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Showing papers in "Journal of Global Optimization in 1997"


Journal ArticleDOI
Rainer Storn1, Kenneth Price
TL;DR: In this article, a new heuristic approach for minimizing possibly nonlinear and non-differentiable continuous space functions is presented, which requires few control variables, is robust, easy to use, and lends itself very well to parallel computation.
Abstract: A new heuristic approach for minimizing possibly nonlinear and non-differentiable continuous space functions is presented. By means of an extensive testbed it is demonstrated that the new method converges faster and with more certainty than many other acclaimed global optimization methods. The new method requires few control variables, is robust, easy to use, and lends itself very well to parallel computation.

24,053 citations


Journal ArticleDOI
TL;DR: A collection of electronically available data instances for the Quadratic Assignment Problem are described, indicating whether or not the problem is solved to optimality and the best known bounds for the problem are supplied.
Abstract: A collection of electronically available data instances for the Quadratic Assignment Problem is described. For each instance, we provide detailed information, indicating whether or not the problem is solved to optimality. If not, we supply the best known bounds for the problem. Moreover we survey available software and describe recent dissertations related to the Quadratic Assignment Problem.

826 citations


Journal ArticleDOI
TL;DR: The new algorithm, CDA, efficiently produces local optima and sometimes produces global optima inLinearly constrained indefinite quadratic problems and a decomposition branch and bound method for globally solving these problems is proposed.
Abstract: Linearly constrained indefinite quadratic problems play an important role in global optimization. In this paper we study d.c. theory and its local approach to such problems. The new algorithm, CDA, efficiently produces local optima and sometimes produces global optima. We also propose a decomposition branch and bound method for globally solving these problems. Finally many numerical simulations are reported.

296 citations


Journal ArticleDOI
TL;DR: This paper presents necessary and sufficient conditions for a convex envelope to be apolyhedral function and illustrates how these conditions may be used inconstructing of convex envelopes.
Abstract: Convex envelopes of multilinear functions on a unit hypercube are polyhedral. This well-known fact makes the convex envelope approximation very useful in the linearization of non-linear 0–1 programming problems and in global bilinear optimization. This paper presents necessary and sufficient conditions for a convex envelope to be a polyhedral function and illustrates how these conditions may be used in constructing of convex envelopes. The main result of the paper is a simple analytical formula, which defines some faces of the convex envelope of a multilinear function. This formula proves to be a generalization of the well known convex envelope formula for multilinear monomial functions.

188 citations


Journal ArticleDOI
TL;DR: A new, global optimization algorithm is proposed which tries to exploit favourabledata constellations, focussing on the continuous problem formulation: maximize a quadratic form over the standard simplex.
Abstract: As is well known, the problem of finding a maximum clique in a graph is NP-hard. Nevertheless, NP-hard problems may have easy instances. This paper proposes a new, global optimization algorithm which tries to exploit favourable data constellations, focussing on the continuous problem formulation: maximize a quadratic form over the standard simplex. Some general connections of the latter problem with dynamic principles of evolutionary game theory are established. As an immediate consequence, one obtains a procedure which consists (a) of an iterative part similar to interior-path methods based on the so-called replicator dynamics; and (b) a routine to escape from inefficient, locally optimal solutions. For the special case of finding a maximum clique in a graph where the quadratic form arises from a regularization of the adjacence matrix, part (b), i.e. escaping from maximal cliques not of maximal size, is accomplished with block pivoting methods based on (large) independent sets, i.e. cliques of the complementary graph. A simulation study is included which indicates that the resulting procedure indeed has some merits.

169 citations


Journal ArticleDOI
TL;DR: Some fundamental characterizations of the SDP relaxation including its equivalence to arelaxation using convex-quadratic valid inequalities for the feasible region of the QP are presented.
Abstract: This paper applies the SDP (semidefinite programming) relaxation originally developed for a 0-1 integer program to a general nonconvex QP (quadratic program) having a linear objective function and quadratic inequality constraints, and presents some fundamental characterizations of the SDP relaxation including its equivalence to a relaxation using convex-quadratic valid inequalities for the feasible region of the QP.

