By Saul I. Gass

Interesting, nontechnical advent covers simple strategies of linear programming and its courting to operations examine; geometric interpretation and challenge fixing, answer thoughts, community difficulties, even more. Appendix bargains distinct statements of definitions, theorems, and methods, extra computational approaches. merely high-school algebra wanted. Bibliography.

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As we shall illustrate, the methods which correspond to one problem may yield more precise information than those of another. Each of the three paradigms has advantages specific to it. PB, by its very form, facilitates our drawing inspiration from the calculus of variations. It is possible within the framework of PB to achieve a unification of the classical calculus of variations and optimal control theory. Indeed, the extension to PB of classical variational methods, which requires consideration of nonsmoothness, turns out to be a powerful tool in optimal control theory.

2 Relation to Derivatives and Subderivatives 31 and provided the convergence is uniform for v in compact sets. (This last condition is automatic if F is Lipschitz near x). " 2. 2. 1 Proposition Let F map a neighborhood of x to Y, and let £ be an element of &(X, Y). The following are equivalent'. (a) F is strictly differentiable at x and DsF(x) = f. (b) F is Lipschitz near x, and for each v in X one has Proof. Assume (a). The equality in (b) holds by assumption, so to prove (b) we need only show that F is Lipschitz near jc.

Then Proof. 2 Relation to Derivatives and Subderivatives 33 (Df(x), v) for all v in X. 5(a). 3 Example That df(x) can contain points other than Df(x) is illustrated by the familiar example on R : f ( x ) := x 2 sin(l/;c). This function is Lipschitz near 0, and it is easy to show that /°(0; v) = \v\. It follows that d/(0) = [-1,1], a set which contains the (nonstrict) derivative Df(0) = 0. 4 Proposition If f is strictly differentiable at x, then f is Lipschitz near x and df(x) = (Dsf(x)}. Conversely, if f is Lipschitz near x and df(x) reduces to a singleton {f}, then f is strictly differentiable at x and Dsf(x) = f.