Lecture2-1-25-2002

Lecture2-1-25-2002 - MAE 552 Heuristic Optimization Lecture...

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MAE 552 – Heuristic Optimization Lecture 2 January 25, 2002
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The optimization problem is then: Find values of the variables that minimize or maximize the objective function while satisfying the constraints. The standard form of the constrained optimization problem can be written as: Minimize: F( x ) objective function Subject to: g j ( x ) 0 j=1,m inequality constraints h k ( x ) 0 k=1,l equality constraints x i lower x i x i upper i=1,n side constraints where x=(x 1 , x 2 , x 3 , x 4 , x 5 ,x n ) design variables
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Conditions for Optimality Unconstrained Problems 1. F(x)=0 The gradient of F(x) must vanish at the optimum 1. Hessian Matrix must be positive definite (i.e. all positive eigenvalues at optimum point). = 2 n 2 2 n 2 1 n 2 n 2 2 2 2 2 1 2 2 n 1 2 2 1 2 2 1 2 x ) x ( F x x ) x ( F x x ) x ( F x x ) x ( F x ) x ( F x x ) x ( F x x ) x ( F x x ) x ( F x ) x ( F H
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Conditions for Optimality Unconstrained Problems A positive definite Hessian at the minimum ensures only that a local minimum has been found The minimum is the global minimum only if it can be shown that the Hessian is positive definite for all possible values of x . This would imply a convex design space. Very hard to prove in practice!!!!
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Conditions for Optimality Constrained Problems Kuhn Tucker Conditions 1. x * is feasible 1. λ j g j = 0 j=1,m 0 ) x ( h ) x ( g ) x ( F * * * j l 1 kj k m j m 1 j j = λ + λ + = + = 3. size in ed unrestrict 0 k m j + λ These conditions only guarantee that x * is a local optimum.
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Conditions for Optimality Constrained Problems In addition to the Kuhn Tucker conditions two other conditions two other conditions must be satisfied to guarantee a global optima. 1. Hessian must be positive definite for all x. 2. Constraints must be convex. A constraint is convex if a line connecting any two points in the feasible space travels always lies in the feasible region of the design space.
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Determining the Complexity of Problems Why are some problems difficult to Solve? 1. The number of possible solutions in the search space is so large as to forbid an exhaustive search for the best answer. 2. The problem is so complex that just to facilitate any answer at all requires that we simplify the model such that any result is essentially useless. 3.
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This note was uploaded on 07/09/2011 for the course MAE 522 taught by Professor Hacker during the Spring '10 term at SUNY Buffalo.

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Lecture2-1-25-2002 - MAE 552 Heuristic Optimization Lecture...

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