Engineering Optimization : Theory & Practice

Introduction to Optimization.



· Historical Development.

· Engineering Applications of Optimization.

· Statement of an Optimization Problem.

· Classification of Optimization Problems.

· Optimization Techniques.

· Engineering Optimization Literature.

· Solution of Optimization Problems Using MATLAB.



Classical Optimization Techniques.



· Single-Variable Optimization.

· Multivariable Optimization with No Constraints.

· Multivariable Optimization with Equality Constraints.

· Multivariable Optimization with Inequality Constraints.

· Convex Programming Problem.



Linear Programming I: Simplex Method.



· Applications of Linear Programming.

· Standard Form of a Linear Programming Problem.

· Geometry of Linear Programming Problems.

· Definitions and Theorems.

· Solution of a System of Linear Simultaneous Equations.

· Pivotal Reduction of a General System of Equations.

· Motivation of the Simplex Method.

· Simplex Algorithm.

· Two Phases of the Simplex Method.

· MATLAB Solution of LP Problems.



Linear Programming II: Additional Topics and Extensions.



· Revised Simplex Method.

· Duality in Linear Programming.

· Decomposition Principle.

· Sensitivity or Postoptimality Analysis.

· Transportation Problem.

· Karmarkar's Interior Method.

· Quadratic Programming.

· MATLAB Solutions.



Nonlinear Programming I: One-Dimensional Minimization Methods.



· Unimodal Function.

· ELIMINATION METHODS.

· Unrestricted Search.

· Exhaustive Search.

· Dichotomous Search.

· Interval Halving Method.

· Fibonacci Method.

· Golden Section Method.

· Comparison of Elimination Methods.



INTERPOLATION METHODS.



· Quadratic Interpolation Method.

· Cubic Interpolation Method.

· Direct Root Methods.

· Practical Considerations.

· MATLAB Solution of One-Dimensional Minimization Problems.



Nonlinear Programming II: Unconstrained Optimization Techniques.



· DIRECT SEARCH METHODS.

· Random Search Methods.

· Grid Search Method.

· Univariate Method.

· Pattern Directions.

· Powell's Method.

· Simplex Method.



INDIRECT SEARCH (DESCENT) METHODS.



· Gradient of a Function.

· Steepest Descent (Cauchy) Method.

· Conjugate Gradient (Fletcher--Reeves) Method.

· Newton's Method.

· Marquardt Method.

· Quasi-Newton Methods.

· Davidon--Fletcher--Powell Method.

· Broyden--Fletcher--Goldfarb--Shanno Method.

· Test Functions.

· MATLAB Solution of Unconstrained Optimization Problems.





Nonlinear Programming III: Constrained Optimization Techniques.



· Characteristics of a Constrained Problem.



DIRECT METHODS.



· Random Search Methods.

· Complex Method.

· Sequential Linear Programming.

· Basic Approach in the Methods of Feasible Directions.

· Zoutendijk's Method of Feasible Directions.

· Rosen's Gradient Projection Method.

· Generalized Reduced Gradient Method.

· Sequential Quadratic Programming.



INDIRECT METHODS.



· Transformation Techniques.

· Basic Approach of the Penalty Function Method.

· Interior Penalty Function Method.

· Convex Programming Problem.

· Exterior Penalty Function Method.

· Extrapolation Techniques in the Interior Penalty Function Method.

· Extended Interior Penalty Function Methods.

· Penalty Function Method for Problems with Mixed Equality and Inequality Constraints.

· Penalty Function Method for Parametric Constraints.

· Augmented Lagrange Multiplier Method.

· Checking the Convergence of Constrained Optimization Problems.

· Test Problems.

· MATLAB Solution of Constrained Optimization Problems.