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assign04soln

# n is optimal obviously x 0 and l x n x 1 so primal

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Unformatted text preview: . . , µn ) ≥ 0, we are to consider the minimization problem min −µ x s.t. x ≥ 0 , l x = 1 By Theorem 5.2, the necessary and suﬃcient conditions for optimality are x≥0, lx=1 µ = −v + ul for some u ∈ R , v ∈ Rn with v ≥ 0 vx=0 Now we check the point x∗ deﬁned by x∗ = i 1 if i = k 0 if i = k , where k ∈ {1, . . . , n} is such that µk ≥ µi for all i = 1, . . . , n, is optimal. Obviously, x∗ ≥ 0 and l x = n x∗ = 1, so primal feasibility conditions i=1 i are satisﬁed. Let v ∈ Rn be deﬁned by vi := µk − µi for i = 1, . . . , n. By the choice of index k, vi ≥ 0 for all i. It is also clear from the deﬁnition that v = µk l − µ, that is, µ = −v + µk l, so dual feasibility conditions are satisﬁed. Finally, v x∗ = n vi x∗ = vk x∗ = (µk − µk )x∗ = 0, so complementarity i k k i=1 slackness is satisﬁed. Therefore, x∗ is optimal for the given optimization problem. (5.7) Given the problem 1 min{−tµ x + x Σx | l x = 1, x ≥ 0} , 2 the optimality conditions for this problem are given by: l x = 1, x ≥ 0, tµ − Σx = cl − u, u ≥ 0, u x = 0. 8 We need to make an additional assumption that only one asset has the greatest expected return, say µ1 > µi for all i = 1. (Otherwise, the result is not necessarily true.) We claim that x∗ = (1, 0, . . . , 0) for all t large enough. We just need to show that x∗ satisﬁes the optimality conditions for some u. Notice that primal feasibility holds for x∗ and that complementary slackness implies that u1 = 0. Dual feasibility implies that c = ui + tµi − σi,1 , ∀i = 1, . . . , n, where σ1,1 . . Σ= . σn,1 ··· ... σ1,n . . . . · · · σn,n Since u1 = 0, we have c = tµ1 − σ1,1 , which implies tµ1 − σ1,1 = ui + tµi − σi,1 , ∀i = 1, . . . , n. Therefore, ui = t(µ1 − µi ) − (σ1,1 − σi,1 ), ∀i = 1, . . . , n. Since we must have ui ≥ 0 for all i, we get the following condition on t: t≥ σ1,1 − σi,1 , ∀i = 2, . . . , n. µ1 − µi Therefore, x∗ = (1, 0, . . . , 0) is optimal when t ≥ max σ1,1 − σi,1 | i = 2, . . . , n . µ1 − µi 9...
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