chap06_8up - Optimization Problems One-Dimensional...

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Optimization Problems One-Dimensional Optimization Multi-Dimensional Optimization Scientific Computing: An Introductory Survey Chapter 6 – Optimization Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c ± 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 74 Optimization Problems One-Dimensional Optimization Multi-Dimensional Optimization Outline 1 Optimization Problems 2 One-Dimensional Optimization 3 Multi-Dimensional Optimization Michael T. Heath Scientific Computing 2 / 74 Optimization Problems One-Dimensional Optimization Multi-Dimensional Optimization Definitions Existence and Uniqueness Optimality Conditions Optimization Given function f : R n R , and set S R n , find x * S such that f ( x * ) f ( x ) for all x S x * is called minimizer or minimum of f It suffices to consider only minimization, since maximum of f is minimum of - f Objective function f is usually differentiable, and may be linear or nonlinear Constraint set S is defined by system of equations and inequalities, which may be linear or nonlinear Points x S are called feasible points If S = R n , problem is unconstrained Michael T. Heath Scientific Computing 3 / 74 Optimization Problems One-Dimensional Optimization Multi-Dimensional Optimization Definitions Existence and Uniqueness Optimality Conditions Optimization Problems General continuous optimization problem: min f ( x ) subject to g ( x ) = 0 and h ( x ) 0 where f : R n R , g : R n R m , h : R n R p Linear programming : f , g , and h are all linear Nonlinear programming : at least one of f , g , and h is nonlinear Michael T. Heath Scientific Computing 4 / 74 Optimization Problems One-Dimensional Optimization Multi-Dimensional Optimization Definitions Existence and Uniqueness Optimality Conditions Examples: Optimization Problems Minimize weight of structure subject to constraint on its strength, or maximize its strength subject to constraint on its weight Minimize cost of diet subject to nutritional constraints Minimize surface area of cylinder subject to constraint on its volume: min x 1 ,x 2 f ( x 1 , x 2 ) = 2 πx 1 ( x 1 + x 2 ) subject to g ( x 1 , x 2 ) = πx 2 1 x 2 - V = 0 where x 1 and x 2 are radius and height of cylinder, and V is required volume Michael T. Heath Scientific Computing 5 / 74 Optimization Problems One-Dimensional Optimization Multi-Dimensional Optimization Definitions Existence and Uniqueness Optimality Conditions Local vs Global Optimization x * S is global minimum if f ( x * ) f ( x ) for all x S x * S is local minimum if f ( x * ) f ( x ) for all feasible x in some neighborhood of x * Michael T. Heath Scientific Computing 6 / 74 Optimization Problems One-Dimensional Optimization Multi-Dimensional Optimization Definitions Existence and Uniqueness Optimality Conditions Global Optimization Finding, or even verifying, global minimum is difficult, in general Most optimization methods are designed to find local minimum, which may or may not be global minimum
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chap06_8up - Optimization Problems One-Dimensional...

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