bv_cvxbook_extra_exercises

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Unformatted text preview: mize tr(AX ) subject to tr X = r 0 X I, with variable X ∈ Sn , is equal to f (A). (b) Show that f is a convex function. (c) Assume A(x) = A0 + x1 A1 + · · · + xm Am , with Ak ∈ Sn . Use the observation in part (a) to formulate the optimization problem minimize f (A(x)), with variable x ∈ Rm , as an SDP. 4.18 An exact penalty function. Suppose we are given a convex problem minimize f0 (x) subject to fi (x) ≤ 0, i = 1, . . . , m with dual (14) maximize g (λ) subject to λ 0. (15) We assume that Slater’s condition holds, so we have strong duality and the dual optimum is attained. For simplicity we will assume that there is a unique dual optimal solution λ⋆ . For fixed t > 0, consider the unconstrained minimization problem minimize f0 (x) + t max fi (x)+ , i=1,...,m (16) where fi (x)+ = max{fi (x), 0}. (a) Show that the objective function in (16) is convex. (b) We can express (16) as minimize f0 (x) + ty subject to fi (x) ≤ y, 0≤y i = 1. . . . , m (17) where the variables are x and y ∈ R. Find the Lagrange dual...
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