A new parallel method for verified global optimization

We present a new parallel method for verified global optimization, using challenge leadership for the dynamic load balancing. The new approach combines advantages of two previous models: the centralized mediator model (see [1]) and the processor farm (see [2]). It has the following properties: centralization of the process; reduction of the number of box exchanges, communications used to send boxes from one processor to another; handling of the box that most probably contains the global minimizer. Numerical results show the efficiency of this method.     

Parallel Implementation of Synchronous Type Artificial Bee Colony Algorithm for Global Optimization

Evolutionary algorithms often need huge running times when solving large-scale optimization problems. One of the solutions for this issue is to introduce parallelization into the algorithm. To benefit from this approach for the artificial bee colony optimization algorithm, we present a new synchronous and parallel version of the algorithm. Performances of the proposed version and the original asynchronous algorithm are compared in terms of efficiency and speedup. Algorithms are competed to solve 20 large-scale global optimization problems. Comparative results show that the proposed parallel algorithm is still efficient as asynchronous version while it requires much less time to solve complex and large problems.     

Accelerating method of global optimization for signomial geometric programming

Signomial geometric programming (SGP) has been an interesting problem for many authors recently. Many methods have been provided for finding locally optimal solutions of SGP, but little progress has been made for global optimization of SGP. In this paper we propose a new accelerating method for global optimization algorithm of SGP using a suitable deleting technique. This technique offers a possibility to cut away a large part of the currently investigated region in which the globally optimal solution of SGP does not exist, and can be seen as an accelerating device for global optimization algorithm of SGP problem. Compared with the method of Shen and Zhang [Global optimization of signomial geometric programming using linear relaxation, Appl. Math. Comput. 150 (2004) 99–114], numerical results show that the computational efficiency is improved obviously by using this new technique in the number of iterations, the required saving list length and the execution time of the algorithm.     

Parallel radial basis function methods for the global optimization of expensive functions

We introduce a master–worker framework for parallel global optimization of computationally expensive functions using response surface models. In particular, we parallelize two radial basis function (RBF) methods for global optimization, namely, the RBF method by Gutmann [Gutmann, H.M., 2001a. A radial basis function method for global optimization. Journal of Global Optimization 19(3), 201–227] (Gutmann-RBF) and the RBF method by Regis and Shoemaker [Regis, R.G., Shoemaker, C.A., 2005. Constrained global optimization of expensive black box functions using radial basis functions, Journal of Global Optimization 31, 153–171] (CORS-RBF). We modify these algorithms so that they can generate multiple points for simultaneous evaluation in parallel. We compare the performance of the two parallel RBF methods with a parallel multistart derivative-based algorithm, a parallel multistart derivative-free trust-region algorithm, and a parallel evolutionary algorithm on eleven test problems and on a 6-dimensional groundwater bioremediation application. The results indicate that the two parallel RBF algorithms are generally better than the other three alternatives on most of the test problems. Moreover, the two parallel RBF algorithms have comparable performances on the test problems considered. Finally, we report good speedups for both parallel RBF algorithms when using a small number of processors.     

A new hybrid classical-quantum algorithm for continuous global optimization problems

Grover’s algorithm can be employed in global optimization methods providing, in some cases, a quadratic speedup over classical algorithms. This paper describes a new method for continuous global optimization problems that uses a classical algorithm for finding a local minimum and Grover’s algorithm to escape from this local minimum. Such algorithms will be useful when quantum computers of reasonable size are available. Simulations with testbed functions and comparisons with algorithms from the literature are presented.     

Global Optimality Conditions and Optimization Methods for Quadratic Knapsack Problems

The quadratic knapsack problem (QKP) maximizes a quadratic objective function subject to a binary and linear capacity constraint. Due to its simple structure and challenging difficulty, it has been studied intensively during the last two decades. This paper first presents some global optimality conditions for (QKP), which include necessary conditions and sufficient conditions. Then a local optimization method for (QKP) is developed using the necessary global optimality condition. Finally a global optimization method for (QKP) is proposed based on the sufficient global optimality condition, the local optimization method and an auxiliary function. Several numerical examples are given to illustrate the efficiency of the presented optimization methods.     

Speedup estimates for some variants of the parallel implementations of the branch-and-bound method

The efficiency of parallel implementations of the branch-and-bound method in discrete optimization problems is considered. A theoretical analysis and comparison of two parallel implementations of this method is performed. A mathematical model of the computation process is constructed and used to obtain estimates of the maximum possible speedup. Examples of problems in which none of these two parallel implementations can speed up the computations are considered.     

A new filled function method for nonlinear equations

In this paper, a new global optimization approach based on the filled function method is proposed for solving box-constrained systems of nonlinear equations. We first convert the nonlinear system into an equivalent global optimization problem, and then propose a new filled function method to solve the converted global optimization problem. Several numerical examples are presented and solved by using different local minimization methods, which illustrate the efficiency of the present approach.     

