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Unformatted text preview: ISyE 6673 Financial Optimization Models Shabbir Ahmed Email: [email protected] Homepage: www.isye.gatech.edu/~sahmed Topic: Optimization Fundamentals Outline • The generic optimization problem • Properties of functions and sets • Existence of optimal solutions • Local vs. global optimality • Convex programs • Improving search 1 The Generic Optimization Problem ( P ) : min f ( x ) , s . t . x ∈ X ⊆ R n , where f : R n 7→ 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 ) ≤ ,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 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 ≤ and g 2 ( x ) = 50 ¯ r > x ≤ . 3 Continuity A function f : R n 7→ R is continuous at x if for every sequence { x i } such that lim i →∞ x i = x we have lim i →∞ f ( x i ) = f ( x ) . A function f : R n 7→ R is continuous if it is continuous at all x ∈ R n . 4 Differentiability A function f : R 7→ R is differentiable at x if lim t ↓ f ( x + t ) f ( x ) t exists. A function f : R n 7→ R is differentiable at x if ∂f ( x ) ∂x i  x = x exists for all i = 1 ,...,n . A function f : R n 7→ R is differentiable (or smooth) if it is differentiable at all x ∈ R n . 5 Gradient The gradient of a differentiable function f : R n 7→ R is defined as ∇ f ( x ) = ∂f ( x ) ∂x 1 ....
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This note was uploaded on 02/04/2012 for the course ECON 6673 taught by Professor Ahmed during the Spring '11 term at Georgia Tech.
 Spring '11
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