Proof of Theorem 8

Since d^t_n ^i.i.d∼ F_n for all area n, q^t(i)defined in (4.16) can be obtained by

q^t(i)= arg min

q(i)≤Q(i)

C¯

q(i), q^t−1(−i)

(A.36)

which implies

C(q¯ ^t) ≤ ¯C(q^t−1), ∀t.

Let ˜q be a limit point of the sequence {q^t}^∞_t=1. The monotonicity of ¯C(q^t) implies that the sequence { ¯C(q^t)}^∞_t=1converges to ¯C( ˜q).

Let {q^t}_t∈T, whereT is an index set, be a subsequence of {q^t}^∞_t=1that converges to ˜q. From the updating rule (A.36) and the monotonicity of ¯C(q^t), we have

C q¯ ^t
≤ ¯C

q(i), q^t−1(−i) , ∀t, ∀i, ∀q(i) ≤ Q(i).

Since ¯C(q)is continuous, implied by Theorem 4, taking the limit as t ∈ T tends to infinity on both sides, we have

C( ˜q) ≤ ¯¯ C(q(i), ˜q(−i)) , ∀q(i) ≤ Q(i)

which means ˜q(i) is an optimal solution of the following optimization

q(i)≤Q(i)min

C¯(q(i), ˜q(−i)) . (A.37)

Therefore, ˜q(i) satisfies the KKT conditions A.32) for A.37 at ˜q. Since (A.30-A.32) hold for all i at ˜q, ˜q satisfies the KKT conditions (A.33-A.35). By the con-vexity of ¯C(q), the KKT conditions are sufficient and necessary for optimality.

Therefore, ˜q is optimal to (4.10).

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