149 citations


Journal ArticleDOI
TL;DR: In this paper Bayesian analysis and Wiener process are used in orderto build an algorithm to solve the problem of globaloptimization and the Bayesian approach is exploited not only in the choice of the Wiener model but also in the estimation of the parameter σ2 of theWiener process.
Abstract: In this paper Bayesian analysis and Wiener process are used in order to build an algorithm to solve the problem of global optimization The paper is divided in two main parts In the first part an already known algorithm is considered: a new (Bayesian) stopping rule is added to it and some results are given, such as an upper bound for the number of iterations under the new stopping rule In the second part a new algorithm is introduced in which the Bayesian approach is exploited not only in the choice of the Wiener model but also in the estimation of the parameter \sigma^2 of the Wiener process, whose value appears to be quite crucial Some results about this algorithm are also given

135 citations


Journal ArticleDOI
TL;DR: It is shown that the feasibility of a system of m linear inequalities over the cone of symmetric positive semidefinite matrices of order n can be tested in mn with arithmetic operations with -bit numbers.
Abstract: We show that the feasibility of a system of m linear inequalities over the cone of symmetric positive semidefinite matrices of order n can be tested in mn^O(min{m,n^2}) arithmetic operations with ln^O(min{m,n^2})-bit numbers, where l is the maximum binary size of the input coefficients. We also show that any feasible system of dimension (m,n) has a solution X such that log||X|| ≤ ln^O(min{m,n^2}).

106 citations


Journal ArticleDOI
TL;DR: By using the topological degree the concept of ’’exceptionalfamily of elements‘‘ specifically for continuous functions is introduced, which has important consequences pertaining to the solvability of the explicit, the implicit and the general order complementarity problems.
Abstract: By using the topological degree we introduce the concept of ’’exceptional family of elements‘‘ specifically for continuous functions. This has important consequences pertaining to the solvability of the explicit, the implicit and the general order complementarity problems. In this way a new direction for research in the complementarity theory is now opened.

96 citations


Journal ArticleDOI
TL;DR: The proposed algorithm is a new version of the well known Price's algorithm and its distinguishing feature is that it tries to employ as much as possible the information about the objective function obtained at previous iterates.
Abstract: We present an algorithm for finding a global minimum of a multimodal, multivariate function whose evaluation is very expensive, affected by noise and whose derivatives are not available. The proposed algorithm is a new version of the well known Price‘s algorithm and its distinguishing feature is that it tries to employ as much as possible the information about the objective function obtained at previous iterates. The algorithm has been tested on a large set of standard test problems and it has shown a satisfactory computational behaviour. The proposed algorithm has been used to solve efficiently some difficult optimization problems deriving from the study of eclipsing binary star light curves.

92 citations


Journal ArticleDOI
TL;DR: The resulting algorithm performs considerably better than the earlier versions of the CRS algorithms and could offer a reasonable alternative to many currently availablestochastic algorithms, especially for problems requiring ’direct search‘type methods.
Abstract: In this paper we propose a new version of the Controlled Random Search (CRS) algorithm of Price. The new algorithm has been tested on thirteen global optimization test problems. Numerical experiments indicate that the resulting algorithm performs considerably better than the earlier versions of the CRS algorithms. The algorithm, therefore, could offer a reasonable alternative to many currently available stochastic algorithms, especially for problems requiring `direct search'' type methods. Also a classification of the CRS algorithms is made based on `global technique'' -- `local technique'' and the relative performance of classes is numerically explored.

