Preface. 1 Introduction to Optimization. 1.1 Introduction. 1.2 Historical Development. 1.3 Engineering Applications of Optimization. 1.4 Statement of an Optimization Problem. 1.5 Classification of Optimization Problems. 1.6 Optimization Techniques. 1.7 Engineering Optimization Literature. 1.8 Solution of Optimization Problems Using MATLAB. References and Bibliography. Review Questions. Problems. 2 Classical Optimization Techniques. 2.1 Introduction. 2.2 Single-Variable Optimization. 2.3 Multivariable Optimization with No Constraints. 2.4 Multivariable Optimization with Equality Constraints. 2.5 Multivariable Optimization with Inequality Constraints. 2.6 Convex Programming Problem. References and Bibliography. Review Questions. Problems. 3 Linear Programming I: Simplex Method. 3.1 Introduction. 3.2 Applications of Linear Programming. 3.3 Standard Form of a Linear Programming Problem. 3.4 Geometry of Linear Programming Problems. 3.5 Definitions and Theorems. 3.6 Solution of a System of Linear Simultaneous Equations. 3.7 Pivotal Reduction of a General System of Equations. 3.8 Motivation of the Simplex Method. 3.9 Simplex Algorithm. 3.10 Two Phases of the Simplex Method. 3.11 MATLAB Solution of LP Problems. References and Bibliography. Review Questions. Problems. 4 Linear Programming II: Additional Topics and Extensions. 4.1 Introduction. 4.2 Revised Simplex Method. 4.3 Duality in Linear Programming. 4.4 Decomposition Principle. 4.5 Sensitivity or Postoptimality Analysis. 4.6 Transportation Problem. 4.7 Karmarkar's Interior Method. 4.8 Quadratic Programming. 4.9 MATLAB Solutions. References and Bibliography. Review Questions. Problems. 5 Nonlinear Programming I: One-Dimensional Minimization Methods. 5.1 Introduction. 5.2 Unimodal Function. ELIMINATION METHODS. 5.3 Unrestricted Search. 5.4 Exhaustive Search. 5.5 Dichotomous Search. 5.6 Interval Halving Method. 5.7 Fibonacci Method. 5.8 Golden Section Method. 5.9 Comparison of Elimination Methods. INTERPOLATION METHODS. 5.10 Quadratic Interpolation Method. 5.11 Cubic Interpolation Method. 5.12 Direct Root Methods. 5.13 Practical Considerations. 5.14 MATLAB Solution of One-Dimensional Minimization Problems. References and Bibliography. Review Questions. Problems. 6 Nonlinear Programming II: Unconstrained Optimization Techniques. 6.1 Introduction. DIRECT SEARCH METHODS. 6.2 Random Search Methods. 6.3 Grid Search Method. 6.4 Univariate Method. 6.5 Pattern Directions. 6.6 Powell's Method. 6.7 Simplex Method. INDIRECT SEARCH (DESCENT) METHODS. 6.8 Gradient of a Function. 6.9 Steepest Descent (Cauchy) Method. 6.10 Conjugate Gradient (Fletcher-Reeves) Method. 6.11 Newton's Method. 6.12 Marquardt Method. 6.13 Quasi-Newton Methods. 6.14 Davidon-Fletcher-Powell Method. 6.15 Broyden-Fletcher-Goldfarb-Shanno Method. 6.16 Test Functions. 6.17 MATLAB Solution of Unconstrained Optimization Problems. References and Bibliography. Review Questions. Problems. 7 Nonlinear Programming III: Constrained Optimization Techniques. 7.1 Introduction. 7.2 Characteristics of a Constrained Problem. DIRECT METHODS. 7.3 Random Search Methods. 7.4 Complex Method. 7.5 Sequential Linear Programming. 7.6 Basic Approach in the Methods of Feasible Directions. 7.7 Zoutendijk's Method of Feasible Directions. 7.8 Rosen's Gradient Projection Method. 7.9 Generalized Reduced Gradient Method. 7.10 Sequential Quadratic Programming. INDIRECT METHODS. 7.11 Transformation Techniques. 7.12 Basic Approach of the Penalty Function Method. 7.13 Interior Penalty Function Method. 7.14 Convex Programming Problem. 7.15 Exterior Penalty Function Method. 7.16 Extrapolation Techniques in the Interior Penalty Function Method. 7.17 Extended Interior Penalty Function Methods. 7.18 Penalty Function Method for Problems with Mixed Equality and Inequality Constraints. 7.19 Penalty Function Method for Parametric Constraints. 