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4084 AOE/ESM Engineering Design Optimization Optimum Design Concepts Methods used for design optimization does not depend on the field of engineering. Broad classification of the optimization Optimality criteria methods (indirect methods) Search methods (direct search methods) Optimality criteria are the conditions a function must satisfy at its minimum point. Study of optimality conditions are necessary regardless of the method of optimization used. September 6, 1999 1 AOE/ESM 4084 Engineering Design Optimization Gradient Vector (the vector of first derivatives): For a function of n variables f(x1, x2, . . . ., xn) f ( x ) c i = ----------- x i f c = f ( x ) = ---- = gradf = x f ( x ) ------------ x 1 f ( x ) ------------ x 2 f ( x ) ------------ x n Geometrically, the gradient vector at point xp is normal to the tangent plane to the function at that point, and points in the direction of maximum increase in the function. September 6, 1999 2 AOE/ESM 4084 Engineering Design Optimization Hessian Matrix (the matrix of second derivatives): For a function of n variables f(x1, x2, . . . ., xn) 2 2 f(x ) f(x ) 2 x 1 x 2 x1 2 2 f(x ) f(x) 2 H = 2f ( x ) = x 2 x 1 x2 2 2 f(x ) f(x ) x n x 1 x n x 2 f = f x i x j x j x i September 6, 1999 3 2 2 f(x) H ij = x i x j 2 2 f(x) x 1 x n 2 f(x) x 2 x n 2 f(x) 2 xn Hessian matrix is always a symmetric matrix AOE/ESM 4084 Engineering Design Optimization Taylor Series Expansion: for a function with one variable, Taylor series expansion about xp df ( x ) p p 2 1 d f(x ) f ( x ) = f ( x ) + -------------- ( x x ) + -- ----------------- ( x x ) + R dx 2 dx 2 p p 2 p At a small distance d from xp x = xp+ d df ( x ) 1 d f( x ) 2 f ( x + d ) = f ( x ) + -------------- d + -- ----------------- d + R dx 2 dx 2 p p p 2 p for a function with n variables p p T p p T p p 1 f ( x ) = f ( x ) + f ( x ) ( x x ) + -- ( x x ) H ( x ) ( x x ) + R 2 change the in value of the function in moving from xp to a p neighboring point d = x - x distance away from it p p p T p 1 T f = f ( x + d ) f ( x ) = f ( x ) d + -- d H ( x ) d + R 2 September 6, 1999 4 AOE/ESM 4084 Engineering Design Optimization Quadratic Forms and Definite Matrices: Quadratic form is a special nonlinear function having only secondorder terms For example; Representations F ( x ) = x 1 + 2x 2 + 3x 3 + 2x 1 x 2 + 2x 2 x 3 + 2x 3 x 1 n n 2 2 2 1 F ( x ) = -2 i = 1j = 1 1 T p ij x i x j = -- x Px 2 Every quadratic form can be put into the following form with a symmetric A matrix 1 T 1 T F ( x ) = -- x Px = -- x Ax 2 2 1 a ij = -- ( p ij + p ji ) 2 Many matrices can be associated with a quadratic function, all of them are asymmetric. There is only one unique symmetric matrix. The symmetric A matrix determines the nature of the quadratic form. September 6, 1999 5 AOE/ESM 4084 Engineering Design Optimization For a given value of x a quadratic form F(x) = 1/2 xTAx may be either positive, negative, or zero A quadratic form is called positive definite if xTAx is always positive except for F(0). It is called negative definite if xTAx < 0 for all x except x = 0. If a quadratic form is xTAx= =0 for all x and there exists one nonzero x (x= =0) with xTAx = 0, then it is called positive semidefinite. If a quadratic form is xTAx= 0 for all x and there exists one nonzero x (x 0) with xTAx = 0, then it is called positive semidefinite. A quadratic form which is positive for some vectors x and negative for others is called indefinite. September 6, 1999 6 AOE/ESM 4084 Engineering Design Optimization Check the eigenvalues of the symmetric n n A matrix associated with the quadratic form F(x) = 1/2 xTAx. F(x) is positive definite if and only if all eigenvalues of A are strictly positive, i.e. i > 0, i = 1 to n. F(x) is positive semidefinite if and only if all eigenvalues of A are non-negative, i.e. i= =0, i = 1 to n. F(x) is negative definite if and only if all eigenvalues of A are strictly negative, i.e. i < 0, i = 1 to n. F(x) is negative semidefinite if and only if all eigenvalues of A are non-positive, i.e. i= =0, i = 1 to n. F(x) is indefinite if some i < 0 and some other i > 0. September 6, 1999 7

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