232, 266].

1,065 citations


Journal ArticleDOI
TL;DR: This paper contains a bibliography of all references central to bilevel and multilevel programming that the authors know of and it is hoped that this bibliography facilitates and encourages their research.
Abstract: This paper contains a bibliography of all references central to bilevel and multilevel programming that the authors know of. It should be regarded as a dynamic and permanent contribution since all the new and appropriate references that are brought to our attention will be periodically added to this bibliography. Readers are invited to suggest such additions, as well as corrections or modifications, and to obtain a copy of the LaTeX and BibTeX files that constitute this manuscript, using the guidelines contained in this paper. To classify some of the references in this bibliography a short overview of past and current research in bilevel and multilevel programming is included. For those who are interested in but unfamiliar with the references in this area, we hope that this bibliography facilitates and encourages their research.

614 citations


Journal ArticleDOI
TL;DR: Advantages and disadvantages of Bayesian approach (average case analysis), comparing it with more usual minimax approach (worst case analysis) are discussed and new interactive version of software for global optimization is discussed.
Abstract: In this paper a review of application of Bayesian approach to global and stochastic optimization of continuous multimodal functions is given. Advantages and disadvantages of Bayesian approach (average case analysis), comparing it with more usual minimax approach (worst case analysis) are discussed. New interactive version of software for global optimization is discussed. Practical multidimensional problems of global optimization are considered

338 citations


Journal ArticleDOI
TL;DR: Empirical comparisons with other algorithms suggest competitive performance by Hide-and-Seek and the sequence of iteration points converges in probability to a global optimum regardless of how rapidly the temperatures converge to zero.
Abstract: Hide-and-Seek is a powerful yet simple and easily implemented continuous simulated annealing algorithm for finding the maximum of a continuous function over an arbitrary closed, bounded and full-dimensional body. The function may be nondifferentiable and the feasible region may be nonconvex or even disconnected. The algorithm begins with any feasible interior point. In each iteration it generates a candidate successor point by generating a uniformly distributed point along a direction chosen at random from the current iteration point. In contrast to the discrete case, a single step of this algorithm may generateany point in the feasible region as a candidate point. The candidate point is then accepted as the next iteration point according to the Metropolis criterion parametrized by anadaptive cooling schedule. Again in contrast to discrete simulated annealing, the sequence of iteration points converges in probability to a global optimum regardless of how rapidly the temperatures converge to zero. Empirical comparisons with other algorithms suggest competitive performance by Hide-and-Seek.

237 citations


Journal ArticleDOI
TL;DR: A global optimization algorithm is proposed based on an efficient partitioning strategy which is guaranteed to attain ε-convergence to the global minimum potential energy configuration of a molecule through the solution of a series of nonlinear convex optimization problems.
Abstract: A global optimization algorithm is proposed for finding the global minimum potential energy conformations of small molecules. The minimization of the total potential energy is formulated on an independent set of internal coordinates involving only torsion (dihedral) angles. Analytical expressions for the Euclidean distances between non-bonded atoms, which are required for evaluating the individual pairwise potential terms, are obtained as functions of bond lengths, covalent bond angles, and torsion angles. A novel procedure for deriving convex lower bounding functions for the total potential energy function is also introduced. These underestimating functions satisfy a number of important theoretical properties. A global optimization algorithm is then proposed based on an efficient partitioning strategy which is guaranteed to attain e-convergence to the global minimum potential energy configuration of a molecule through the solution of a series of nonlinear convex optimization problems. Moreover, lower and upper bounds on the total finite number of required iterations are also provided. Finally, this global optimization approach is illustrated with a number of example problems.

174 citations


Journal ArticleDOI
TL;DR: Analysis of branch and bound methods for enclosing all unconstrained global minimizers of a nonconvex nonlinear twice-continuously differentiable objective function shows that the problem is highly related to the behavior of the objective function near the global minimizer and to the order of the corresponding interval extension.
Abstract: We consider branch and bound methods for enclosing all unconstrained global minimizers of a nonconvex nonlinear twice-continuously differentiable objective function. In particular, we consider bounds obtained with interval arithmetic, with the “midpoint test,” but no acceleration procedures. Unless the lower bound is exact, the algorithm without acceleration procedures in general gives an undesirable cluster of boxes around each minimizer. In a previous paper, we analyzed this problem for univariate objective functions. In this paper, we generalize that analysis to multi-dimensional objective functions. As in the univariate case, the results show that the problem is highly related to the behavior of the objective function near the global minimizers and to the order of the corresponding interval extension.

