bv_cvxbook_extra_exercises

# The solution of an lp need not be unique all we say

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Unformatted text preview: : R → R is increasing and convex. Then the problem ˜ minimize f0 (x) = φ(f0 (x)) (12) subject to fi (x) ≤ 0, i = 1, . . . , m is convex and equivalent to it; in fact, it has the same optimal set as (11). In this problem we explore the connections between the duals of the two problems (11) and (12). We assume fi are diﬀerentiable, and to make things speciﬁc, we take φ(a) = exp a. (a) Suppose λ is feasible for the dual of (11), and x minimizes ¯ m λi fi ( x ) . f0 ( x ) + i=1 Show that x also minimizes ¯ m exp f0 (x) + ˜ λ i fi ( x ) i=1 ˜ ˜ for appropriate choice of λ. Thus, λ is dual feasible for (12). (b) Let p⋆ denote the optimal value of (11) (so the optimal value of (11) is exp p⋆ ). From λ we obtain the bound p⋆ ≥ g ( λ) , 32 ˜ where g is the dual function for (11). From λ we obtain the bound exp p⋆ ≥ g (λ), where g is ˜˜ ˜ the dual function for (12). This can be expressed as p⋆ ≥ log g (λ). ˜˜ How do these bounds compare? Are they the same, or is one better than the other? 4.12 Variable bounds and dual feasibilit...
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## This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.

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