Showing papers in "Journal of Global Optimization in 2000"
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TL;DR: A finite new algorithm is proposed for clustering m given points in n-dimensional real space into k clusters by generating k planes that constitute a local solution to the nonconvex problem of minimizing the sum of squares of the 2-norm distances between each point and a nearest plane.
Abstract: A finite new algorithm is proposed for clustering m given points in n-dimensional real space into k clusters by generating k planes that constitute a local solution to the nonconvex problem of minimizing the sum of squares of the 2-norm distances between each point and a nearest plane. The key to the algorithm lies in a formulation that generates a plane in n-dimensional space that minimizes the sum of the squares of the 2-norm distances to each of m1 given points in the space. The plane is generated by an eigenvector corresponding to a smallest eigenvalue of an n × n simple matrix derived from the m1 points. The algorithm was tested on the publicly available Wisconsin Breast Prognosis Cancer database to generate well separated patient survival curves. In contrast, the k-mean algorithm did not generate such well-separated survival curves.
424 citations
TL;DR: An algorithm of the basin-hopping type is presented which has found all of the current putative global minima in the literature up to 110 atoms, as well as discovered a new global minimum for the 98-atom cluster of a novel geometrical class.
Abstract: Molecular conformation problems arising in computational chemistry require the global minimization of a non-convex potential energy function representing the interactions of, for example, the component atoms in a molecular system. Typically the number of local minima on the potential energy surface grows exponentially with system size, and often becomes enormous even for relatively modestly sized systems. Thus the simple multistart strategy of randomly sampling local minima becomes impractical. However, for many molecular conformation potential energy surfaces the local minima can be organized by a simple adjacency relation into a single or at most a small number of funnels. A distinguished local minimum lies at the bottom of each funnel and a monotonically descending sequence of adjacent local minima connects every local minimum in the funnel with the funnel bottom. Thus the global minimum can be found among the comparatively small number of funnel bottoms, and a multistart strategy based on sampling funnel bottoms becomes viable. In this paper we present such an algorithm of the basin-hopping type and apply it to the Lennard–Jones cluster problem, an intensely studied molecular conformation problem which has become a benchmark for global optimization algorithms. Results of numerical experiments are presented which confirm both the multifunneling character of the Lennard–Jones potential surface as well as the efficiency of the algorithm. The algorithm has found all of the current putative global minima in the literature up to 110 atoms, as well as discovered a new global minimum for the 98-atom cluster of a novel geometrical class.
220 citations
TL;DR: The primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase, and are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached.
Abstract: A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization FFT where F is some non-negative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual affine-scaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.
204 citations
TL;DR: A deterministic global optimization approach based on a branch and bound framework is introduced to address the nonlinear optimal control problem to global optimality and only mild conditions on the differentiability of the dynamic system are required.
Abstract: The accurate solution of optimal control problems is crucial in many areas of engineering and applied science. For systems which are described by a nonlinear set of differential-algebraic equations, these problems have been shown to often contain multiple local minima. Methods exist which attempt to determine the global solution of these formulations. These algorithms are stochastic in nature and can still get trapped in local minima. There is currently no deterministic method which can solve, to global optimality, the nonlinear optimal control problem. In this paper a deterministic global optimization approach based on a branch and bound framework is introduced to address the nonlinear optimal control problem to global optimality. Only mild conditions on the differentiability of the dynamic system are required. The implementa-tion of the approach is discussed and computational studies are presented for four control problems which exhibit multiple local minima.
160 citations
TL;DR: It is shown that the Steiner ratio is 1/ 4, that is, the minimum spanning tree yields a polynomial-time approximation with performance ratio exactly 4, and there exists a poynomial- time approxi-mation scheme under certain conditions.
Abstract: Given n terminals in the Euclidean plane and a positive constant, find a Steiner tree interconnecting all terminals with the minimum number of Steiner points such that the Euclidean length of each edge is no more than the given positive constant. This problem is NP-hard with applications in VLSI design, WDM optical networks and wireless communications. In this paper, we show that (a) the Steiner ratio is 1/ 4, that is, the minimum spanning tree yields a polynomial-time approximation with performance ratio exactly 4, (b) there exists a polynomial-time approximation with performance ratio 3, and (c) there exists a polynomial-time approxi-mation scheme under certain conditions.