7.20 Augmented Lagrange Multiplier Method. 7.21 Checking the Convergence of Constrained Optimization Problems. 7.22 Test Problems. 7.23 MATLAB Solution of Constrained Optimization Problems. References and Bibliography. Review Questions. Problems. 8 Geometric Programming. 8.1 Introduction. 8.2 Posynomial. 8.3 Unconstrained Minimization Problem. 8.4 Solution of an Unconstrained Geometric Programming Program Using Differential Calculus. 8.5 Solution of an Unconstrained Geometric Programming Problem Using Arithmetic-Geometric Inequality. 8.6 Primal-Dual Relationship and Sufficiency Conditions in the Unconstrained Case. 8.7 Constrained Minimization. 8.8 Solution of a Constrained Geometric Programming Problem. 8.9 Primal and Dual Programs in the Case of Less-Than Inequalities. 8.10 Geometric Programming with Mixed Inequality Constraints. 8.11 Complementary Geometric Programming. 8.12 Applications of Geometric Programming. References and Bibliography. Review Questions. Problems. 9 Dynamic Programming. 9.1 Introduction. 9.2 Multistage Decision Processes. 9.3 Concept of Suboptimization and Principle of Optimality. 9.4 Computational Procedure in Dynamic Programming. 9.5 Example Illustrating the Calculus Method of Solution. 9.6 Example Illustrating the Tabular Method of Solution. 9.7 Conversion of a Final Value Problem into an Initial Value Problem. 9.8 Linear Programming as a Case of Dynamic Programming. 9.9 Continuous Dynamic Programming. 9.10 Additional Applications. References and Bibliography. Review Questions. Problems. 10 Integer Programming. 10.1 Introduction 588. INTEGER LINEAR PROGRAMMING. 10.2 Graphical Representation. 10.3 Gomory's Cutting Plane Method. 10.4 Balas' Algorithm for Zero-One Programming Problems. INTEGER NONLINEAR PROGRAMMING. 10.5 Integer Polynomial Programming. 10.6 Branch-and-Bound Method. 10.7 Sequential Linear Discrete Programming. 10.8 Generalized Penalty Function Method. 10.9 Solution of Binary Programming Problems Using MATLAB. References and Bibliography. Review Questions. Problems. 11 Stochastic Programming. 11.1 Introduction. 11.2 Basic Concepts of Probability Theory. 11.3 Stochastic Linear Programming. 11.4 Stochastic Nonlinear Programming. 11.5 Stochastic Geometric Programming. References and Bibliography. Review Questions. Problems. 12 Optimal Control and Optimality Criteria Methods. 12.1 Introduction. 12.2 Calculus of Variations. 12.3 Optimal Control Theory. 12.4 Optimality Criteria Methods. References and Bibliography. Review Questions. Problems. 13 Modern Methods of Optimization. 13.1 Introduction. 13.2 Genetic Algorithms. 13.3 Simulated Annealing. 13.4 Particle Swarm Optimization. 13.5 Ant Colony Optimization. 13.6 Optimization of Fuzzy Systems. 13.7 Neural-Network-Based Optimization. References and Bibliography. Review Questions. Problems. 14 Practical Aspects of Optimization. 14.1 Introduction. 14.2 Reduction of Size of an Optimization Problem. 14.3 Fast Reanalysis Techniques. 14.4 Derivatives of Static Displacements and Stresses. 14.5 Derivatives of Eigenvalues and Eigenvectors. 14.6 Derivatives of Transient Response. 14.7 Sensitivity of Optimum Solution to Problem Parameters. 14.8 Multilevel Optimization. 14.9 Parallel Processing. 14.10 Multiobjective Optimization. 14.11 Solution of Multiobjective Problems Using MATLAB. References and Bibliography. Review Questions. Problems. A Convex and Concave Functions. B Some Computational Aspects of Optimization. B.1 Choice of Method. B.2 Comparison of Unconstrained Methods. B.3 Comparison of Constrained Methods. B.4 Availability of Computer Programs. B.5 Scaling of Design Variables and Constraints. B.6 Computer Programs for Modern Methods of Optimization. References and Bibliography. C Introduction to MATLAB(R) . C.1 Features and Special Characters. C.2 Defining Matrices in MATLAB. C.3 CREATING m-FILES. C.4 Optimization Toolbox. Answers to Selected Problems. Index .

