By Ivar Ekeland, Roger Témam

Not anyone operating in duality could be with no replica of Convex research and Variational difficulties. This publication includes various advancements of endless dimensional convex programming within the context of convex research, together with duality, minmax and Lagrangians, and convexification of nonconvex optimization difficulties within the calculus of diversifications (infinite dimension). it is usually the idea of convex duality utilized to partial differential equations; no different reference offers this in a scientific means. The minmax theorems contained during this ebook have many beneficial purposes, particularly the strong keep watch over of partial differential equations in finite time horizon. First released in English in 1976, this SIAM Classics in utilized arithmetic variation includes the unique textual content in addition to a brand new preface and a few extra references.

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Chap. (1) The problem: is termed the dual problem of ^ with respect to

* are finite Proof. 9). e. the T-regularization of #($**er(Fx 10).

18), we have G(u) ^ inf G + g. 1 to G. There exists i/A e V such that For greater simplicity, we shall denote by ^ the cone (MA, G(wA)) + #(e/A) in F x R. It is a closed convex set with non-empty interior and epi G is a closed convex set. 24) and the Hahn-Banach theorem, we can separate ^A and epi G by a closed affine hyperplane #f of V x R with equation : 32 FUNDAMENTALS OF CONVEX ANALYSIS Since ^ is closed and convex, it is the closure of its (non-empty) interior and so 3tf also separates ^A and epi G.

It therefore converges to a limit (d,u) which belongs to S since the latter is closed. It only remains to show that (a,w) is the required maximal element. For this we take any / e / and write the inequality and, passing to the limit iny: Thus (d,u) ^ (fli,Mi) for all /, which establishes the result. 2. Application to non-convex functions Let F be a mapping of V into R with — oo < inf F < +00. To say that F(u) = inf F amounts to saying that 0 is the subgradient of F at u. 1. c. function of V into R, with — oo < inf F< +00, and let there be a point u where 30 FUNDAMENTALS OF CONVEX ANALYSIS For all 1 > 0, there is au^eV such that prooof.

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