bv_cvxbook_extra_exercises

# Be as specic as you can c we can also minimize the

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Unformatted text preview: plex in the variable statement. (In particular, you do not have to manually form or solve the SOCP from part (b).) 15 3.10 Linear programming with random cost vector. We consider the linear program minimize cT x subject to Ax b. Here, however, the cost vector c is random, normally distributed with mean E c = c0 and covariance E(c − c0 )(c − c0 )T = Σ. (A, b, and x are deterministic.) Thus, for a given x ∈ Rn , the cost cT x is a (scalar) Gaussian variable. We can attach several diﬀerent meanings to the goal ‘minimize cT x’; we explore some of these below. (a) How would you minimize the expected cost E cT x subject to Ax b? (b) In general there is a tradeoﬀ between small expected cost and small cost variance. One way to take variance into account is to minimize a linear combination E cT x + γ var(cT x) (3) of the expected value E cT x and the variance var(cT x) = E(cT x)2 − (E cT x)2 . This is called the ‘risk-sensitive cost’, and the parameter γ ≥ 0 is called the risk-aversion parameter, since it sets the relative values of cost variance and...
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