Engineering Optimization : Theory and Practice

A comprehensive study of various mathematical programming methods for structural optimization is presented. In recent years, many modern optimization techniques and convergence results have been developed in the field of mathematical programming. The aim of this paper is twofold: (a) to discuss the applicability of modern optimization techniques to structural design problems, and (b) to present mathematical programming methods from a unified and design engineers' viewpoint. Theoretical aspects are considered here, while numerical results of test problems are discussed in a companion paper. Special features possessed by structural optimization problems, together with recent developments in mathematical programming (recursive quadratic programming methods, global convergence theory), have formed a basis for conducting the study. Some improvements of existing methods are noted and areas for future investigation are discussed.

A study of mathematical programming methods for structural optimization. Part I: Theory

Design sensitivity analysis plays a central role in structural optimization, since virtually all optimization methods require computation of derivatives of structural response quantities with respect to design variables. Three fundamentally different approaches to design sensitivity analysis are presented. These have been used extensively in the structural optimization literature. They are the virtual load method, the state space method, and the design space method. An analysis of these methods indicates that the state space and design space methods are more general than the virtual load method. Moreover, the virtual load method, when applicable, is a special case of the state space method. Any one of these procedures may be incorporated into an optimality criterion or a mathematical programming method for structural optimization.

Methods of Design Sensitivity Analysis in Structural Optimization

Structural optimization with frequency constraints - A review

Various aspects of structural optimization techniques under transient loads are extensively reviewed. The main themes of the paper are treatment of time-dependent constraints, calculation of design sensitivity, and approximation. Each subject is reviewed with corresponding papers that have been published since the 1970s. The treatment of time-dependent constraints in both the direct method and the transformation method is discussed. Two ways of calculating design sensitivity of a structure under transient loads are discussed—direct differentiation method and adjoint variable method. The approximation concept mainly focuses on the response surface method in crashworthiness and local approximation with the intermediate variables. Especially, a method using the equivalent static load is discussed as an approximation method. It takes advantage of the well-established static response optimization. The structural optimization in flexible multibody dynamic systems is reviewed in the viewpoint of the above three themes.

A review of optimization of structures subjected to transient loads

The extended interior penalty function formulation is implemented. A rational method for determining the transition between the interior and extended parts is set forth. The formulation includes a straightforward method for avoiding design points with some negative components, which are physically meaningless in structural analysis. The technique, when extended to problems involving parametric constraints, can facilitate closed form integration of the penalty terms over the most important parts of the parameter interval. The method lends itself well to the use of approximation concepts, such as design variable linking, constraint deletion and Taylor series expansions of response quantities in terms of design variables. Examples demonstrating the algorithm, in the context of planar orthogonal frames subjected to ground motion, are included.

On implementation of the extended interior penalty function

Dynamic response quantities, treated parametrically in time, are included in the problem for the first time. The method is made efficient by the use of the following approximation concepts: (1)Design variable linking; (2)time parametric constraint deletion; and (3)use of the first-order Taylor series expansions of the dynamic response functions with respect to the design variables. It is found that the feasible design space in an optimum structural design problem in the dynamic response regime is usually disjoint. The Davidon-Fletcher-Powell algorithm is used for the unconstrained minimizations. To successfully implement the exterior penalty function with approximate constraints, dummy boundaries and a new concept of move limits are introduced. It is found that the sinusoidal contributions in the relation between design variables and dynamic response functions can cause infeasible minima of the exterior penalty function.

Juan H. Cassis

Papers

On implementation of the extended interior penalty function

Optimum Structural Design with Dynamic Constraints