143 citations
TL;DR: It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in \realn can be reformulated into certain smooth/Convex unconstrained dual problems in\realm with m≤slant n and without duality gap, and some NP-hard concave minimization problems can be transformed into unconstrains convex minimization dual problems.
Abstract: This paper presents, within a unified framework, a potentially powerful canonical dual transformation method and associated generalized duality theory in nonsmooth global optimization. It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in \realn can be reformulated into certain smooth/convex unconstrained dual problems in \realm with m≤slant n and without duality gap, and some NP-hard concave minimization problems can be transformed into unconstrained convex minimization dual problems. The extended Lagrange duality principles proposed recently in finite deformation theory are generalized suitable for solving a large class of nonconvex and nonsmooth problems. The very interesting generalized triality theory can be used to establish nice theoretical results and to develop efficient alternative algorithms for robust computations.
114 citations
TL;DR: A parallel greedy randomized adaptive search procedure (GRASP) for the Steiner problem in graphs and the main contribution of the parallel algorithm concerns the fact that larger speedups of the same order of the number of processors are obtained exactly for the most difficult problems.
Abstract: In this paper, we present a parallel greedy randomized adaptive search procedure (GRASP) for the Steiner problem in graphs. GRASP is a two-phase metaheuristic. In the first phase, solutions are constructed using a greedy randomized procedure. Local search is applied in the second phase, leading to a local minimum with respect to a specified neighborhood. In the Steiner problem in graphs, feasible solutions can be characterized by their non-terminal nodes (Steiner nodes) or by their key-paths. According to this characterization, two GRASP procedures are described using different local search strategies. Both use an identical construction procedure. The first uses a node-based neighborhood for local search, while the second uses a path-based neighborhood. Computational results comparing the two procedures show that while the node-based variant produces better quality solutions, the path-based variant is about twice as fast. A hybrid GRASP procedure combining the two neighborhood search strategies is then proposed. Computational experiments with a parallel implementation of the hybrid procedure are reported, showing that the algorithm found optimal solutions for 45 out of 60 benchmark instances and was never off by more than 4% of the optimal solution value. The average speedup results observed for the test problems show that increasing the number of processors reduces elapsed times with increasing speedups. Moreover, the main contribution of the parallel algorithm concerns the fact that larger speedups of the same order of the number of processors are obtained exactly for the most difficult problems.
98 citations
TL;DR: It will be proved that, under suitable assumptions, the annealing algorithm for continuous global optimization is convergent.
Abstract: In this paper a simulated annealing algorithm for continuous global optimization will be considered. The algorithm, in which a cooling schedule based on the distance between the function value in the current point and an estimate of the global optimum value is employed, has been first introduced in Bohachevsky, Johnson and Stein (1986) [2], but without any proof of convergence. Here it will be proved that, under suitable assumptions, the algorithm is convergent
93 citations
TL;DR: This work introduces various notions of well-posedness for a family of variational inequalities and for an optimization problem with constraints defined byVariational inequalities having a unique solution and presents an application to an exact penalty method.
Abstract: We introduce various notions of well-posedness for a family of variational inequalities and for an optimization problem with constraints defined by variational inequalities having a unique solution. Then, we give sufficient conditions for well-posedness of these problems and we present an application to an exact penalty method.
92 citations
TL;DR: Duality theorems are proved for Mond–Weir and general Mond-Weir type duality under the above generalized type I assumptions.
Abstract: In this paper, we are concerned with the multiobjective programming problem with inequality constraints. We introduce new classes of generalized type I vector-valued functions. Duality theorems are proved for Mond–Weir and general Mond–Weir type duality under the above generalized type I assumptions.
90 citations
TL;DR: It is shown that these problems can be solved in an efficient manner by adapting a branch and bound algorithm proposed by Androulakis–Maranas–Floudas for nonconvex problems containing products of two variables.
Abstract: This paper is concerned with a practical algorithm for solving low rank linear multiplicative programming problems and low rank linear fractional programming problems. The former is the minimization of the sum of the product of two linear functions while the latter is the minimization of the sum of linear fractional functions over a polytope. Both of these problems are nonconvex minimization problems with a lot of practical applications. We will show that these problems can be solved in an efficient manner by adapting a branch and bound algorithm proposed by Androulakis–Maranas–Floudas for nonconvex problems containing products of two variables. Computational experiments show that this algorithm performs much better than other reported algorithms for these class of problems.
