optimization第二講-預(yù)覽頁(yè)
【正文】 principal minor of |H*| alternate in sign in the following way:,It is a maximum if all the principal minors of |H*| are strictly negative.,Note: Both theorems only give sufficient conditions.,16,第十六頁(yè)，共三十九頁(yè)。,Weierstrass’s Theorem: If f is a continuous function, and X is a nonempty, closed, and bounded set, then f has both a minimum and a maximum on X.,But when can we be sure that a minimum and a maximum really exist!?,X is nonempty if it contains at least one element. Otherwise the problem does not make sense. If there is no value, there is also no maximum value.,20,第二十頁(yè)，共三十九頁(yè)。,To give a formal derivation, we need some more mathematics.,Convex set Consider some set X. Take any two points in X. Draw a line between these points. If the entire line is within X, and this is true for any two points in the set, then the set is convex.,Convex set,Not a convex set,24,第二十四頁(yè)，共三十九頁(yè)。,To give a formal derivation, we need some more mathematics.,Quasiconvexity Consider some function f(x). Take some point x1. Consider the set X0 consisting of all points x0 that have f(x0) ≤ f(x1). If this set is convex, and this is true for all possible x1, then the function is quasiconvex.,x1,This function is quasiconvex, but not convex!,28,第二十八頁(yè)，共三十九頁(yè)。,Theorem 13.8: Uniqueness,In a constrained maximization problem,where f and all the g’s are increasing, then if f is strictly quasiconcave and the g’s are convex, or f is quasiconcave and the g’s are strict convex, then a locally optimal solution is unique and also globally optimal. Example: the consumer problem! Utility function is increasing and strictly quasiconcave, Budget constraint is increasing and convex. The theorem says that solving the FOCs yields a unique and global optimum.,32,第三十二頁(yè)，共三十九頁(yè)。,Example,A representative student spends 60 hours per week studying. She takes two subjects. Her objective: allocate time between the two subjects such that the average grade is maximized. Subjects differ with respect to their production function.,Objective function:,36,第三十六頁(yè)，共三十九頁(yè)。Let’s maximize thi
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