Showing papers by "John E. Dennis published in 1998"

Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems

Abstract: This paper proposes an alternate method for finding several Pareto optimal points for a general nonlinear multicriteria optimization problem. Such points collectively capture the trade-off among the various conflicting objectives. It is proved that this method is independent of the relative scales of the functions and is successful in producing an evenly distributed set of points in the Pareto set given an evenly distributed set of parameters, a property which the popular method of minimizing weighted combinations of objective functions lacks. Further, this method can handle more than two objectives while retaining the computational efficiency of continuation-type algorithms. This is an improvement over continuation techniques for tracing the trade-off curve since continuation strategies cannot easily be extended to handle more than two objectives.

A Rigorous Framework for Optimization of Expensive Functions by Surrogates

Abstract: The goal of the research reported here is to develop rigorous optimization algorithms to apply to some engineering design problems for which design 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.

Optimization Using Surrogate Objectives on a Helicopter Test Example

Abstract: This paper presents results for a 31 variable helicopter rotor design example. Results are given for several numerical methods. This is a brief description of a portion of the Boeing/IBM/Rice University collaboration whose purpose is to develop effective numerical methods for managing the use of approximation concepts or response surface methodology in design optimization.

Trust-Region Interior-Point SQP Algorithms for a Class of Nonlinear Programming Problems

Abstract: In this paper, a family of trust-region interior-point sequential quadratic programming (SQP) algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise, e.g., from the discretization of optimal control problems. The algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial differential equations. The algorithms keep strict feasibility with respect to the bound constraints by using an affine scaling method proposed, for a different class of problems, by Coleman and Li [SIAM J. Optim., 6 (1996), pp. 418--445] and they exploit trust-region techniques for equality-constrained optimization. Thus, they allow the computation of the steps using a variety of methods, including many iterative techniques. Global convergence of these algorithms to a first-order Karush--Kuhn--Tucker (KKT) limit point is proved under very mild conditions on the trial steps. Under reasonable, but more stringent, conditions on the quadratic model and on the trial steps, the sequence of iterates generated by the algorithms is shown to have a limit point satisfying the second-order necessary KKT conditions. The local rate of convergence to a nondegenerate strict local minimizer is q-quadratic. The results given here include, as special cases, current results for only equality constraints and for only simple bounds. Numerical results for the solution of an optimal control problem governed by a nonlinear heat equation are reported.

Managing surrogate objectives to optimize a helicopter rotor design - Further experiments

Abstract: It is common engineering practice to use response surface approximations as surrogates for an expensive objective function in engineering design. The rationale is to reduce the number of detailed, costly analyses required during optimization. In earlier work, we developed a rigorous and eeective scheme for managing the interplay between the use of surrogates in the optimization and scheduled progress checks with the expensive analysis so that the process converges to a solution of the original design problem. In this paper, we will report our latest numerical tests with a helicopter rotor design problem which has proved to be a fruitful laboratory for experimentation. The results given here support the use of an ANOVA decomposition on a DACE model to identify the most important optimization variables in an optimal design problem. Introduction The use of optimization tools with computer simulations to drive engineering design underpins MDO. However, there still are many important problems for which existing methods are either unreliable ad hoc procedures or impractical. Legacy simulation codes that fail on some plausible inputs and run slowly on the rest are far from the clean, innnitely diier-entiable functions for which optimization specialists design and analyze powerful algorithms. A research collaboration, involving Boeing Applied Research & Technology and Rice University, has developed methods that interpose a computationally clean surrogate function between the optimizer and the simulation, to provide a reasonably accurate model of the true simulation and a function that the optimizer can evaluate quickly. The resulting methods have led to new theories and better practical solutions. We present numerical results here for two versions of one of the target problems for our collaboration, the design of a lower vibration helicopter rotor blade.

Managing the Choice of Surrogate Variables and the Use of Approximation Models to Optimize Expensive Functions

Abstract: : We propose research in three basic areas. First, it is standard in engineering practice to use approximation models of expensive simulations to drive nonlinear programming algorithms. An open question, which we will investigate using well-established notions from the literature on trust-region methods, is how to manage the interplay between optimization and the fidelity of the approximation models to insure that the process converges to a reasonable solution of the original design problem. It is also standard in engineering design to reduce the dimension of the optimization problem to be solved by using a technique known as variable linking. We plan to generalize this concept using some recent ideas from the optimization community to design robust ways to decompose a single iteration of any optimization method into a multi-phase process. First, an algorithm is applied to solve subproblems posed on subspaces of lower dimension, and then to solve the full problem on the affine hull defined by the current iterate and the solutions found for the subproblems. Crucial questions remain about the convergence of a practical implementation of this process. Finally, we will continue our research to extend and analyze pattern search methods. Pattern search methods can be successfully applied when only ranking (ordinal) information is available and when derivatives are either unavailable or unreliable. Since these are situations that occur in the design problems of interest, we propose to continue our investigation of pattern search methods. In particular, we will examine robust extensions to handle problems with bound constraints.