A History of Mathematical Notations Volume II, Notations by Florian Cajori

By Florian Cajori

367 pp.

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3) with u = 0. 3), F(x, u) is convex in (x, u). On the other hand, if F is convex the function is convex; since the function x -* K(x, y) is the infimum of this in u, it too is convex (Theorem 1). 1)) as (D) maximize g over y, where The next result is perhaps the central theorem about dual problems. It relates g to the optimal value function which, as we recall from Theorem 1, is convex in u if F is convex in (x, u). THEOREM 7. The function g in (D) is closed concave. In fact, taking conjugates in the concave sense we have g = ( —

4). Instead of K = K, we have from CONJUGATE DUALITY AND OPTIMIZATION 23 In this case the discrepancy between K and K is "trivial," but other examples are known where, even in finite-dimensional spaces, the two functions can differ at special points where both are finite-valued, or where one is finite and the other infinite. In infinite-dimensional spaces, the relationship can be quite complicated. The details of the relationship can be found in [13], [16]. The point we wish to make here is that the continuity properties of Lagrangian functions are not so elementary.

It is easy to see that the Slater condition is equivalent to having Oe core dom (p. For the properties in Theorem 17' all to hold, we have from Corollary 18'A 46 R. TYRRELL ROCKAFELLAR the sufficient condition that for some E > 0 and a > inf (P), the set is equicontinuous in X. If X = R", this can according to Theorem 18' (d) be weakened to the condition that for some real number a the set is nonempty and bounded. Corollary 16A leads to a very complete characterization of the existence of an optimal multiplier vector y in the case where inf (P) is finite.

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