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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 41 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 42 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 > 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 bardbl z x bardbl 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 43 Implicit constraints the standard form optimization problem has an implicit constraint x D = m intersectiondisplay i =0 dom f i p intersectiondisplay i =1 dom h i , 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 ) = k i =1 log( b i a T i x ) is an unconstrained problem with implicit constraints a T i x < b i Convex optimization problems 44 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 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 45 Convex optimization problem standard form convex optimization problem minimize f ( x ) subject to f i ( x ) , i = 1 ,...,m a T i 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 Ax = b important property: feasible set of a convex optimization problem is convex Convex optimization problems 46 example...
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 Fall '08
 Tarraf,D

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