micro-fall2007-8

micro-fall2007-8 - Detour The Theory of Non-Linear...

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Detour: The Theory of Non-Linear Optimization Optimization Problems An optimization problem consists of a choice set A a function f There are two types of optimization problems: Maximization problems: max x 2 A f ( x ) Minimization problems: min x 2 A f ( x ) Ani Guerdjikova ECON 313 Fall 2007 91 / 126
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Detour: The Theory of Non-Linear Optimization Constrained versus unconstrained optimization The problem is unconstrained , if A coincides with the domain of f . constrained , if A is a proper subset of f . The solution of the problem consists of the value of x , at which f reaches its optimal value , f ( x ) . Ani Guerdjikova ECON 313 Fall 2007 92 / 126
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Detour: The Theory of Non-Linear Optimization Theorem If a function f reaches a maximum at a point x , then the function g ( x ) = ± f ( x ) reaches a minimum at x . This implies: Each minimization problem can be solved by rewriting it as an equivalent maximization problem : Instead of f ( x ) s.t. x 2 A solve ± f ( x ) s.t. x 2 A Ani Guerdjikova ECON 313 Fall 2007 93 / 126
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Detour: The Theory of Non-Linear Optimization Proof: Observe that, a solution x of max x 2 A ± f ( x ) ± f ( x ) ² ± f ( x ) for all x 2 A . Hence, (multiplying by ± 1): f ( x ) ³ f ( x ) for all x 2 A . Therefore, x is a solution of the minimization problem min x 2 A f ( x ) . Ani Guerdjikova ECON 313 Fall 2007 93 / 126
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Detour: The Theory of Non-Linear Optimization Uniqueness A given optimization problem can have no solution a unique solution multiple solutions Ani Guerdjikova ECON 313 Fall 2007 94 / 126
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micro-fall2007-8 - Detour The Theory of Non-Linear...

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