Journal ArticleDOI
TL;DR: This paper summarizes the current state of knowledge concerning putative global minima of the potential energy function for Lennard-Jones clusters, an intensely studied molecular conformation problem.
Abstract: This paper summarizes the current state of knowledge concerning putative global minima of the potential energy function for Lennard-Jones clusters, an intensely studied molecular conformation problem. Almost all known exceptions to global optimality of the well-known Northby multilayer icosahedral conformations for microclusters are shown to be minor variants of that geometry. The truly exceptional case of face-centered cubic lattice conformations is examined and connections are made with the macrocluster problem. Several types of algorithms and their limitations are explored, and a new variation on the growth sequence idea is presented and shown to be effective for both small and large clusters.

Journal ArticleDOI
TL;DR: It is shown that there existsubdifferentials which may be smaller than the Michel–Penot subdifferential and which have certain useful calculus which hold for the Clarke sub differential only in the regular case.
Abstract: Certain useful basic results of the gradient (in the smooth case), the Clarke subdifferential, the Michel–Penot subdifferential, which is also known as the "small" subdifferential, and the directional derivative (in the nonsmooth case) are stated and discussed One of the advantages of the Michel–Penot subdifferential is the fact that it is in general "smaller" than the Clarke subdifferential In this paper it is shown that there exist subdifferentials which may be smaller than the Michel–Penot subdifferential and which have certain useful calculus It is further shown that in the case of quasidifferentiability, the Michel–Penot subdifferential enjoys calculus which hold for the Clarke subdifferential only in the regular case

Journal ArticleDOI
TL;DR: This paper addresses a global optimization approach to a water distribution network design problem, employing an arc-based formulation that is linear except forcertain complicating head-loss constraints and developing a first globaloptimization scheme for this model.
Abstract: In this paper, we address a global optimization approach to a water distribution network design problem. Traditionally, a variety of local optimization schemes have been developed for such problems, each new method discovering improved solutions for some standard test problems, with no known lower bound to test the quality of the solutions obtained. A notable exception is a recent paper by Eiger et al. (1994) who present a first global optimization approach for a loop and path-based formulation of this problem, using a semi-infinite linear program to derive lower bounds. In contrast, we employ an arc-based formulation that is linear except for certain complicating head-loss constraints and develop a first global optimization scheme for this model. Our lower bounds are derived through the design of a suitable Reformulation-Linearization Technique (RLT) that constructs a tight linear programming relaxation for the given problem, and this is embedded within a branch-and-bound algorithm. Convergence to an optimal solution is induced by coordinating this process with an appropriate partitioning scheme. Some preliminary computational experience is provided on two versions of a particular standard test problem for the literature for which an even further improved solution is discovered, but one that is verified for the first time to be an optimum, without any assumed bounds on the flows. Two other variants of this problem are also solved exactly for illustrative purposes and to provide researchers with additional test cases having known optimal solutions. Suggestions on a more elaborate study involving several algorithmic enhancements are presented for future research.

Journal ArticleDOI
TL;DR: A deterministic global optimization method is described for identifying the global minimum potential energy conformation of oligopeptides and provides valuable information on upper and lower bounds of theglobal minimum energy structure and low energy conformers close to the globalminimum one.
Abstract: A deterministic global optimization method is described for identifying the global minimum potential energy conformation of oligopeptides. The ECEPP/3 detailed potential energy model is utilized for describing the energetics of the atomic interactions posed in the space of the peptide dihedral angles. Based on previous work on the microcluster and molecular structure determination [21, 22, 23, 24], a procedure for deriving convex lower bounding functions for the total potential energy function is developed. A procedure that allows the exclusion of domains of the (o, ψ) space based on the analysis of experimentally determined native protein structures is presented. The reduced disjoint sub-domains are appropriately combined thus defining the starting regions for the search. The proposed approach provides valuable information on (i) the global minimum potential energy conformation, (ii) upper and lower bounds of the global minimum energy structure and (iii) low energy conformers close to the global minimum one. The proposed approach is illustrated with Ac-Ala4-Pro-NHMe, Met-enkephalin, Leu-enkephalin, and Decaglycine.

