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

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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
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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 .
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