TL;DR: First, the existence of solutions to a parabolic hemivariational inequality which contains nonlinear evolution operator is established and the upper semicontinuity property of the solution set of the inequality is proved.
Abstract: In this paper we study the optimal control of systems driven by parabolic hemivariational inequalities. First, we establish the existence of solutions to a parabolic hemivariational inequality which contains nonlinear evolution operator. Introducing a control variable in the second member and in the multivalued term, we prove the upper semicontinuity property of the solution set of the inequality. Then we use this result and the direct method of the calculus of variations to show the existence of optimal admissible state–control pairs.
TL;DR: The existence of solutions of the generalized vector equilibrium problem in the setting of Hausdorff topological vector spaces is proved.
Abstract: In this paper we prove the existence of solutions of the generalized vector equilibrium problem in the setting of Hausdorff topological vector spaces As applications, we present some relevant particular cases: a generalized vector variational-like inequality in Hausdorff topological vector spaces, and equilibrium problem in the case of pseudomonotone real functions, and a generalized weak Pareto optima problem
TL;DR: Variants of interval branch-and-bound algorithms for global optimization where the bisection step was substituted by the subdivision of the current, actual interval into many subintervals in a single iteration step are investigated.
Abstract: We have investigated variants of interval branch-and-bound algorithms for global optimization where the bisection step was substituted by the subdivision of the current, actual interval into many subintervals in a single iteration step. The convergence properties of the multisplitting methods, an important class of multisection procedures are investigated in detail. We also studied theoretically the convergence improvements caused by multisection on algorithms which involve the accelerating tests (like e.g. the monotonicity test). The results are published in two papers, the second one contains the numerical test result.
TL;DR: It is shown that any proper positively homogeneous function annihilating at the origin is a pointwise minimum of sublinear functions (MSL function) and an application to a locally Lipschitz extremum problem without constraint qualifications is presented.
Abstract: In this paper we show that any proper positively homogeneous function annihilating at the origin is a pointwise minimum of sublinear functions (MSL function). By means of a generalized Gordan's theorem for inequality systems with MSL functions, we present an application to a locally Lipschitz extremum problem without constraint qualifications.
TL;DR: The paper proposes a modified basis reduction method for direct application to the two-surface plasticity model of bounded kinematic hardening material and shows an enlargement of the load carrying capacity due to bounded hardening.
Abstract: Limit and shakedown analysis are effective methods for assessing the load carrying capacity of a given structure. The elasto–plastic behavior of the structure subjected to loads varying in a given load domain is characterized by the shakedown load factor, defined as the maximum factor which satisfies the sufficient conditions stated in the corresponding static shakedown theorem. The finite element dicretization of the problem may lead to very large convex optimization. For the effective solution a basis reduction method has been developed that makes use of the special problem structure for perfectly plastic material. The paper proposes a modified basis reduction method for direct application to the two-surface plasticity model of bounded kinematic hardening material. The considered numerical examples show an enlargement of the load carrying capacity due to bounded hardening.
TL;DR: A new approach for solving the general NPLNFP in a continuous formulation by adapting a dynamic domain contraction (DDC) algorithm is presented and preliminary computational results on a wide range of test problems are reported.
Abstract: We consider the Nonconvex Piecewise Linear Network Flow Problem (NPLNFP) which is known to be {\mathcal NP}-hard Although exact methods such as branch and bound have been developed to solve the NPLNFP, their computational requirements increase exponentially with the size of the problem Hence, an efficient heuristic approach is in need to solve large scale problems appearing in many practical applications including transportation, production-inventory management, supply chain, facility expansion and location decision, and logistics In this paper, we present a new approach for solving the general NPLNFP in a continuous formulation by adapting a dynamic domain contraction A Dynamic Domain Contraction (DDC) algorithm is presented and preliminary computational results on a wide range of test problems are reported The results show that the proposed algorithm generates solutions within 0 to 094 % of optimality in all instances that the exact solutions are available from a branch and bound method
TL;DR: This paper presents a simple analysis of Simulated Annealing that will provide a time bound for convergence with overwhelming probability, and gives a simpler and more general proof of convergence for Nested Annealing, a heuristic algorithm developed in [12].
