Opoptimaztion concept & application, Study notes of Optimization Techniques in Engineering

Download Opoptimaztion concept & application and more Study notes Optimization Techniques in Engineering in PDF only on Docsity!

Optimization Concepts and Applications

in Engineering

Second Edition

Ashok D. Belegundu

The Pennsylvania State University

Tirupathi R. Chandrupatla

Rowan University

To our families

Preface

This book is a revised and enhanced edition of the first edition. The authors have identified a clear need for teaching engineering optimization in a manner that inte- grates theory, algorithms, modeling, and hands-on experience based on their exten- sive experience in teaching, research, and interactions with students. They have strived to adhere to this pedagogy and reinforced it further in the second edition, with more detailed explanations, an increased number of solved examples and end- of-chapter problems, and source codes on multiple platforms. The development of the software, which parallels the theory, has helped to explain the implementation aspects in the text with greater insight and accuracy. Students have integrated the optimization programs with simulation codes in their theses. The programs can be tried out by researchers and practicing engineers as well. Programs on the CD-ROM have been developed in Matlab, Excel VBA, VBScript, and Fortran. A battery of methods is available for the user. This leads to effective solution of problems since no single method can be successful on all problems. The book deals with a variety of optimization problems: unconstrained, con- strained, gradient, and nongradient techniques; duality concepts; multiobjective optimization; linear, integer, geometric, and dynamic programming with applica- tions; and finite element–based optimization. Matlab graphics and optimization toolbox routines and the Excel Solver optimizer are presented in detail. Through solved examples, problem-solving strategies are presented for handling problems where the number of variables depends on the number of discretization points in a mesh and for handling time-dependent constraints. Chapter 8 deals exclusively with treatment of the objective function itself as opposed to methods for minimizing it. This book can be used in courses at the graduate or senior-undergraduate level and as a learning resource for practicing engineers. Specifically, the text can be used

xi

xii Preface

in courses on engineering optimization, design optimization, structural optimiza- tion, and nonlinear programming. The book may be used in mechanical, aerospace, civil, industrial, architectural, chemical, and electrical engineering, as well as in applied mathematics. In deciding which chapters are to be covered in a course, the instructor may note the following. Chapters 1, 2, 3.1–3.5, and 8 are fundamental. Chapters 4, 9, and 11 focus on linear problems, whereas Chapters 5–7 focus on non- linear problems. Even if the focus is on nonlinear problems, Sections 4.1–4.6 present important concepts related to constraints. Chapters 10 and 12 are specialized top- ics. Thus, for instance, a course on structural optimization (i.e., finite element-based optimization) may cover Chapters 1–2, 3.1–3.5, 4.1–4.6, 5, 6, 7.7–7.10, 8, and 12. We are grateful to the students at our respective institutions for motivating us to develop this book. It has been a pleasure working with our editor, Peter Gordon.

Preliminary Concepts

1.1 Introduction

Optimization is the process of maximizing or minimizing a desired objective func- tion while satisfying the prevailing constraints. Nature has an abundance of exam- ples where an optimum system status is sought. In metals and alloys, the atoms take positions of least energy to form unit cells. These unit cells define the crystalline structure of materials. A liquid droplet in zero gravity is a perfect sphere, which is the geometric form of least surface area for a given volume. Tall trees form ribs near the base to strengthen them in bending. The honeycomb structure is one of the most compact packaging arrangements. Genetic mutation for survival is another example of nature’s optimization process. Like nature, organizations and businesses have also strived toward excellence. Solutions to their problems have been based mostly on judgment and experience. However, increased competition and consumer demands often require that the solutions be optimum and not just feasible solutions. A small savings in a mass-produced part will result in substantial savings for the corporation. In vehicles, weight minimization can impact fuel efficiency, increased payloads, or performance. Limited material or labor resources must be utilized to maximize profit. Often, optimization of a design process saves money for a company by simply reducing the developmental time. In order for engineers to apply optimization at their work place, they must have an understanding of both the theory and the algorithms and techniques. This is because there is considerable effort needed to apply optimization techniques on practical problems to achieve an improvement. This effort invariably requires tuning algorithmic parameters, scaling, and even modifying the techniques for the specific application. Moreover, the user may have to try several optimization methods to find one that can be successfully applied. To date, optimization has been used more as a design or decision aid, rather than for concept generation or detailed design. In

