Infinite-Dimensional Optimization and Convexity
University of Chicago Press, Sep 15, 1983 - Business & Economics - 166 pages
In this volume, Ekeland and Turnbull are mainly concerned with existence theory. They seek to determine whether, when given an optimization problem consisting of minimizing a functional over some feasible set, an optimal solution—a minimizer—may be found.
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affine functional everywhere Ax,y Banach space Borel function calculus of variations Cauchy-Schwarz inequality closed hyperplane continuous affine functional continuous linear functional converges convex and l.s.c. convex set critical point critical value defined Definition dual problem duality theory epi F epi G equation example exist F by Proposition F is continuous F is convex F is l.s.c. F o A)(x F(xe F(xQ finite-dimensional follows from Proposition functional everywhere less G(Ax Hamilton-Jacobi equation Hamiltonian system Hence F Hilbert space implies inf F inf F(x inffP INFINITE-DIMENSIONAL OPTIMIZATION l.s.c. by Proposition l.s.c. function least one solution Lebesgue measure Lemma less than F lim infn F(xn linear subspace measurable functions metric space minimizing sequence optimal control pointwise supremum Proposition 2.2 Proposition II.4.2 reflexive Rudin sequence xn subject to x(0 subset Suppose supremum supx t)dt u(dt weak topology weakly closed xn(t