Abstract: Simulated Annealing is a family of randomized algorithms used to solve many combinatorial optimization problems. In practice they have been applied to solve some presumably hard (e.g., NP-complete) problems. The level of performance obtained has been promising [2,5,6,14]. The success of this heuristic technique has motivated analysis of this algorithm from a theoretical point of view. In particular, people have looked at the convergence of this algorithm. They have shown (see e.g., [10]) that this algorithm converges in the limit to a globally optimal solution with probability 1. However few of these convergence results specify a time limit within which the algorithm is guaranteed to converge (with some high probability, say). We present, for the first time, a simple analysis of SA that will provide a time bound for convergence with overwhelming probability. The analysis will hold no matter what annealing schedule is used. Convergence of Simulated Annealing in the limit will follow as a corollary to our time convergence proof. In this paper we also look at optimization problems for which the cost function has some special properties. We prove that for these problems the convergence is much faster. In particular, we give a simpler and more general proof of convergence for Nested Annealing, a heuristic algorithm developed in [12]. Nested Annealing is based on defining a graph corresponding to the given optimization problem. If this graph is `small separable', they [12] show that Nested Annealing will converge `faster'. For an arbitrary optimization problem, we may not have any knowledge about the `separability' of its graph. In this paper we give tight bounds for the `separability' of a random graph. We then use these bounds to analyze the expected behavior of Nested Annealing on an arbitrary optimization problem. The `separability' bounds we derive in this paper are of independent interest and have the potential of finding other applications.
TL;DR: Analysis of variants of interval branch-and-bound algorithms for global optimization where the bisection step was substituted by the subdivision of the current, actual interval into many subintervals in a single iteration step indicates that multisection can substantially improve the efficiency of interval global optimization procedures.
Abstract: We have investigated variants of interval branch-and-bound algorithms for global optimization where the bisection step was substituted by the subdivision of the current, actual interval into many subintervals in a single iteration step. The results are published in two papers, the first one contains the theoretical investigations on the convergence properties. An extensive numerical study indicates that multisection can substantially improve the efficiency of interval global optimization procedures, and multisection seems to be indispensable in solving hard global optimization problems in a reliable way.
TL;DR: This work uses Bayesian decision theory to address a variable selection problem arising in attempts to indirectly measure the quality of hospital care, by comparing observed mortality rates to expected values based on patient sickness at admission.
Abstract: We use Bayesian decision theory to address a variable selection problem arising in attempts to indirectly measure the quality of hospital care, by comparing observed mortality rates to expected values based on patient sickness at admission. Our method weighs data collection costs against predictive accuracy to find an optimal subset of the available admission sickness variables. The approach involves maximizing expected utility across possible subsets, using Monte Carlo methods based on random division of the available data into N modeling and validation splits to approximate the expectation. After exploring the geometry of the solution space, we compare a variety of stochastic optimization methods –- including genetic algorithms (GA), simulated annealing (SA), tabu search (TS), threshold acceptance (TA), and messy simulated annealing (MSA) –- on their performance in finding good subsets of variables, and we clarify the role of N in the optimization. Preliminary results indicate that TS is somewhat better than TA and SA in this problem, with MSA and GA well behind the other three methods. Sensitivity analysis reveals broad stability of our conclusions.
TL;DR: To accelerate the convergence of the algorithm, the bounding operation is reinforced using a Lagrangian relaxation, which is a concave minimization but yields a tighter bound than the usual linear programming relaxation in O(mn log n) additional time.
Abstract: We propose a branch-and-bound algorithm of Falk–Soland's type for solving the minimum cost production-transportation problem with concave production costs. To accelerate the convergence of the algorithm, we reinforce the bounding operation using a Lagrangian relaxation, which is a concave minimization but yields a tighter bound than the usual linear programming relaxation in O(mn log n) additional time. Computational results indicate that the algorithm can solve fairly large scale problems.
TL;DR: How the performance of a standard semismooth Newton-type method as a basic solver for complementarity problems can be improved by incorporating two well-known global optimization algorithms, namely a tunneling and a filled function method are described.