95 citations


Journal ArticleDOI
TL;DR: The molecular conformation problem is discussed, and a concave quadratic global minimization approach for solving it is described, based on a quadRatic assignment formulation of a discrete approximation to the original problem.
Abstract: The molecular conformation problem is discussed, and a concave quadratic global minimization approach for solving it is described. This approach is based on a quadratic assignment formulation of a discrete approximation to the original problem.

92 citations


Journal ArticleDOI
TL;DR: Some of the most commonly used potential energy functions are introduced and different optimization methods used in the minimization of nonconvex potentialEnergy functions are discussed.
Abstract: The minimization of potential energy functions plays an important role in the determination of ground states or stable states of certain classes of molecular clusters and proteins. In this paper we introduce some of the most commonly used potential energy functions and discuss different optimization methods used in the minimization of nonconvex potential energy functions. A very complete bibliography is also given.

86 citations


Journal ArticleDOI
TL;DR: This paper identifies classes of nonconvex problems involving either sums or products of ratios of linear terms which may be treated by analysis in a transformed space and presents an algorithm that locates global solutions by computing both upper and lower bounds on the solution and then solving a sequence of linear programming sub-problems.
Abstract: The solution of a particular nonconvex program is usually very dependent on the structure of the problem. In this paper we identify classes of nonconvex problems involving either sums or products of ratios of linear terms which may be treated by analysis in a transformed space. In each class, the image space is defined by a mapping which associates a new variable with each original ratio of linear terms. In the image space, optimization is easy in certain directions, and the overall solution may be realized by sequentially optimizing in these directions. In addition to these ratio problems, we also show how to use image space analysis to treat the subclass of problems whose objective is to optimize a product of linear terms. For each class of nonconvex problems, we present an algorithm that locates global solutions by computing both upper and lower bounds on the solution and then solving a sequence of linear programming sub-problems. We also demonstrate the algorithms described in this paper by solving several example problems.

76 citations


Journal ArticleDOI
TL;DR: Simple data structures are introduced which reduce the time complexity of the Northby algorithm for lattice search from O(n5/3) per move toO(n2/3" per move for ann-atom cluster involving full Lennard-Jones potential function and show that in some cases, the relaxation of a lattice local minimizers with a worse potential function value may lead to a local minimizer with a better potential functionvalue.
Abstract: In 1987, Northby presented an efficient lattice based search and optimization procedure to compute ground states ofn-atom Lennard-Jones clusters and reported putative global minima for 13⩽n⩽150. In this paper, we introduce simple data structures which reduce the time complexity of the Northby algorithm for lattice search fromO(n5/3) per move toO(n2/3) per move for ann-atom cluster involving full Lennard-Jones potential function. If nearest neighbor potential function is used, the time complexity can be further reduced toO(logn) per move for ann-atom cluster. The lattice local minimizers with lowest potential function values are relaxed by a powerful Truncated Newton algorithm. We are able to reproduce the minima reported by Northby. The improved algorithm is so efficient that less than 3 minutes of CPU time on the Cray-XMP is required for each cluster size in the above range. We then further improve the Northby algorithm by relaxingevery lattice local minimizer found in the process. This certainly requires more time. However, lower energy configurations were found with this improved algorithm forn=65, 66, 75, 76, 77 and 134. These findings also show that in some cases, the relaxation of a lattice local minimizer with a worse potential function value may lead to a local minimizer with a better potential function value.

73 citations


Journal ArticleDOI
TL;DR: An algorithm for generalized convex multiplicative programming problems, a special class of nonconvex minimization problems in which the objective function is expressed as a sum ofp products of two convex functions, is discussed.
Abstract: This paper discusses an algorithm for generalized convex multiplicative programming problems, a special class of nonconvex minimization problems in which the objective function is expressed as a sum ofp products of two convex functions. It is shown that this problem can be reduced to a concave minimization problem with only 2p variables. An outer approximation algorithm is proposed for solving the resulting problem.

Journal ArticleDOI
TL;DR: A new dynamic programming method for the single item capacitated dynamic lot size model with non-negative demands and no backlogging is developed, which builds the Optimal value function in piecewise linear segments.
Abstract: We develop a new dynamic programming method for the single item capacitated dynamic lot size model with non-negative demands and no backlogging. This approach builds the Optimal value function in piecewise linear segments. It works very well on the test problems, requiring less than 0.3 seconds to solve problems with 48 periods on a VAX 8600. Problems with the time horizon up to 768 periods are solved. Empirically, the computing effort increases only at a quadratic rate relative to the number of periods in the time horizon.

