2_lecture2 - ISyE 6673 Financial Optimization Models...

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ISyE 6673 Financial Optimization Models Shabbir Ahmed E-mail: [email protected] Homepage: www.isye.gatech.edu/~sahmed Topic: Optimization Fundamentals
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Outline The generic optimization problem Properties of functions and sets Existence of optimal solutions Local vs. global optimality Convex programs Improving search 1
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The Generic Optimization Problem ( P ) : min f ( x ) , s . t . x X R n , where f : R n R is the objective function and X is the constraint set or the set of feasible solutions . W.l.o.g our discussion will be in terms of minimization problems. max { f ( x ) | x X } ≡ - min {- f ( x ) | x X } . Typically the constraint set is defined by inequality and equality constraints, as well as the domain of the values of the decision variables, for e.g. X = { x D | g i ( x ) 0 , i = 1 , . . . , m, h j ( x ) = 0 , j = 1 , . . . , p } . Example D = R n + The effort in solving ( P ) depends on the structure of f and X . 2
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Example For our portfolio optimization example: m = n + 2 and p = 0 with f ( x ) = x Q x g 1 ( x ) = - x 1 . . . g n ( x ) = - x n g n +1 ( x ) = e x - 1000 g n +2 ( x ) = 50 - ¯ r x We could also have D = R n + and have two explicit constraints g 1 ( x ) = e x - 1000 0 and g 2 ( x ) = 50 - ¯ r x 0 . 3
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Continuity A function f : R n R is continuous at x 0 if for every sequence { x i } such that lim i →∞ x i = x 0 we have lim i →∞ f ( x i ) = f ( x 0 ) . A function f : R n R is continuous if it is continuous at all x R n . 4
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Differentiability A function f : R R is differentiable at x 0 if lim t 0 f ( x 0 + t ) - f ( x 0 ) t exists. A function f : R n R is differentiable at x 0 if ∂f ( x ) ∂x i | x = x 0 exists for all i = 1 , . . . , n . A function f : R n R is differentiable (or smooth) if it is differentiable at all x R n . 5
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Gradient The gradient of a differentiable function f : R n R is defined as f ( x 0 ) = ∂f ( x ) ∂x 1 . . . ∂f ( x ) x n x = x 0 . 6
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Hessian The Hessian of a (twice) differentiable function f : R n R is defined as 2 f ( x 0 ) = 2 f ( x ) 2 x 1 2 f ( x ) ∂x 1 ∂x 2 . . .
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