Abstract: We investigate the theoretical and numerical properties of two global optimization techniques for the solution of mixed complementarity problems. More precisely, using a standard semismooth Newton-type method as a basic solver for complementarity problems, we describe how the performance of this method can be improved by incorporating two well-known global optimization algorithms, namely a tunneling and a filled function method. These methods are tested and compared with each other on a couple of very difficult test examples.
TL;DR: This is the first time, to the authors' knowledge, that these tools are used to guarantee the finiteness of a simplicial branch-and-bound approach.
Abstract: In this paper simplicial branch-and-bound algorithms for concave minimization problems are discussed Some modifications of the basic algorithm are presented, mainly consisting in rules to start local searches, introduction of cuts and updates of the original objective function While some of these tools are not new in the literature, it is the first time, to the authors' knowledge, that they are used to guarantee the finiteness of a simplicial branch-and-bound approach
TL;DR: A stochastic algorithm to solve numerically the problem of finding the global minimizers of a real valued function subject to lower and upper bounds is presented.
Abstract: We present a stochastic algorithm to solve numerically the problem of finding the global minimizers of a real valued function subject to lower and upper bounds This algorithm looks for the global minimizers following the paths of a suitable system of stochastic differential equations Numerical experience on several test problems known in literature is shown
TL;DR: The p-th power Lagrangian method developed in this paper offers a success guarantee for the dual search in generating an optimal solution of the primal integer programming problem in an equivalent setting via two key transformations.
Abstract: Although the Lagrangian method is a powerful dual search approach in integer programming, it often fails to identify an optimal solution of the primal problem. The p-th power Lagrangian method developed in this paper offers a success guarantee for the dual search in generating an optimal solution of the primal integer programming problem in an equivalent setting via two key transformations. One other prominent feature of the p-th power Lagrangian method is that the dual search only involves a one-dimensional search within [0,1]. Some potential applications of the method as well as the issue of its implementation are discussed.
TL;DR: It is shown how various structures in optimal design of experiments problems determine the structure of corresponding challenging global optimization problems.
Abstract: In this paper we show that optimal design of experiments, a specific topic in statistics, constitutes a challenging application field for global optimization. This paper shows how various structures in optimal design of experiments problems determine the structure of corresponding challenging global optimization problems. Three different kinds of experimental designs are discussed: discrete designs, exact designs and replicationfree designs. Finding optimal designs for these three concepts involves different optimization problems.
TL;DR: The algorithm of Uniform Covering by Probabilistic Rejection is discussed as an approach to the practical realisation of PAS, which is a global optimisation ideal with a desirable complexity.
Abstract: The problem of generating a random sample over a level set, called Uniform Covering, is considered A variant is discussed of an algorithm known as Pure Adaptive Search which is a global optimisation ideal with a desirable complexity The algorithm of Uniform Covering by Probabilistic Rejection is discussed as an approach to the practical realisation of PAS Consequences for the complexity and practical performance in comparison to other algorithms are illustrated
TL;DR: The computational burden usually suffered by multivariate covering methods is significantly reduced, and this generalization is extended to the (non-differentiable) d.c.c., in this case the covering method of Breiman and Cutler, showing that it is a particular case of the standard outer approximation approach.
Abstract: Covering methods constitute a broad class of algorithms for solving multivariate Global Optimization problems. In this note we show that, when the objective f is d.c. and a d.c. decomposition for f is known, the computational burden usually suffered by multivariate covering methods is significantly reduced. With this we extend to the (non-differentiable) d.c. case the covering method of Breiman and Cutler, showing that it is a particular case of the standard outer approximation approach. Our computational experience shows that this generalization yields not only more flexibility but also faster convergence than the covering method of Breiman-Cutler.
TL;DR: It is shown that a simpler performance index, that retains the behavior of the original performance index near optimal values of the functional, can be used.
Abstract: We develop a technique to utilize the Cole–Hopf transformation to solve an optimal control problem for Burgers' equation. While the Burgers' equation is transformed into a simpler linear equation, the performance index is transformed to a complicated rational expression. We show that a simpler performance index, that retains the behavior of the original performance index near optimal values of the functional, can be used.