Fundamentals of Optimization Techniques with Algorithms

Front Cover
Fundamentals of Optimization Techniques With Algorithms
Copyright Page
Dedication
Contents
Preface
Acknowledgments
1. Introduction to optimization
1.1 Optimal problem formulation
1.1.1 Design variables
1.1.2 Constraints
1.1.3 Objective function
1.1.4 Variable bounds
1.2 Engineering applications of optimization
1.3 Optimization techniques
Further reading
2. Linear programming
2.1 Formulation of the problem
Practice set 2.1
2.2 Graphical method
2.2.1 Working procedure
Practice set 2.2
2.3 General LPP
2.3.1 Canonical and standard forms of LPP
Practice set 2.3
2.4 Simplex method
2.4.1 Reduction of feasible solution to a basic feasible solution
2.4.2 Working procedure of the simplex method
Practice set 2.4
2.5 Artificial variable techniques
2.5.1 Big M method
2.5.2 Two-phase method
Practice set 2.5
2.6 Duality Principle
2.6.1 Formulation of a dual problem
2.6.1.1 Formulation of a dual problem when the primal has equality constraints
2.6.1.2 Duality principle
Practice set 2.6
2.7 Dual simplex method
2.7.1 Working procedure for a dual simplex method
Practice set 2.7
Further reading
3. Single-variable nonlinear optimization
3.1 Classical method for single-variable optimization
3.2 Exhaustive search method
3.3 Bounding phase method
3.4 Interval halving method
3.5 Fibonacci search method
3.6 Golden section search method
3.7 Bisection method
3.8 Newton–Raphson method
3.9 Secant method
3.10 Successive quadratic point estimation method
Further reading
4. Multivariable unconstrained nonlinear optimization
4.1 Classical method for multivariable optimization
4.1.1 Definition: rth differential of a function f(X)
4.1.2 Necessary condition
4.1.3 Sufficient condition
4.2 Unidirectional search method
4.3 Evolutionary search method
4.3.1 Box’s evolutionary optimization method
4.4 Simplex search method
4.5 Hooke–Jeeves pattern search method
4.5.1 Exploratory move
4.5.2 Pattern move
4.6 Conjugate direction method
4.6.1 Parallel subspace property
4.6.2 Extended parallel subspace property
4.7 Steepest descent method
4.7.1 Cauchy’s (steepest descent) method
4.8 Newton’s method
4.9 Marquardt’s method
Practice set
Further reading
5. Multivariable constrained nonlinear optimization
5.1 Classical methods for equality constrained optimization
5.1.1 Solution by direct substitution
5.1.2 Solution by the method of constrained variation
5.1.3 Solution by the method of Lagrange multipliers
5.1.3.1 Necessary conditions
5.1.3.2 Sufficient condition
5.2 Classical methods for inequality constrained optimization
5.3 Random search method
5.4 Complex method
5.4.1 Iterative procedure
5.5 Sequential linear programming
5.6 Zoutendijk’s method of feasible directions
5.7 Sequential quadratic programming
5.7.1 Derivation
5.7.2 Solution procedure
5.8 Penalty function method
5.9 Interior penalty function method
5.10 Convex programming problem
5.11 Exterior penalty function method
Practice set
Further reading
6. Geometric programming
6.1 Posynomial
6.2 Unconstrained geometric programming program
6.2.1 Arithmetic–geometric inequality
6.2.2 Primal–dual relationship and sufficiency conditions in the unconstrained case
6.2.3 Primal and dual problems
6.2.4 Computational procedure
6.3 Constrained optimization
6.3.1 Solution of a constrained geometric programming problem
6.3.2 Optimum design variables
6.3.3 Primal and dual programs in the case of less-than inequalities
6.4 Geometric programming with mixed inequality constraints
Practice set
Further reading
7. Dynamic programming
7.1 Characteristics of dynamic programming
7.2 Terminologies
7.3 Developing optimal decision policy
7.4 Multiplicative separable return function and single additive constraint
7.5 Additive separable return function and single additive constraint
7.6 Additively separable return function and single multiplicative constraint
7.7 Dynamic programming approach for solving a linear programming problem
7.8 Types of multilevel decision problem
7.8.1 Concept of suboptimization and the principle of optimality
7.8.2 Formulation of water tank optimization problem into a dynamic programming problem and the solution procedure
7.8.3 Procedure
Practice set
Further reading
8. Integer programming
8.1 Integer linear programming
8.1.1 Types of integer programming problems
8.1.2 Enumeration and concept of cutting plane solution
8.1.3 Gomory’s all integer cutting plane method
8.1.3.1 Method for constructing additional constraint (cut)
8.1.3.2 Procedure
8.1.3.3 Steps of Gomory’s all integer programming algorithm
8.1.4 Gomory’s mixed-integer cutting plane method
8.1.4.1 Method for constructing additional constraint (cut)
8.1.5 Branch and bound method
8.1.5.1 Procedure
8.1.6 Applications of zero-one integer programming
8.2 Integer nonlinear programming
8.2.1 Representation of an integer variable by an equivalent system of binary variables
Practice set
Further reading
9. Multiobjective optimization
9.1 Global criterion method
9.1.1 Methods for a priori articulation information given
9.2 Utility function method
9.3 Inverted utility method
9.4 Bounded objective function method
9.4.1 Methods for mixed ordinal and cardinal information given
9.5 Lexicographic model
9.6 Goal programming method
9.6.1 Practice set
Further reading
10. Nature-inspired optimization
10.1 Genetic algorithm
10.1.1 Genetic operations on binary strings
10.1.1.1 Selection
10.1.1.2 Crossover
10.1.1.3 Mutation
10.1.2 Analysis of GA
10.2 Neural network-based optimization
10.2.1 Architecture of ANN
10.2.2 Paradigms of learning
10.2.3 Learning processes
10.2.4 Activation functions
10.2.5 Applications of ANN in optimization
10.3 Ant colony optimization
10.4 Particle swarm optimization
Further reading
Index
Back Cover