1. 1 University of Palestine Operations Research ITGD4207 WIAM_H-Whba 320050030 Dr. Sana’a Wafa Al-Sayegh 2 nd Semester 2008-2009

2. 2 Reference Textbooks SE- 521: Nonlinear Programming and Applications NLPintro

3. 3 Chapter _18 Nonlinear Programming: Classical optimization Theory and algorithms

4. Learning objectives What is Nonlinear programming Formulation of NLP Application What is an Optimization Problem Terminology Classifiers Optimization Problems Exercise Quiz 4

5. 5 Mathematical Programming Linear program Nonlinear programming *** Quadratic program Integer programming Dynamic programming Stochastic programming

6. 6 Formulation of NLP Min f(X) S.t : g i (X) ≤ 0 for I = 1, …, m h i (X) = 0 for I = 1, …, l X ε Χ

7. Application of NLP Estimation : Regression Production- Inventory Optimal control Highway construction Mechanical design Electrical Network Water resources 7

8. Application of NLP Estimation : Regression Production- Inventory Optimal control Highway construction Mechanical design Electrical Network Water resources 8

9. What is an Optimization Problem? Optimization problems involve the selection of values of a number of interrelated variables by focusing attention on one or more selection criteria designed to measure the quality of the selection 9

10. Optimization Problem Statement Select the values of x From a set of possible values X that satisfy a group of algebraic constraints In such a way that will optimize the value f(x) 10

11. Terminology Decision Variables = x Objective Function(s) = f(x) Feasible Region = X Constraints defining X Inequality constraints : g(x)<=0 Equality constraints : h(x)=0 11

12. Optimization Problems Three Classifiers 12 Optimization Problems Decision Variables ObjectiveConstraints

13. 13 Optimization Problems Constraints Unconstrained Constrained Linear Non linear Simple bounds GUB Network Block diag onal Decision Variables (DV) Discrete Continuous Finite Infinite Objective One Differentiable Non Differentiable Many Closed Form From Simulation Linear Non linear Deterministic Stochas tic © 2006, 2007 Zilliant, Inc. -- CONFIDENTIAL

14. Exercise 1. Linear Programming 2. Goal Programming 3. Network Programming 4. Integer Programming 5. Non differentiable Optimization 6. Global Optimization 7. Dynamic Programming 8. Stochastic Programming 9. Quadratic Programming 10. Fractional Programming 11. Geometric Programming 12. Multi Objective Optimization Identify the attributes that define following optimization problems and prove one simple example. Display the results in table format

15. Quiz What are the Three Classifiers for optimization problem? 15