Journal ArticleDOI
TL;DR: A general class of branch and bound algorithms forsolving a wide class of nonlinear programs with branching only in asubset of the problem variables is presented, and this technique may dramatically reduce the number of iterations and time required for convergence to tolerance while retaining proven exact convergence in the infinite limit.
Abstract: A general class of branch and bound algorithms for solving a wide class of nonlinear programs with branching only in a subset of the problem variables is presented. By reducing the dimension of the search space, this technique may dramatically reduce the number of iterations and time required for convergence to e tolerance while retaining proven exact convergence in the infinite limit. This presentation includes specifications of the class of nonlinear programs, a statement of a class of branch and bound algorithms, a convergence proof, and motivating examples with results.

Journal ArticleDOI
TL;DR: A class of parallel characteristical algorithms for global optimization of one-dimensional multiextremal functions and a generalization for the multidimensional case is considered.
Abstract: A class of parallel characteristical algorithms for global optimization of one-dimensional multiextremal functions is introduced. General convergence and efficiency conditions for the algorithms of the class introduced are established. A generalization for the multidimensional case is considered. Examples of parallel characteristical algorithms and numerical experiments are presented.

Journal ArticleDOI
TL;DR: A new multi-dimensional method to solve unconstrained global optimization problems with Lipschitzian first derivatives is proposed, based on apartition scheme that subdivides the search domain into a set of hypercubes in the course of optimization.
Abstract: In this paper we propose a new multi-dimensional method to solve unconstrained global optimization problems with Lipschitzian first derivatives. The method is based on a partition scheme that subdivides the search domain into a set of hypercubes in the course of optimization. This partitioning is regulated by the decision rule that provides evaluation of the "importance" of each generated hypercube and selection of some partition element to perform the next iteration. Sufficient conditions of global convergence for the new method are investigated. Results of numerical experiments are also presented.

Journal ArticleDOI
TL;DR: The proposed global optimization method is related to the previous stochastic/perturbation global optimization methods for finding minimum energy configurations, but has several key differences that are important to its success.
Abstract: We present a new global optimization approach for solving exactly or inexactly constrained distance geometry problems. Distance geometry problems are concerned with determining spatial structures from measurements of internal distances. They arise in the structural interpretation of nuclear magnetic resonance data and in the prediction of protein structure. These problems can be naturally formulated as global optimization problems which generally are large and difficult. The global optimization method that we present is related to our previous stochastic/perturbation global optimization methods for finding minimum energy configurations, but has several key differences that are important to its success. Our computational results show that the method readily solves a set of artificial problems introduced by More and Wu that have up to 343 atoms. On a set of considerably more difficult protein fragment problems introduced by Hendrickson, the method solves all the problems with up to 377 atoms exactly, and finds nearly exact solution for all the remaining problems which have up to 777 atoms. These preliminary results indicate that this approach has very good promise for helping to solve distance geometry problems.

Journal ArticleDOI
TL;DR: It is proved that applying RLT directly to the original polynomial program produces a bound that dominates in the sense of being at least as tight as the value obtained when RLT is applied to the joint collection of all equivalent quadratic problems that could be constructed by recursively defining additional variables as suggested by Shor.
Abstract: In this paper, we compare two strategies for constructing linear programming relaxations for polynomial programming problems using a Reformulation-Linearization Technique (RLT). RLT involves an automatic reformulation of the problem via the addition of certain nonlinear implied constraints that are generated by using the products of the simple bounding restrictions (among other products), and a subsequent linearization based on variable redefinitions. We prove that applying RLT directly to the original polynomial program produces a bound that dominates in the sense of being at least as tight as the value obtained when RLT is applied to the joint collection of all equivalent quadratic problems that could be constructed by recursively defining additional variables as suggested by Shor.

