Convex Analysis and Variational Problems
No one working in duality should be without a copy of Convex Analysis and Variational Problems. This book contains different developments of infinite dimensional convex programming in the context of convex analysis, including duality, minmax and Lagrangians, and convexification of nonconvex optimization problems in the calculus of variations (infinite dimension). It also includes the theory of convex duality applied to partial differential equations; no other reference presents this in a systematic way. The minmax theorems contained in this book have many useful applications, in particular the robust control of partial differential equations in finite time horizon. First published in English in 1976, this SIAM Classics in Applied Mathematics edition contains the original text along with a new preface and some additional references.
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a.e. x e affine function apply Proposition Borel calculus of variations Carathéodory Carathéodory function Chapter closed convex compact set converges convex functions convex set Corollary deduce defined denote differentiable dual problem duality epi F equation Euler equation exists extremality relations Fatou's lemma finite function F G(Au Gâteaux-differentiable grad u(x gradu Hä(Q hence hypersurfaces hypotheses implies indicator function Inf L(u infimum infº Inſ L(ii Lemma Let F linear Lipschitz Lº(Q mapping minimal hypersurfaces minimizing sequence non-convex non-empty non-negative norm normal integrand obtain optimization problem possesses a solution problem 9 Proof Proposition 2.3 Remark ſ f(x saddle point satisfies Section ſº strictly convex subdifferentiable sup 9 supremum T-regularization Theorem 1.2 topology unique solution variational inequalities weakly whence x e Q