Optimum Engineering Design Mathematical Methods Professor Kamran Iqbal
Optimum Engineering Design: Mathematical Methods Professor Kamran Iqbal, Ph. D, PE Professor of Systems Engineering College of Engineering and Information Technology University of Arkansas at Little Rock kxiqbal@ualr. edu
Course Objectives • After successfully completing this course participants will be able to: • Formulate the optimum engineering design problem in a given scenario • Solve the optimum design problem in the following cases: – Solve nonlinear optimization problems using optimality criteria and numerical techniques – Solve linear optimization problems using linear programming (LP) techniques – Solve discrete optimization problems using integer programming (IP) techniques – Solve mathematical programming problems using LINGO and Matlab softwares
Course Prerequisites • Participation in this course requires the knowledge of – Multivariable calculus – Linear algebra • Additionally, familiarity with mathematical analysis, scientific reasoning, and basic computer and programming skills will be assumed of participants
Course Materials • Introduction to Optimum Design, 4 e / J. S. Arora, Academic Press • Fundamental Engineering Optimization Methods / K. Iqbal; download from http: //bookboon. com/en/fundamental-engineering -optimization-methods-ebook
Numerical Methods •
The Iterative Method •
Line Search Problem •
Example: Quadratic Function •
Computer Methods for Line Search Problem • Interval reduction methods – Golden search – Fibonacci search • Approximate search methods – Arjimo’s rule • Quadrature curve fitting
Interval Reduction Methods •
The Bracketing Algorithm •
Fibonacci’s Method •
Fibonacci Algorithm •
Golden Section Method •
Integrated Bracketing and Golden Section Algorithm •
Approximate Search Methods •
Approximate Line Search •
Approximate Line Search •
Quadratic Curve Fitting •
Example: Approximate Search •
Example: Approximate Search •
Computer Methods for Finding the Search Direction • Gradient based methods – Steepest descent method – Conjugate gradient method – Quasi Newton methods • Hessian based methods – Newton’s method
Steepest Descent Method •
Steepest Descent Algorithm •
Example: Steepest Descent •
Steepest Descent Method •
Preconditioning •
Conjugate Gradient Method •
Conjugate Directions Method •
Conjugate Gradient Method •
Conjugate Gradient Algorithm •
Preconditioning •
Modified Conjugate Gradient Algorithm •
CG Rate of Convergence •
Example •
Newton’s Method •
Marquardt Modification to Newton’s Method •
Modified Newton’s Method •
Newton’s Algorithm •
Rate of Convergence • Rate of Convergence. Newton’s method achieves quadratic rate of convergence in the close neighborhood of the optimal point, and superlinear rate of convergence otherwise. • The main drawback of the Newton’s method is its computational cost: the Hessian matrix needs to be computed at every step, and a linear system of equations needs to be solved to obtain the update. • Due to its high computational and storage costs, classic Newton’s method is rarely used in practice.
Quasi Newton’s Methods •
Quasi-Newton Methods •
Quasi-Newton Algorithm •
Example: Quasi-Newton Method •
Trust-Region Methods •
Trust-Region Methods •
Trust-Region Algorithm •
Computer Methods for Constrained Problems • • Penalty and Barrier methods (SUMT) Augmented Lagrangian method (AL) Sequential linear programming (SLP) Sequential quadratic programming (SQP)
Penalty and Barrier Methods •
Penalty and Barrier Methods •
The Augmented Lagrangian Method •
The Augmented Lagrangian Method •
Augmented Lagrangian Algorithm •
Example: Augmented Lagrangian •
Example: Augmented Lagrangian •
Sequential Linear Programming •
Sequential Linear Programming •
Sequential Linear Programming •
SLP Example •
SLP Example •
Sequential Quadratic Programming •
Sequential Quadratic Method •
Descent Function Approach •
SQP Algorithm •
SQP with Approximate Line Search •
SQP Example •
SQP Example •
The Active Set Strategy •
The Active Set Strategy •
SQP With Hessian Update •
Modified SQP Algorithm •
Example: SQP with Hessian Update •
Example: SQP with Hessian Update •
Conclusion • Engineering design optimization is an exciting subject • Engineering design problems can be cast as mathematical optimization problems • Engineering graduates should be familiar with fundamental optimization methods (analytical tools to solve the mathematical optimization problem) • The numerical aspects of the optimization methods are equally important, but they were not covered
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