Journal ArticleDOI
TL;DR: A modification to the QP subproblem is given and a modified QP method is provided which guarantees that the associated constraint region isn’t empty and for which a robust convergence theory is established.
Abstract: The sequential quadratic programming method developed by Wilson, Han and Powell may fail if the quadratic programming subproblems become infeasible or if the associated sequence of search directions is unbounded. In [1], Han and Burke give a modification to this method wherein the QP subproblem is altered in a way which guarantees that the associated constraint region is nonempty and for which a robust convergence theory is established. In this paper, we give a modification to the QP subproblem and provide a modified SQP method. Under some conditions, we prove that the algorithm either terminates at a Kuhn–Tucker point within finite steps or generates an infinite sequence whose every cluster is a Kuhn–Tucker point. Finally, we give some numerical examples.

Journal ArticleDOI
TL;DR: It is proved that the minimum inter-particle distance at any global minimizer of Lennard-Jones clusters is bounded away from zero by a positive constant which is independent of the number of particles.
Abstract: In computer simulations of molecular conformation and protein folding, a significant part of computing time is spent in the evaluation of potential energy functions and force fields. Therefore many algorithms for fast evaluation of potential energy functions and force fields are proposed in the literature. However, most of these algorithms assume that the particles are uniformly distributed in order to guarantee good performance. In this paper, we prove that the minimum inter-particle distance at any global minimizer of Lennard-Jones clusters is bounded away from zero by a positive constant which is independent of the number of particles. As a by-product, we also prove that the global minimum of an n particle Lennard-Jones cluster is bounded between two linear functions. Our first result is useful in the design of fast algorithms for potential function and force field evaluation. Our second result can be used to decide how good a local minimizer is.

Journal ArticleDOI
TL;DR: An aspiration based simulated annealing algorithm for continuousvariables that appears to offer a useful alternative to some of the currently availablestochastic algorithms for global optimization.
Abstract: An aspiration based simulated annealing algorithm for continuous variables has been proposed. The new algorithm is similar to the one given by Dekkers and Aarts (1991) except that a kind of memory is introduced into the procedure with a self-regulatory mechanism. The algorithm has been applied to a set of standard global optimization problems and a number of more difficult, complex, practical problems and its performance compared with that of the algorithm of Dekkers and Aarts (1991). The new algorithm appears to offer a useful alternative to some of the currently available stochastic algorithms for global optimization.

Journal ArticleDOI
TL;DR: In this paper, the sum of a convex function and a product of k nonnegative convex functions over a set is reduced to a k-dimensional quasiconcave minimization problem which is solved by a conical branch-and-bound algorithm.
Abstract: We present a new method for minimizing the sum of a convex function and a product of k nonnegative convex functions over a convex set. This problem is reduced to a k-dimensional quasiconcave minimization problem which is solved by a conical branch-and-bound algorithm. Comparative computational results are provided on test problems from the literature.

Journal ArticleDOI
TL;DR: A Maximum Entropy based approach to the restoration of degraded images as an alternative to restoration techniques using inverse Wiener filtering and a variety of experimental results supporting the imaging model are included.
Abstract: We present a Maximum Entropy based approach to the restoration of degraded images as an alternative to restoration techniques using inverse Wiener filtering. The method we discuss applies in particular to images corrupted by a relatively high system noise. A variety of experimental results supporting our imaging model are included.

