Lecture 4

# Lecture 4 - Nonlinear Programming 1 The problem and the...

This preview shows pages 1–3. Sign up to view the full content.

Nonlinear Programming 1 The problem and the solution concept We are interested in solving a general constrained optimization problem minimize f ( x ) subject to x C, (1) where f is the objective function and C R N is the constraint set . Such an optimization problem is called a linear programming problem when both the ob- jective function and the constraint are linear. Otherwise, it is called a nonlinear programming problem . If both the objective function and the constraint happen to be convex, it is called a convex programming problem . Thus Linear Programming Convex Programming Nonlinear Programming . We focus on minimization because maximizing f is the same as minimizing f . If f is continuous and C is compact, by the Bolzano-Weierstrass theorem we know there is a solution, but the theorem does not tell you how to compute it. In this section you will learn how to derive necessary conditions for optimality for general nonlinear programming problems. Oftentimes, the necessary conditions alone will pin down the solution to a few candidates, so you only need to compare these candidates. ¯ x C is said to be a global solution if f ( x ) f x ) for all x C . ¯ x is a local solution if there exists ϵ > 0 such that f ( x ) f x ) whenever x C and x ¯ x < ϵ . If ¯ x is a global solution, clearly it is also a local solution. 2 Tangent cone and normal cone Let ¯ x C be any point. The tangent cone of C at ¯ x is defined by T C x ) = y R N ( ) { α k } 0 , { x k } C, y = lim k →∞ α k ( x k ¯ x ) . That is, y T C x ) if y points to the same direction as the the limiting direction of { x k ¯ x } . Intuitively, the tangent cone of C at ¯ x consists of all directions that can be approximated by that from ¯ x to another point in C . Lemma 1. T C x ) is a nonempty closed cone. Proof. Setting α k = 0 for all k we get 0 T C x ). If y T C x ), then y = lim α k ( x k ¯ x ) for some { α k } 0 and { x k } C . Then for β 0 we have β y = lim βα k ( x k ¯ x ) T C x ), so T C x ) is a cone. To show that T C x ) is closed, let { y l } T C x ) and y l ¯ y . For each l we can take a sequence such 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2014W Econ 172B Operations Research (B) Alexis Akira Toda that α k,l 0, lim k →∞ x k,l = ¯ x , and y l = lim k →∞ α k,l ( x k,l ¯ x ). Hence we can take k l such that x k l ,l ¯ x < 1 /l and y l α k l ,l ( x k l ,l ¯ x ) < 1 /l . Then x k l ,l ¯ x and ¯ y α k l ,l ( x k l ,l ¯ x ) ∥ ≤ ∥ ¯ y y l + y l α k l ,l ( x k l ,l ¯ x ) ∥ → 0 , so ¯ y T C x ). The dual cone of T C x ) is called the normal cone at ¯ x and is denoted by N C x ). By the definition of the dual cone, we have N C x ) = z R N ( y T C x )) y, z ⟩ ≤ 0 .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern