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Unformatted text preview: Convex Optimization — Boyd & Vandenberghe 4. Convex optimization problems • optimization problem in standard form • convex optimization problems • quasiconvex optimization • linear optimization • quadratic optimization • geometric programming • generalized inequality constraints • semidefinite programming • vector optimization 4–1 Optimization problem in standard form minimize f ( x ) subject to f i ( x ) ≤ , i = 1 , . . . , m h i ( x ) = 0 , i = 1 , . . . , p • x ∈ R n is the optimization variable f : R n R is the objective or cost function • → f i : R n R , i = 1 , . . . , m , are the inequality constraint functions • → h i : R n R are the equality constraint functions • → optimal value: ⋆ p = inf { f ( x )  f i ( x ) ≤ , i = 1 , . . . , m, h i ( x ) = 0 , i = 1 , . . . , p } ⋆ • p = ∞ if problem is infeasible (no x satisfies the constraints) ⋆ • p = −∞ if problem is unbounded below Convex optimization problems 4–2 Optimal and locally optimal points x is feasible if x ∈ dom f and it satisfies the constraints a feasible x is optimal if f ( x ) = p ⋆ ; X opt is the set of optimal points x is locally optimal if there is an R > 0 such that x is optimal for minimize (over z ) f ( z ) subject to f i ( z ) ≤ , i = 1 , . . . , m, h i ( z ) = 0 , i = 1 , . . . , p z − x 2 ≤ R examples (with n = 1 , m = p = 0 ) ⋆ • f ( x ) = 1 /x , dom f = R ++ : p = 0 , no optimal point ⋆ • f ( x ) = − log x , dom f = R ++ : p = −∞ ⋆ • f ( x ) = x log x , dom f = R ++ : p = − 1 /e , x = 1 /e is optimal f ( x ) = x 3 − 3 x , p ⋆ = −∞ , local optimum at x = 1 • Convex optimization problems 4–3 Implicit constraints the standard form optimization problem has an implicit constraint m p x ∈ D = dom f i ∩ dom h i , i =0 i =1 • we call D the domain of the problem • the constraints f i ( x ) ≤ , h i ( x ) = 0 are the explicit constraints • a problem is unconstrained if it has no explicit constraints ( m = p = 0 ) example : minimize f ( x ) = − i k =1 log( b i − a i T x ) is an unconstrained problem with implicit constraints a i T x < b i Convex optimization problems 4–4 Feasibility problem find x subject to f i ( x ) ≤ , i = 1 , . . . , m h i ( x ) = 0 , i = 1 , . . . , p can be considered a special case of the general problem with f ( x ) = 0 : minimize 0 subject to f i ( x ) ≤ , i = 1 , . . . , m h i ( x ) = 0 , i = 1 , . . . , p ⋆ • p = 0 if constraints are feasible; any feasible x is optimal ⋆ • p = ∞ if constraints are infeasible Convex optimization problems 4–5 Convex optimization problem standard form convex optimization problem minimize f ( x ) subject to f i ( x ) ≤ , i = 1 , . . . , m a i T x = b i , i = 1 , . . . , p • f , f 1 , . . . , f m are convex; equality constraints are aﬃne • problem is quasiconvex if f is quasiconvex (and f 1 , . . . , f m convex) often written as minimize f ( x ) subject to f i ( x ) ≤ , i = 1 , . . . , m , ....
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This note was uploaded on 01/31/2012 for the course ECE 202 taught by Professor Boyde during the Fall '06 term at Goldsmiths.
 Fall '06
 boyde
 C Programming

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