Journal ArticleDOI
TL;DR: This paper proposes a global search phase in which an aperiodic and bounded trace function is added to the search to first identify promising regions for local search, using a trace to aid in identifying promising regions before committing to local searches.
Abstract: In this paper we present a method called NOVEL (Nonlinear Optimization via External Lead) for solving continuous and discrete global optimization problems. NOVEL addresses the balance between global search and local search, using a trace to aid in identifying promising regions before committing to local searches. We discuss NOVEL for solving continuous constrained optimization problems and show how it can be extended to solve constrained satisfaction and discrete satisfiability problems. We first transform the problem using Lagrange multipliers into an unconstrained version. Since a stable solution in a Lagrangian formulation only guarantees a local optimum satisfying the constraints, we propose a global search phase in which an aperiodic and bounded trace function is added to the search to first identify promising regions for local search. The trace generates an information-bearing trajectory from which good starting points are identified for further local searches. Taking only a small portion of the total search time, this elegant approach significantly reduces unnecessary local searches in regions leading to the same local optimum. We demonstrate the effectiveness of NOVEL on a collection of continuous optimization benchmark problems, finding the same or better solutions while satisfying the constraints. We extend NOVEL to discrete constraint satisfaction problems (CPSs) by showing an efficient transformation method for CSPs and the associated representation in finite- difference equations in NOVEL. We apply NOVEL to solve Boolean satisfiability instances in circuit fault detection and circuit synthesis applications, and show comparable performance when compared to the best existing method.

Journal ArticleDOI
TL;DR: An algorithm is designed for the cubic lattice model using tabu search and it is shown that this method outperforms previously reported approaches for the same model.
Abstract: We apply tabu search techniques to the problem of determining the optimal configuration of a chain of protein sequences on a cubic lattice. The problem under study is difficult to solve because of the large number of possible conformations and enormous amount of computations required. Tabu search is an iterative heuristic procedure which has been shown to be a remarkably effective method for solving combinatorial optimization problems. In this paper, an algorithm is designed for the cubic lattice model using tabu search. The algorithm has been tested on a chain of 27 monomers. Computational results show that our method outperforms previously reported approaches for the same model.

Journal ArticleDOI
TL;DR: An analgorithm is obtained which tries to exploit favourable data constellations in a systematic way, and to avoid the worst-case behaviourof such NP-hard problems whenever possible.
Abstract: Consider the problem of maximizing a quadratic form over the standard simplex. Problems of this type occur, e.g., in the search for the maximum (weighted) clique in an undirected graph. In this paper, copositivity-based escape procedures from inefficient local solutions are rephrased into lower-dimensional subproblems which are again of the same type. As a result, an algorithm is obtained which tries to exploit favourable data constellations in a systematic way, and to avoid the worst-case behaviour of such NP-hard problems whenever possible. First results on finding large cliques in DIMACS benchmark graphs are encouraging.

Journal ArticleDOI
TL;DR: It is shown that one can globally minimize the total cost of production and transportation by solving a Hitchcock transportation problem with m sources and n terminals and a minimum linear-cost flow problem withm+n nodes.
Abstract: In this paper, we propose a primal-dual algorithm for solving a class of production-transportation problems. Among m(≥ 2) sources two factories exist, which produce given goods at some concave cost and supply them to n terminals. We show that one can globally minimize the total cost of production and transportation by solving a Hitchcock transportation problem with m sources and n terminals and a minimum linear-cost flow problem with m+n nodes. The number of arithmetic operations required by the algorithm is pseudo-polynomial in the problem input length.

Journal ArticleDOI
TL;DR: A fundamental role is played by the probability of reaching the global minimum, whose asymptoticalbehavior allows to provide useful information on the efficiency of repeated trials.
Abstract: While searching for the global minimum of a cost function we have often to decide if a restart from a different initial point would be more advantageous than continuing current optimization. This is a particular case of the efficiency comparison between repeated minimizations and single extended search having the same total length. A theoretical approach for the treatment of this general problem forms the subject of the present paper. A fundamental role is played by the probability of reaching the global minimum, whose asymptotical behavior allows to provide useful information on the efficiency of repeated trials. The second part of this work is devoted to a detailed analysis of three optimization algorithms whose evolution is independent of the cost function to be minimized: pure random search, grid search and random walk. These three examples give an interesting validation of the theoretical results and provide a general procedure which can be employed in the study of more complex optimization problems.