nonlinear programming

nonlinear programming - Nonlinear Programming Nonlinear...

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Nonlinear Programming Nonlinear Programming McCarl and Spreen McCarl and Spreen Chapter 12 Chapter 12

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Optimality Conditions Optimality Conditions Unconstrained optimization – multivariate calculus problem. For Y=f(X), the optimum occurs at the point where f '(X) =0 and f’''(X) meets second order conditions A relative minimum occurs where f '(X) =0 and f’''(X) >0 A relative maximum occurs where f '(X) =0 and f’''(X) <0
Concavity and Second Derivative Concavity and Second Derivative f’’(x)<0 f’’(x)>0 f’’(x)<0 f‘’(x)>0 local max and global max local max local min local min and global min

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Multivariate Case Multivariate Case To find an optimum point, set the first partial derivatives (all of them) to zero. At the optimum point, evaluate the matrix of second partial derivatives (Hessian matrix) to see if it is positive definite (minimum) or negative definite (maximum). Check characteristic roots or apply determinental test to principal minors.
Determinental Test for a Maximum – Determinental Test for a Maximum – Negative Definite Hessian Negative Definite Hessian f11 f12 f13 f21 f22 f23 f31 f32 f33 f11 f12 f21 f22 f11 < 0 > 0 < 0 These would all be positive for a minimum. (matrix positive definite)

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Global Optimum Global Optimum A univariate function with a negative second derivative everywhere guarantees a global maximum at the point (if there is one) where f’(X)=0. These functions are called “concave down” or sometimes just “concave.” A univariate function with a positive second derivative everywhere guarantees a global minimum (if there is one) at the point where f’(X)=0. These functions are called “concave up” or sometimes “convex.”
Multivariate Global Optimum Multivariate Global Optimum If the Hessian matrix is positive definite (or negative definite) for all values of the variables, then any optimum point found will be a global minimum (maximum).

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Constrained Optimization Constrained Optimization Equality constraints – often solvable by calculus Inequality constraints – sometimes solvable by numerical methods
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This note was uploaded on 11/15/2011 for the course AGEC 7100 taught by Professor Duffy,p during the Fall '08 term at Auburn University.

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nonlinear programming - Nonlinear Programming Nonlinear...

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