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Unformatted text preview: P r e l i m i n a r y d r a f t o n l y : p l e a s e c h e c k f o r fi n a l v e r s i o n ARE211, Fall 2007 LECTURE #22: TUE, NOV 13, 2007 PRINT DATE: AUGUST 21, 2007 (NPP2) Contents 6. Nonlinear Programming Problems and the Kuhn Tucker conditions (cont) 1 6.2. Necessary and sufficient conditions for a solution to an NPP (cont) 1 6.2.1. Preliminaries: the problem of the vanishing gradient 2 6.2.2. Preliminaries: The relationship between quasi-concavity and the Hessian of f 4 6.2.3. Preliminaries: The relationship between (strict) quasi-concavity and (strict) concavity 6 6.3. Sufficient Conditions for a solution to the NPP 7 6. Nonlinear Programming Problems and the Kuhn Tucker conditions (cont) 6.2. Necessary and sufficient conditions for a solution to an NPP (cont) So far we’ve only established necessary conditions for a solution to the NPP. But we can’t stop here. We could have found a minimum on the constraint set, and the same KKT conditions would be satisfied. In this lecture we focus on finding sufficient conditions for a solution, and in particular, conditions under which the KKT conditions will be both necessary and almost but not quite sufficient for a solution. The basic sufficiency conditions we’re going to rely on are that the objective function f is strictly quasi-concave while the constraint functions are quasi-convex. But there are a lot of subtleties that we need to address. We begin with some preliminary issues. 1 2 LECTURE #22: TUE, NOV 13, 2007 PRINT DATE: AUGUST 21, 2007 (NPP2) 6.2.1. Preliminaries: the problem of the vanishing gradient. In addition to the usual quasi-concavity, quasi-convexity conditions, we have to deal with the familiar annoyance posed by the example: max x 3 on [- 1 , 1]. Obviously this problem has a solution at x = 1. However, at x = 0, the KKT condi- tions are satisfied, i.e., gradient is zero, and so can be written as a nonnegative linear combination of the constraints vectors, with weights zero applied to each of the constraints, niether of which is satisfied with equality. That is, f prime ( x ) = 0 which is the sum of the gradients of two constraints at zero, each weighted by zero. So without imposing a restriction that excludes function such as this, we cannot say that satisfying the KKT conditions is sufficient for a max when the objective and constraint functions have the right “quasi” properties. To exclude this case, we could assume that f has a non-vanishing gradient. But this restriction throws the baby out with the bath-water: e.g., the problem max x (1- x ) s.t. x ∈ [0 , 1] has a global max at 0.5, at which point the gradient vanishes. So we want to exclude precisely those functions that have vanishing gradients at x ’s which are not unconstrained maxima. The following condition on f —called pseudoconcavity in S&B (the original name) and M.K.9 in MWG—does just this, in addition to implying quasi-concavity....
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- Fall '07
- Optimization, NPP, dx, Necessary and sufficient condition, Level set