Unformatted text preview: e consider a (possibly nonconvex) QCQP, with nonnegative variable
x ∈ Rn ,
minimize f0 (x)
subject to fi (x) ≤ 0, i = 1, . . . , m
where fi (x) = (1/2)xT Pi x + qi x + ri , with Pi ∈ Sn , qi ∈ Rn , and ri ∈ R, for i = 0, . . . , m. We do
not assume that Pi 0, so this need not be a convex problem. Suppose that qi 0, and Pi have nonpositive oﬀ-diagonal entries, i.e., they satisfy
(Pi )jk ≤ 0, j = k, j, k = 1, . . . , n, for i = 0, . . . , m. (A matrix with nonpositive oﬀ-diagonal entries is called a Z -matrix.) Explain
how to reformulate this problem as a convex problem.
Hint. Change variables using yj = φ(xj ), for some suitable function φ.
3.27 Aﬃne policy. We consider a family of LPs, parametrized by the random variable u, which is
uniformly distributed on U = [−1, 1]p ,
minimize cT x
subject to Ax b( u ) , where x ∈ Rn , A ∈ Rm×n , and b(u) = b0 + Bu ∈ Rm is an aﬃne function of u. You can think of
ui as representing a deviation of the ith parameter from its nominal value. The parameters might
represent (deviations in) levels of resources available, or other varying limits.
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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.
- Fall '13
- The Aeneid