# Convex analysis by Nickel S.

By Nickel S.

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Y m , extremals are then given by Eqs. 7). 2) in the form 1 1 n x ( F ) F " - ' dF. 9) 0 It is assumed that x is a function of F . 10) x z ( F ) d F = 1. 1 1 ) 0 1 1 0 We consider the Lagrangian A(X,y , y', A) = nxy"-' t h l x t AzxZ. 1 2 ) The Euler equation gives ny"-' + Al t 2 A z x = 0. 11). 9). In such a derivation, the existence of the constants X I and h2 is not guaranteed. In Chapter V we shall consider the problem from another point of view and will also demonstrate the existence of the solution.

And Kemp, C. M. (1971). Variational Methods in Economics. American Elsevier, New York. , and Rubin, H. (1954). On minimizing an expectation subject t o certain side conditions, Tech. Rep. No. 25, Appl. Math. Statist. , Stanford, California. , and Studden, W. J . (1966). Tchebycheff Systems: With Applications in Analysis and Sratistics. Wiley (lnterscience), New York. Kiefer, J . (1959). Optimal experimental designs, J. Roy. Statist. Ser. B 21, 273-319. , and Pearson, E. S. (1936, 1938). Contributions to the theory of testing statistical hypotheses.

5, we consider a few connections between mathematical programming methods and statistics. The scope of these applications is very large, and only a few cases are discussed. The application of the NeymanPearson lemma in developing the duality of nonlinear programming problems has been investigated by Francis and Wright (1969) and many others. We give an introduction to this theory. There are many applications of mathematical programming methods t o statistics. Some of the moment problems can also be reduced to those of programming problems, and the duality theory for these problems leads to interesting results.