lecture3 - IE417: Nonlinear Programming: Lecture 3 Jeff...

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Unformatted text preview: IE417: Nonlinear Programming: Lecture 3 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 24th January 2006 Jeff Linderoth IE417:Lecture 3 Todays Outline Review Line Search and Trust Region Steepest Descent and Newtons Method Rates of Convergence Problem Time Jeff Linderoth IE417:Lecture 3 Theorems Theorems Theorems Descent Direction If f : R n R is continuously differentiable, and d R n such that d T f ( x ) < , then T > such that f ( x + d ) < f ( x ) (0 , T ) . d is called a descent direction of f at x . Proof. Taylors Theorem and Continuity of f ( x ) Jeff Linderoth IE417:Lecture 3 (Unconstrained) Optimality Conditions First Order Necessary Condition If f : R n R is a continuously differentiable function, and x * is a local minimizer of f , then f ( x * ) = 0 Proof. Contradiction. d =- f ( x * ) is a descent direction. Second Order Necessary Condition If f : R n R is a twice continuously differentiable function, and...
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This note was uploaded on 02/29/2008 for the course IE 417 taught by Professor Linderoth during the Spring '08 term at Lehigh University .

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lecture3 - IE417: Nonlinear Programming: Lecture 3 Jeff...

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