SE- 521: Nonlinear Programming and Applications S. O. Duffuaa

Course Description  Formulation of engineering problems as nonlinear programs; optimality conditions for nonlinear programs; algorithms for unconstrained optimization, algorithms for constrained nonlinear program; methods of feasible directions, sequential unconstrained minimization techniques SUMT, comparisons of algorithms for nonlinear programs. Case studies.

Course Objectives w 1. Applications : Enhance the students' modeling and formulation abilities of Engineering and other problems using nonlinear programs. w 2. Algorithms : Expose students to several nonlinear programming algorithms and enable them to apply and analyze rate of convergence and compare nonlinear programming algorithms. w 3. Theory: Apply first and second order optimality conditions and be familiar with basic convergence theory.

Outcomes w 1. Formulate nonlinear programs. w 2. Select an appropriate algorithm to solve nonlinear programs. w 3. Present the basic conditions on local and global minimum for convex and non- convex functions. w 4. Derive first order and second order optimality conditions. w 5. State the drawbacks and the limitations of known algorithms. w 6. Derive the rate of convergence of several nonlinear programming algorithm.

Terms w Text Book “Nonlinear Programming Theory and Algorithms” by M.S. Bazaraas, H. d. Sherali and C. M. Shetty 2nd Edition, Wiley, w Pre-requisite SE 305 or ( Math 280 and Advanced Calculus) w References 1. Introduction to linear and nonlinear programming, by D. Luenberger 2nd h Ed. Addison Wesley, w Practical Methods o Optimization, by Fletcher, R, 2nd Edition, Wiley, w Issues of Journal of Optimization Theory and Applications.

Grading Policy w Homework 20% w Class participation 10% w Project 15% w Midterm 25% w Final Exam 30%

Syllabus w Background, basic definitions and applications. w Background, basic definitions and applications w Constraints qualification and Lagrangian Duality. w Algorithms maps and applications.

Syllabus w Zangwill convergence theorem w Line search techniques. w Multidimensional search techniques. w Methods of conjugate directions. w Penalty and barrier functions methods. w Reduced gradient algorithm.

Mathematical Programming w Linear program w Nonlinear programming *** w Quadratic program w Integer programming w Dynamic programming w Stochastic programming

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

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

Application of NLP w Portfolio selection w Machine cutting w Quality control w Facility location w Electric Generators Maintenance

Application of NLP w Problems on chapter 1 1.1, 1.2, 1.3, 1.13, 1.14 Due next Tuesday next week.

Mathematical Background Convex Sets w A set is convex iff x 1 and x 2 ε S then λ x 1 + (1-λ) ε S w Examples of convex sets Rectangle Square Triangle Half spaces Epigraph of convex functions w.

Convex Sets w If S 1 and S 2 convex then S 1 ∩ S 2 is convex. w S 1 U S 2 is not necessary convex why? w S 1 + S 2 is convex w S 1 - S 2 is convex w S where S is a solution of a linear program w See page 35 text.

Convex Sets w Convex hull of a set X denoted as H(S) or conv (S). X ε H(S) iff X can be written as a convex combination of points in S. w The convex hull is the minimal convex set that contains S.

Closure and Interior w A point X in the closure of S iff S∩N ε (x) ≠ Φ  X is in the interior of S iff N ε (x) ⊂ S w Theorem Let S be a convex set in E n with a nonempty interior Let X 1 εCl S and X 2 εint S. Then λ X 1 + (1-λ) X 2 εint S for λ ε(0,1).

Weiestrass Thorem w Let S a nonempty compact set, and f : S → E 1. Then, the problem min { f(x) : X ε S}. Attains its minimum. That is there is a minimizing solution to this problem. w Discuss the three cases min does not exist.

Convex Functions w Let f: S → E1 where S is convex set in En, the function f is convex if f(λ X 1 + (1-λ) X 2 ) ≤ λf(X 1 ) + (1-λ) f(X 2 )