1

1.2 Historical Sketch 3

Jeeves [1961], Powell’s method of conjugate directions [1964], the simplex method of Nelder and Meade [1965], and the method of Box [1965]. Special methods that exploit some particular structure of a problem were also developed. Dynamic pro- gramming originated from the work of Bellman who stated the principle of optimal policy for system optimization [1952]. Geometric programming originated from the work of Duffin, Peterson, Zener [1967]. Lasdon [1970] drew attention to large-scale systems. Pareto optimality was developed in the context of multiobjective optimiza- tion. More recently, there has been focus on stochastic methods, which are better able to determine global minima. Most notable among these are genetic algo- rithms [Holland 1975, Goldberg 1989], simulated annealing algorithms that origi- nated from Metropolis [1953], and differential evolution methods [Price and Storn, http://www.icsi.berkeley.edu/∼storn/code.html]. In operations research and industrial engineering, use of optimization tech- niques in manufacturing, production, inventory control, transportation, scheduling, networks, and finance has resulted in considerable savings for a wide range of busi- nesses and industries. Several operations research textbooks are available to the reader. For instance, optimization of airline schedules is an integer program that can be solved using the branch and bound technique [Nemhauser 1997]. Shortest path routines have been used to reroute traffic due to road blocks. The routines may also be applied to route messages on the Internet. The use of nonlinear optimization techniques in structural design was pio- neered by Schmit [1960]. Early literature on engineering optimization are Johnson [1961], Wilde [1967], Fox [1971], Siddall [1972], Haug and Arora [1979], Morris [1982], Reklaitis, Ravindran and Ragsdell [1983], Vanderplaats [1984], Papalam- bros and Wilde [1988], Banichuk [1990], Haftka and Gurdal [1991]. Several authors have added to this collection including books on specialized topics such as struc- tural topology optimization [Bendsoe and Sigmund 2004], design sensitivity anal- ysis [Haug, Choi and Komkov 1986], optimization using evolutionary algorithms [Deb 2001] and books specifically targeting chemical, electrical, industrial, com- puter science, and other engineering systems. We refer the reader to the bibliog- raphy at end of this book. These, along with several others that have appeared in the last decade, have made an impact in educating engineers to apply optimization techniques. Today, applications are everywhere, from identifying structures of pro- tein molecules to tracing of electromagnetic rays. Optimization has been used for decades in sizing airplane wings. The challenge is to increase its utilization in bring- ing out the final product. Widely available and relatively easy to use optimization software packages, popular in universities, include the MATLAB optimization toolbox and the EXCEL SOLVER. Also available are GAMS modeling packages (http://gams.nist. gov/) and CPLEX software (http://www.ilog.com/). Other resources include Web sites maintained by Argonne national labs (http://www-fp.mcs.anl.gov/OTC/Guide/

4 Preliminary Concepts

f

− f

x *

x Figure 1.1. Maximization of f is equivalent to minimization of − f.

SoftwareGuide/) and by SIAM (http://www.siam.org/). GAMS is tied to a host of optimizers. Structural and simulation-based optimization software packages that can be procured from companies include ALTAIR (http://www.altair.com/), GENESIS (http://www.vrand.com/), iSIGHT (http://www.engineous.com/), modeFRONTIER (http://www.esteco.com/), and FE-Design (http://www.fe-design.de/en/home.html). Optimization capability is offered in analysis commercial packages such as ANSYS and NASTRAN.

1.3 The Nonlinear Programming Problem

Most engineering optimization problems may be expressed as minimizing (or max- imizing) a function subject to inequality and equality constraints, which is referred to as a nonlinear programming (NLP) problem. The word “programming” means “planning.” The general form is

minimize f ( x ) subject to gi ( x ) ≤ 0 i = 1 ,... , m and h (^) j ( x ) = 0 j = 1 ,... ,  and x L^ ≤ x ≤ x U

where x = ( x 1 , x 2 ,... , xn ) T^ is a column vector of n real-valued design variables. In Eq. (1.1), f is the objective or cost function, g ’s are inequality constraints , and h ’s are equality constraints. The notation x^0 for the starting point, x ∗^ for optimum, and x k for the (current) point at the k th iteration will be generally used.

Maximization versus Minimization

Note that maximization of f is equivalent to minimization of − f (Fig. 1.1). Problems may be manipulated so as to be in the form (1.1). Vectors x L^ , x U represent explicit lower and upper bounds on the design variables, respectively, and