An efficient algorithm for range computation of polynomials using the Bernstein form

We present a novel optimization algorithm for computing the ranges of multivariate polynomials using the Bernstein polynomial approach. The proposed algorithm incorporates four accelerating devices, namely the cut-off test, the simplified vertex test, the monotonicity test, and the concavity test, and also possess many new features, such as, the generalized matrix method for Bernstein coefficient computation, a new subdivision direction selection rule and a new subdivision point selection rule. The features and capabilities of the proposed algorithm are compared with those of other optimization techniques: interval global optimization, the filled function method, a global optimization method for imprecise problems, and a hybrid approach combining simulated annealing, tabu search and a descent method. The superiority of the proposed method over the latter methods is illustrated by numerical experiments and qualitative comparisons.     

Combination method for parallel computation in ODE

In this paper, a 4th order parallel computation method with four processes for solving ODEs is discussed. This method is the Runge-Kutta method combined with a linear multistep method, which overcomes the difficulties of the 4th order parallel Runge-Kutta method discussed in [1]. The concept of critical speedup for parallel methods is also defined, and speedups of some methods are analyzed by using this concept.     

A parallel method for finding the global minimum of univariate functions

We describe a new parallel method for solving global optimization problems. The formulation of the decision rules of this method is presented. We examine convergence conditions of the proposed algorithm and establish conditions which guarantee a considerable speedup with respect to the sequential version of the algorithm. We also present some numerical experiments executed on Alliant FX/80 for one class of multiextremal functions.The authors are greatly indebted to R. G. Strongin who stimulated the fulfillment of this research. They also would like to thank the anonymous referees for their useful suggestions.The research of the first author was partially supported by Grant 9494/NC/89 from the Italian Government under the Italian-Soviet Agreement about the Cultural and Scientific Exchange in 1990–1991. He thanks the Systems Department, University of Calabria, where he was a Visitor.     

Restart strategies for GRASP with path-relinking heuristics

GRASP with path-relinking is a hybrid metaheuristic, or stochastic local search (Monte Carlo) method, for combinatorial optimization. A restart strategy in GRASP with path-relinking heuristics is a set of iterations {i ₁, i ₂, …} on which the heuristic is restarted from scratch using a new seed for the random number generator. Restart strategies have been shown to speed up stochastic local search algorithms. In this paper, we propose a new restart strategy for GRASP with path-relinking heuristics. We illustrate the speedup obtained with our restart strategy on GRASP with path-relinking heuristics for the maximum cut problem, the maximum weighted satisfiability problem, and the private virtual circuit routing problem.     

Application of parallel heuristic algorithms for speeding up parallel implementations of the branch-and-bound method

A scheme for the parallel implementation of the combined branch-and-bound method and heuristic algorithms is proposed. Results of computations for the one-dimensional Boolean knapsack problem are presented that demonstrate the efficiency of the proposed approach. The main factors that affect the speedup of the solution when local optimization is used are discussed.     

Parallel Asynchronous Global Search and the Nested Optimization Scheme

In this paper, a parallel asynchronous information algorithm for solving multidimensional Lipschitz global optimization problems, where times for evaluating the objective function can be different from point to point, is proposed. This method uses the nested optimization scheme and a new parallel asynchronous global optimization method for solving core univariate subproblems generated by the nested scheme. The properties of the scheme related to parallel computations are investigated. Global convergence conditions for the new method and theoretical conditions of speed up, which can be reached by using asynchronous parallelization in comparison with the pure sequential case, are established. Numerical experiments comparing sequential, synchronous, and asynchronous algorithms are also reported.     

On parallel hashing and integer sorting

The problem of sorting n integers from a restricted range [1…m], where m is a superpolynomial in n, is considered. An o(n log n) randomized algorithm is given. Our algorithm takes O(n log log m) expected time and O(n) space. (Thus, for m = n^polylog(n) we have an O(n log log n) algorithm.) The algorithm is parallelizable. The resulting parallel algorithm achieves optimal speedup. Some features of the algorithm make us believe that it is relevant for practical applications. A result of independent interest is a parallel hashing technique. The expected construction time is logarithmic using an optimal number of processors, and searching for a value takes O(1) time in the worst case. This technique enables drastic reduction of space requirements for the price of using randomness. Applicability of the technique is demonstrated for the parallel sorting algorithm and for some parallel string matching algorithms. The parallel sorting algorithm is designed for a strong and nonstandard model of parallel computation. Efficient simulations of the strong model on a CRCW PRAM are introduced. One of the simulations even achieves optimal speedup. This is probably the first optimal speedup simulation of a certain kind.     

