The goal of the research reported here is to develop rigorous optimization algorithms to apply to some engineering design problems for which direct application of traditional optimization approaches is not practical. This paper presents and analyzes a framework for generating a sequence of approximations to the objective function and managing the use of these approximations as surrogates for optimization. The result is to obtain convergence to a minimizer of an expensive objective function subject to simple constraints. The approach is widely applicable because it does not require, or even explicitly approximate, derivatives of the objective. Numerical results are presented for a 31-variable helicopter rotor blade design example and for a standard optimization test example. Key Words: Approximation concepts, surrogate optimization, response surfaces, pattern search methods, derivative-free optimization, design and analysis of computer experiments (DACE), computational engineering. # ...

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. The molecule problem is that of determining the relative locations of a set of objects in Euclidean space relying only upon a sparse set of pairwise distance measurements. This NP--hard problem has applications in the determination of molecular conformation. The molecule problem can be naturally expressed as a continuous, global optimization problem, but it also has a rich combinatorial structure. This paper investigates how that structure can be exploited to simplify the optimization problem. In particular, we present a novel divide--and--conquer algorithm in which a large global optimization problem is replaced by a sequence of smaller ones. Since the cost of the optimization can grow exponentially with problem size, this approach holds the promise of a substantial improvement in performance. Our algorithmic development relies upon some recently published results in graph theory. We describe an implementation of this algorithm and report some results of its performance on a sample ...

Identifying the conformations that a molecular cluster or polymer assumes in nature is an important problem with many practical applications in biology and medicine. It is believed that the naturally occurring molecular conformations minimize or nearly minimize the potential energy of the molecular cluster or polymer. The problem of finding the molecular configuration(s) with the lowest potential energy is a challenging global optimization problem with a potentially huge number of local solutions over a very large parameter space. We have developed a method to solve these types of problems, and have experimented with both molecular cluster and polymer applications, with very promising results. We have implemented the method on powerful massively parallel computers because of the enormous computational requirements of solving these types of problems. The parallel algorithms are interesting asynchronous, multi-level parallel algorithms. 1 Introduction The general problem of finding the ...

Neural networks are usually trained using local, gradient-based procedures. Such methods frequently find suboptimal solutions being trapped in local minima. Optimization of neural structures and global minimization methods applied to network cost functions have strong influence on all aspects of network performance. Recently genetic algorithms are frequently combined with neural methods to select best architectures and avoid drawbacks of local minimization methods. Many other global minimization methods are suitable for that purpose, although they are used rather rarely in this context. This paper provides a survey of such global methods, including some aspects of genetic algorithms.

Global optimization is a very hard problem especially when the number of variables is large (greater than several hundred). Recently, some methods including simulated annealing, branch and bound, and an interval Newton's method have made it possible to solve global optimization problems with several hundred variables. However, this is a small number of variables when one considers that integer programming can tackle problems with thousands of variables, and linear programming is able to solve problems with millions of variables. The goal of this research is to examine the present state of the art for algorithms to solve the unconstrained global optimization problem (GOP) and then to suggest some new approaches that allow problems of a larger size to be solved with an equivalent amount of computer time. This algorithm is then implemented using portable C++ and the software will be released for general use. This new algorithm is given with some theoretical results under which the algorit...

. The concern of this work is global optimization using genetic algorithms (GAs). In this work we propose a synergy between the cluster analysis technique, popular in classical stochastic global optimization, and the GA to accomplish global optimization. This synergy minimizes redundant searches around local optima and enhances the capability of the GA to explore new areas in the search space. The proposed methodology demonstrates superior performance when compared with the simple GA on benchmark cases. We also report our solution of the optimal pump configurations synthesis problem. *Author to whom all correspondence should be addressed. Tel.: (505) 665 0570, FAX: (505) 665 4479, E-mail: vmh@lanl.gov 1 INTRODUCTION Multi-modal objective functions are common in engineering applications. Also, very little or no a priori information about the analytical properties of the objective function in some optimization applications in engineering stress the need to be able to search for the glo...

this paper both involve the determination of the structure of clusters of atoms or molecules, but each application uses a di#erent potential energy function. The first potential is given by the sum of the pairwise interactions between atoms described by the Lennard-Jones function, and the second is the empirical water dimer potential energy surface function (RWK2-M) described in [10]. Problems in determining molecular structure lead to optimization problems because the naturally occurring structure usually minimizes the potential energy of the system. These problems become global optimization problems because typically such functions have very many local minimizers.

. In the optimal plant location and sizing problem it is desired to optimize a cost function involving plant sizes, locations, and production schedules in the face of supply-demand and plant capacity constraints. We will use simulated annealing (SA) and a genetic algorithm (GA) to solve this problem. We will compare these techniques with respect to computational expenses, constraint handling capabilities, and the quality of the solution obtained in general. Simulated Annealing is a combinatorial stochastic optimization technique which has been shown to be effective in obtaining fast suboptimal solutions for computationally hard problems. The technique is especially attractive since solutions are obtained in polynomial time for problems where an exhaustive search for the global optimum would require exponential time. We propose a synergy between the cluster analysis technique, popular in classical stochastic global optimization, and the GA to accomplish global optimization. This synergy...

The goal of these lectures is to acquaint the audience with some approaches to a class of nasty optimization problems involving nonconvex nonlinear extended-valued functions. Such functions arise often in multidisciplinary optimization (MDO). The first three lectures are meant to set the context for applying our algorithms. The context determines the form of the algorithms, and to present this context requires a bit more than just a short list of assumptions. Briefly though, the objective function and constraints depend not only on the optimization variables, but also on some ancillary variables such as the solutions of some coupled systems by stand-alone solvers for partial differential equations, table look-ups, and other nonsmooth simulation codes. This has important algorithmic implications. First, the function and constraint values may be very expensive. Second, the functions may be nondifferentiable and discontinuous. In fact, they are often treated as extended valued since a function call may not return a value even if all the specified constraints are satisfied. The approach we treat in these lectures has been successful for some real problems in engineering design. We hope to convince engineers and mathematicians alike that not only are the algorithms given here useful, but the mathematics involved is interesting and relevant. We hope to convince mathematicians that good applied problems produce good mathematics, and that contrary to what they may have heard, they will suffer no loss of virtue as a direct result of considering them.