Journal ArticleDOI
TL;DR: A method for global minimization of a functionf(x), x εA ⊂Rn by using presampled global points in A is presented, and the working of a sequential version in Fortran is illustrated.
Abstract: A method for global minimization of a functionf(x), x eA ⊂R n by using presampled global points inA is presented. The global points are obtained by uniform sampling, discarding points too near an already accepted point to obtain a very uniform covering. The accepted points and their nearest-neighbours matrix are stored on a file. When optimzing a given function these pre-sampled points and the matrix are read from file. Then the function value of each point is computed and itsk nearest neighbours that have larger function values are marked. The points for which all its neighbours are marked are extracted as promising starting points for local minimizations. Results from a parallel implementation are presented. The working of a sequential version in Fortran is illustrated.

Journal ArticleDOI
TL;DR: The new TMSL method performs much better (in some cases several times) than the Multilevel Single Linkage method in terms of number of function evaluations but is not quite so competitive with respect to CPU time.
Abstract: An iterative topographical Multilevel Single Linkage (TMSL) method has been introduced. The approach uses topographical information on the objective function, in particular theg-nearest-neighbour graph. The algorithm uses evenly distributed points from a Halten sequence of uniform limiting density. We discuss the implementation of the algorithm and compare its performance with other well-known algorithms. The new algorithm performs much better (in some cases several times) than the Multilevel Single Linkage method in terms of number of function evaluations but is not quite so competitive with respect to CPU time.

Journal ArticleDOI
TL;DR: It is established that a suitable random perturbation of the gradient method with a fixed parameter generates a bounded minimizing sequence and leads to a global minimum: the perturbations avoids convergence to local minima.
Abstract: The paper deals with the global minimization of a differentiable cost function mapping a ball of a finite dimensional Euclidean space into an interval of real numbers. It is established that a suitable random perturbation of the gradient method with a fixed parameter generates a bounded minimizing sequence and leads to a global minimum: the perturbation avoids convergence to local minima. The stated results suggest an algorithm for the numerical approximation of global minima: experiments are performed for the problem of fitting a sum of exponentials to discrete data and to a nonlinear system involving about 5000 variables. The effect of the random perturbation is examined by comparison with the purely deterministic gradient method.

Journal ArticleDOI
TL;DR: This work shows how it is possible to guarantee ε-global solutions for a certain important class of the phase and chemical equilibrium problem, namely when the liquid phase can be modeled using neither the Non-Random Two-Liquid (NRTL) equation, or the UNIversal QUAsi Chemical (UNIQUAC) equation.
Abstract: An increasingly popular approach when solving the phase and chemical equilibrium problem is to pose it as an optimization problem. However, difficulties are encountered due to the highly nonlinear nature of the models used to represent the behavior of the fluids, and because of the existence of multiple local solutions. This work shows how it is possible to guarantee e-global solutions for a certain important class of the phase and chemical equilibrium problem, namely when the liquid phase can be modeled using neither the Non-Random Two-Liquid (NRTL) equation, or the UNIversal QUAsi Chemical (UNIQUAC) equation. Ideal vapor phases are easily incorporated into the global optimization framework. A numberof interesting properties are described which drastically alter the structure of the respective problems. For the NRTL equation, it is shown that the formulation can be converted into a biconvex optimization problem. The GOP algorithm of Floudas and Visweswaran [8, 9] can then be used to obtain e-global solutions in this case. For the UNIQUAC equation, the new properties show how the objective function can be transformed into the difference of two convex functions (i.e. a D.C. programming problem is obtained), where the concave portion is separable. A branch and bound algorithm based on that of Falk and Soland [6] is used to guarantee convergence to an e-global solution. Examples are presented which demonstrate the performance of both algorithms.

