bv_cvxbook_extra_exercises

# Thus is dual feasible for 12 b let p denote the

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Unformatted text preview: -one matrix Z zT z 1 ≈ λ1 v1 t1 v1 t1 T . (Here we assume the eigenvalues are sorted in decreasing order). Then we take x = sign(v1 ) as our guess of good solution of (8). (c) We can also give a probabilistic interpretation of the relaxation (9). Suppose we interpret z and Z as the ﬁrst and second moments of a random vector v ∈ Rn (i.e., z = E v , Z = E vv T ). Show that (9) is equivalent to the problem minimize E Av − b 2 2 2 subject to E vk = 1, k = 1, . . . , n, where we minimize over all possible probability distributions of v . This interpretation suggests another heuristic method for computing suboptimal solutions of (8) based on the result of (9). We choose a distribution with ﬁrst and second moments E v = z , E vv T = Z (for example, the Gaussian distribution N (z, Z − zz T )). We generate a number of samples v from the distribution and round them to feasible solutions x = sign(˜). ˜ v We keep the solution with the lowest objective value as our guess of the optimal solution of...
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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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