Natalia Alexandrov

Bio: Natalia Alexandrov is an academic researcher from Langley Research Center. The author has contributed to research in topics: Optimization problem & Multidisciplinary design optimization. The author has an hindex of 17, co-authored 47 publications receiving 2113 citations.

A Trust Region Framework for Managing the Use of Approximation Models in Optimization

Abstract: This paper presents an analytically robust, globally convergent approach to managing the use of approximation models of various fidelity in optimization. By robust global behavior we mean the mathematical assurance that the iterates produced by the optimization algorithm, started at an arbitrary initial iterate, will converge to a stationary point or local optimizer for the original problem. The approach we present is based on the trust region idea from nonlinear programming and is shown to be provably convergent to a solution of the original high-fidelity problem. The proposed method for managing approximations in engineering optimization suggests ways to decide when the fidelity, and thus the cost, of the approximations might be fruitfully increased or decreased in the course of the optimization iterations. The approach is quite general. We make no assumptions on the structure of the original problem, in particular, no assumptions of convexity and separability, and place only mild requirements on the approximations. The approximations used in the framework can be of any nature appropriate to an application; for instance, they can be represented by analyses, simulations, or simple algebraic models. This paper introduces the approach and outlines the convergence analysis.

Approximation and Model Management in Aerodynamic Optimization with Variable-Fidelity Models

Abstract: This workdiscussesan approach,e rst-orderapproximation and modelmanagementoptimization (AMMO), for solving design optimization problems that involve computationally expensive simulations. AMMO maximizes the use of lower-e delity, cheaper models in iterative procedures with occasional, but systematic, recourse to highere delity, more expensive models for monitoring the progress of design optimization. A distinctive feature of the approach is thatit is globally convergent to a solution oftheoriginal, high-e delity problem. VariantsofAMMObased on three nonlinear programming algorithms are demonstrated on a three-dimensional aerodynamic wing optimization problemand atwo-dimensionalairfoiloptimizationproblem. Euleranalysisonmeshesof varying degrees of ree nement provides a suite of variable-e delity models. Preliminary results indicate threefold savings in terms of high-e delity analyses for the three-dimensional problem and twofold savings for the two-dimensional problem.

Optimization with variable-fidelity models applied to wing design

Abstract: This work discusses an approach, the Approximation Management Framework (AMF), for solving optimization problems that involve computationally expensive simulations. AMF aims to maximize the use of lower-fidelity, cheaper models in iterative procedures with occasional, but systematic, recourse to higher-fidelity, more expensive models for monitoring the progress of the algorithm. The method is globally convergent to a solution of the original, high-fidelity problem. Three versions of AMF, based on three nonlinear programming algorithms, are demonstrated on a 3D aerodynamic wing optimization problem and a 2D airfoil optimization problem. In both cases Euler analysis solved on meshes of various refinement provides a suite of variable-fidelity models. Preliminary results indicate threefold savings in terms of high-fidelity analyses in case of the 3D problem and twofold savings for the 2D problem.

Analytical and Computational Aspects of Collaborative Optimization for Multidisciplinary Design

Abstract: Analytical features of multidisciplinary optimization (MDO) problem formulations have significant practical consequences for the ability of nonlinear programming algorithms to solve the resulting computational optimization problems reliably and efficiently. We explore this important but frequently overlooked fact using the notion of disciplinary autonomy. Disciplinary autonomy is a desirable goal in formulating and solving MDO problems; however, the resulting system optimization problems are frequently difficult to solve. We illustrate the implications of MDO problem formulation for the tractability of the resulting design optimization problem by examining a representative class of MDO problem formulations known as collaborative optimization. We also discuss an alternative problem formulation, distributed analysis optimization, that yields a more tractable computational optimization problem.

An Overview of First-Order Model Management for Engineering Optimization

Abstract: First-order approximation/model management optimization (AMMO) is a rigorous methodology for solving high-fidelity optimization problems with minimal expense in high-fidelity function and derivative evaluation. AMMO is a general approach that is applicable to any derivative based optimization algorithm and any combination of high-fidelity and low-fidelity models. This paper gives an overview of the principles that underlie AMMO and puts the method in perspective with other similarly motivated methods. AMMO is first illustrated by an example of a scheme for solving bound-constrained optimization problems. The principles can be easily extrapolated to other optimization algorithms. The applicability to general models is demonstrated on two recent computational studies of aerodynamic optimization with AMMO. One study considers variable-resolution models, where the high-fidelity model is provided by solutions on a fine mesh, while the corresponding low-fidelity model is computed by solving the same differential equations on a coarser mesh. The second study uses variable-fidelity physics models, with the high-fidelity model provided by the Navier-Stokes equations and the low-fidelity model—by the Euler equations. Both studies show promising savings in terms of high-fidelity function and derivative evaluations. The overview serves to introduce the reader to the general concept of AMMO and to illustrate the basic principles with current computational results.

Efficient Global Optimization of Expensive Black-Box Functions

Abstract: In many engineering optimization problems, the number of function evaluations is severely limited by time or cost. These problems pose a special challenge to the field of global optimization, since existing methods often require more function evaluations than can be comfortably afforded. One way to address this challenge is to fit response surfaces to data collected by evaluating the objective and constraint functions at a few points. These surfaces can then be used for visualization, tradeoff analysis, and optimization. In this paper, we introduce the reader to a response surface methodology that is especially good at modeling the nonlinear, multimodal functions that often occur in engineering. We then show how these approximating functions can be used to construct an efficient global optimization algorithm with a credible stopping rule. The key to using response surfaces for global optimization lies in balancing the need to exploit the approximating surface (by sampling where it is minimized) with the need to improve the approximation (by sampling where prediction error may be high). Striking this balance requires solving certain auxiliary problems which have previously been considered intractable, but we show how these computational obstacles can be overcome.

A Taxonomy of Global Optimization Methods Based on Response Surfaces

Abstract: This paper presents a taxonomy of existing approaches for using response surfaces for global optimization. Each method is illustrated with a simple numerical example that brings out its advantages and disadvantages. The central theme is that methods that seem quite reasonable often have non-obvious failure modes. Understanding these failure modes is essential for the development of practical algorithms that fulfill the intuitive promise of the response surface approach.