Convex Optimization — Boyd & Vandenberghe 4. Convex ...

Convex Optimization Boyd & Vandenberghe4. 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 optimization4 1 Optimization problem in standard formminimizef0(x)subject tofi(x) 0, i= 1, .. , mhi(x) = 0, i= 1, .. , p x Rnis the Optimization variable f0:Rn Ris the objective or cost function fi:Rn R,i= 1, .. , m, are the inequality constraint functions hi:Rn Rare the equality constraint functionsoptimal value:p = inf{f0(x)|fi(x) 0, i= 1.}

• minimization over nonnegative orthant minimize f 0(x) subject to x 0 x is optimal if and only if x ∈ domf 0, x 0, ˆ ∇f 0(x)i ≥ 0 xi = 0 ∇f 0(x)i = 0 xi > 0 Convex optimization problems 4–10

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