Journal ArticleDOI
TL;DR: A two-level simulated annealing algorithm for solving certain class of hard combinatorial optimization problems is proposed and the full Lennard-Jones potential function is presented, with satisfactory results for problems for cluster sizes as large as 100,000.
Abstract: In this paper, we propose a new kind of simulated annealing algorithm calledtwo-level simulated annealing for solving certain class of hard combinatorial optimization problems. This two-level simulated annealing algorithm is less likely to get stuck at a non-global minimizer than conventional simulated annealing algorithms. We also propose a parallel version of our two-level simulated annealing algorithm and discuss its efficiency. This new technique is then applied to the Molecular Conformation problem in 3 dimensional Euclidean space. Extensive computational results on Thinking Machines CM-5 are presented. With the full Lennard-Jones potential function, we were able to get satisfactory results for problems for cluster sizes as large as 100,000. A peak rate of over 0.8 giga flop per second in 64-bit operations was sustained on a partition with 512 processing elements. To the best of our knowledge, ground states of Lennard-Jones clusters of size as large as these have never been reported before.

Journal ArticleDOI
TL;DR: A global optimization method for general indefinite quadratic problems, which consist of maximizing a non-concave quadRatic function over a polyhedron inn-dimensional Euclidean space, which is shown to be finite and exact in non-degenerate situations.
Abstract: Here we propose a global optimization method for general, i.e. indefinite quadratic problems, which consist of maximizing a non-concave quadratic function over a polyhedron inn-dimensional Euclidean space. This algorithm is shown to be finite and exact in non-degenerate situations. The key procedure uses copositivity arguments to ensure escaping from inefficient local solutions. A similar approach is used to generate an improving feasible point, if the starting point is not the global solution, irrespective of whether or not this is a local solution. Also, definiteness properties of the quadratic objective function are irrelevant for this procedure. To increase efficiency of these methods, we employ pseudoconvexity arguments. Pseudoconvexity is related to copositivity in a way which might be helpful to check this property efficiently even beyond the scope of the cases considered here.

Journal ArticleDOI
TL;DR: New conditions are imposed on the ordering cone such that for a set which is closed and bounded in the usual sense or with respect to the cone, the set of efficient points is nonempty and the domination property holds.
Abstract: We study the existence of efficient points in a locally convex space ordered by a convex cone. New conditions are imposed on the ordering cone such that for a set which is closed and bounded in the usual sense or with respect to the cone, the set of efficient points is nonempty and the domination property holds.

Journal ArticleDOI
TL;DR: Algorithms that use only the Lipschitz constant and can be modified to use second derivative bounds or gradient calculations to work faster on a more restricted class are provided.
Abstract: Optimization methods for a given class are easily modified to utilize additional information and work faster on a more restricted class. In particular algorithms that use only the Lipschitz constant (e.g. Mladineo, Piyavskii, Shubert and Wood) can be modified to use second derivative bounds or gradient calculations. The algorithm of Breiman & Cutler can be modified to use Lipschitz bounds. Test cases illustrating accelerations to various algorithms are provided.

Journal ArticleDOI
TL;DR: It is suggested that weighted least squares scaling, a basic method in multidimensional scaling, as a class of test functions for global optimization, are easy to code, cheap to calculate, and have important applications in data analysis.
Abstract: We suggest weighted least squares scaling, a basic method in multidimensional scaling, as a class of test functions for global optimization. The functions are easy to code, cheap to calculate, and have important applications in data analysis. For certain data these functions have many local minima. Some characteristic features of the test functions are investigated.

Journal ArticleDOI
TL;DR: Computational results suggest that the new method, based on a branch-and-bound approach based on Karmakar's interior point method, is very efficient.
Abstract: This paper deals with the problem of identifying a hidden Boolean function ℱ: 0, 1' → 0, 1 from positive and negative examples. This problem is of paramount importance in many real life applications of artificial intelligence. The method proposed in this paper is based on a branch-and-bound approach. This approach is an expansion of some earlier work (Triantaphyllouet al., 1994). Computational results, comparing the new method with one based on Karmakar's interior point method, suggest that the new method is very efficient.

Journal ArticleDOI
TL;DR: Two improvements for the algorithm of Breiman and Cutler are presented and better envelopes can be built up using positive quadratic forms and improvements in convergence rates are demonstrated empirically on standard test functions.
Abstract: Two improvements for the algorithm of Breiman and Cutler are presented. Better envelopes can be built up using positive quadratic forms. Better utilization of first and second derivative information is attained by combining both global aspects of curvature and local aspects near the global optimum. The basis of the results is the geometric viewpoint developed by the first author and can be applied to a number of covering type methods. Improvements in convergence rates are demonstrated empirically on standard test functions.