Generalized speculative computation of parallel simulated annealing

Simulated annealing is known to be highly sequential due to dependences between iterations. While the conventional speculative computation with a binary tree has been found effective for parallel simulated annealing, its performance is limited to (logp)-fold speedup due to parallel execution of logp iterations onp processors. This report presents a new approach to parallel simulated annealing, calledgeneralized speculative computation (GSC). The GSC is synchronous, maintaining the same decision sequence as sequential simulated annealing. The use of two loop indices encoded in a single integer eliminates broadcasting of central data structure to all processors. The master-slave parallel programming paradigm simplifies controlling the activities ofp iterations which are executed in parallel onp processors. To verify the performance of GSC, we implemented 100-city to 500-city Traveling Salesman Problems on the AP1000 massively parallel multiprocessor. Execution results on the AP1000 demonstrate that the GSC approach can indeed be an effective method for parallel simulated annealing as it gave over 20-fold speedup on 100 processors.     

A global optimization method for the molecular replacement problem in X-ray crystallography

The primary technique for determining the three-dimensional structure of a protein molecule is X-ray crystallography, from which the molecular replacement (MR) problem often arises as a critical step. The MR problem is a global optimization problem to locate an optimal position of a model protein so that at this position the model will produce calculated intensities closest to those observed from an X-ray crystallography experiment involving a protein with unknown but similar atomic structure. Improving the applicability and robustness of MR methods is an important research topic because commonly used traditional MR methods, though often successful, have their limitations in solving difficult problems.We introduce a new global optimization strategy that combines a coarse-grid search, using a surrogate function, with extensive multi-start local optimization. A new MR code, called SOMoRe, based on this strategy is developed and tested on four realistic problems, including two difficult problems that traditional MR codes failed to solve directly. SOMoRe was able to solve each test problem without any complication, and SOMoRe solved an MR problem using a less complete model than the models required by three other programs. These results indicate that the new method is promising and should enhance the applicability and robustness of the MR methodology.     

A faster LU decomposition for parallelC programs

This report introduces a faster parallel LU decomposition algorithm that gives a speedup almost equal to the number of nodes used. The new algorithm takes an advantage of an importantC feature that lays out a matrix using a row major scheme and is based on the currently widely used LU decomposition algorithm with one major modification to eliminate most of the communication overhead. Empirical results are included in this report. For example, solving a dense matrix that contains 100,000,000 elements gives a speedup of 50 when executed on 50 nodes of an Intel Paragon in parallel.     

A parallel adaptive finite volume method for nanoscale double-gate MOSFETs simulation

We propose in this paper a quantum correction transport model for nanoscale double-gate metal-oxide-semiconductor field effect transistor (MOSFET) device simulation. Based on adaptive finite volume, parallel domain decomposition, monotone iterative, and a posteriori error estimation methods, the model is solved numerically on a PC-based Linux cluster with MPI libraries. Quantum mechanical effect plays an important role in semiconductor nanoscale device simulation. To model this effect, a physical-based quantum correction equation is derived and solved with the hydrodynamic transport model. Numerical calculation of the quantum correction transport model is implemented with the parallel adaptive finite volume method which has recently been proposed by us in deep-submicron semiconductor device simulation. A 20 nm double-gate MOSFET is simulated with the developed quantum transport model and computational technique. Compared with a classical transport model, it is found that this model can account for the quantum mechanical effects of the nanoscale double-gate MOSFET quantitatively. Various biasing conditions have been verified on the simulated device to demonstrate its accuracy. Furthermore, for the same tested problem, the parallel adaptive computation shows very good computational performance in terms of the mesh refinements, the parallel speedup, the load-balancing, and the efficiency.     

An efficient algorithm for large scale global optimization of continuous functions

A fast descent algorithm, resorting to a “stretching” function technique and built on one hybrid method (GRSA) which combines simulated annealing (SA) algorithm and gradient based methods for large scale global optimizations, is proposed. Unlike the previously proposed method in which the original objective functions remain unchanged during the whole course of optimization, the new method firstly constructs an auxiliary function on one local minimizer obtained by gradient based methods and then SA is executed on this constructed auxiliary function instead of on the original objective function in order that we can improve the jumping ability of SA algorithm to escape from the currently discovered local minimum to a better one from which the gradient based methods restart a new local search. The above procedure is repeated until a global minimum is detected. In addition, corresponding to the adopted “stretching” technique, a new next trial point generating scheme is designed. It is verified by simulation especially on large scale problems that the convergence speed is greatly accelerated, which is its main difference from many other reported methods that mostly cope with functions with less than 50 variables and does not apply to large scale optimization problems. Furthermore, the new algorithm functions as a global optimization procedure with a high success probability and high solution precision.