Journal ArticleDOI
TL;DR: A branch and bound algorithm is developed which takes the specific structure into account and combines an outer approximation technique with a subdivision procedure, via a reverse convex transformation.
Abstract: In this paper the linear two-level problem is considered. The problem is reformulated to an equivalent quasiconcave minimization problem, via a reverse convex transformation. A branch and bound algorithm is developed which takes the specific structure into account and combines an outer approximation technique with a subdivision procedure.

Journal ArticleDOI
TL;DR: A new algorithm is proposed for dynamic lot size models (LSM) in which production and inventory cost functions are only assumed to be piecewise linear, and there are no assumptions of convexity, concavity or monotonicity.
Abstract: We propose a new algorithm for dynamic lot size models (LSM) in which production and inventory cost functions are only assumed to be piecewise linear. In particular, there are no assumptions of convexity, concavity or monotonicity. Arbitrary capacities on both production and inventory may occur, and backlogging is allowed. Thus the algorithm addresses most variants of the LSM appearing in the literature. Computational experience shows it to be very effective on NP-hard versions of the problem. For example, 48 period capacitated problems with production costs defined by eight linear segments are solvable in less than 2.5 minutes of Vax 8600 cpu time.

Journal ArticleDOI
TL;DR: A new finite algorithm for globally minimizing a concave function over a compact polyhedron that does not require the objective function to be separable or even analytically defined, requires no nonlinear computations, and requires no determinations of convex envelopes or underestimating functions.
Abstract: In this article we present a new finite algorithm for globally minimizing a concave function over a compact polyhedron. The algorithm combines a branch and bound search with a new process called neighbor generation. It is guaranteed to find an exact, extreme point optimal solution, does not require the objective function to be separable or even analytically defined, requires no nonlinear computations, and requires no determinations of convex envelopes or underestimating functions. Linear programs are solved in the branch and bound search which do not grow in size and differ from one another in only one column of data. Some preliminary computational experience is also presented.

Journal ArticleDOI
TL;DR: In this article, a special integral transformation is introduced to transform the objective function into a class of gradually deformed, but easier functions, and an optimization procedure is then applied to the new functions successively, to trace their solutions back to the original function.
Abstract: This paper presents our recent work on developing parallel algorithms and software for solving the global minimization problem for molecular conformation, especially protein folding. Global minimization problems are difficult to solve when the objective functions have many local minimizers, such as the energy functions for protein folding. In our approach, to avoid directly minimizing a "difficult" function, a special integral transformation is introduced to transform the function into a class of gradually deformed, but "smoother" or "easier" functions. An optimization procedure is then applied to the new functions successively, to trace their solutions back to the original function. The method can be applied to a large class of nonlinear partially separable functions including energy functions for molecular conformation and protein folding. Mathematical theory for the method, as a special continuation approach to global optimization, is established. Algorithms with different solutions tracing strategies are developed. Different levels of parallelism are exploited for the implementation of the algorithms on massively parallel architectures. Keywords: global/local minimization, numerical continuation, parallel computation, protein folding

Journal ArticleDOI
TL;DR: For bicriterion quasiconvex optimization problems, a constructive procedure for an approximation of the efficient outcomes is presented and the accuracy of the approximation is estimated.
Abstract: For bicriterion quasiconvex optimization problems, we present a constructive procedure for an approximation of the efficient outcomes. Performing this procedure we can estimate the accuracy of the approximation. Conversely, if we prescribe an accuracy for the approximation, we can calculate the number of points which have to be computed by a certain scalarization method to remain under the given accuracy. Finally, we give a numerical example.

Journal ArticleDOI
TL;DR: In this paper, a branch-and-bound method was proposed to find a global optimizer for the Weibull distribution, which uses first order conditions and projection to reduce the problem to a univariate optimization problem.
Abstract: Much work has been devoted to the problem of finding maximum likelihood estimators for the three-parameter Weibull distribution. This problem has not been clearly recognized as a global optimization one and most methods from the literature occasionally fail to find a global optimum. We develop a global optimization algorithm which uses first order conditions and projection to reduce the problem to a univariate optimization one. Bounds on the resulting function and its first order derivative are obtained and used in a branch-and-bound scheme. Computational experience is reported. It is also shown that the solution method we propose can be extended to the case of right censored samples.

Journal ArticleDOI
TL;DR: Some sufficient conditions under which a generalized linear complementarity problem (GLCP) can be solved as a pure linear complementity problem are introduced.
Abstract: We introduce some sufficient conditions under which a generalized linear complementarity problem (GLCP) can be solved as a pure linear complementarity problem. We also establish that the GLCP is in general a NP-Hard problem.