Approaches to the Theory of Optimization
Optimization is concerned with finding the best (optimal) solution to mathematical problems that may arise in economics, engineering, the social sciences and the mathematical sciences. As is suggested by its title, this book surveys various ways of penetrating the subject. The author begins with a selection of the type of problem to which optimization can be applied and the remainder of the book develops the theory, mainly from the viewpoint of mathematical programming. To prevent the treatment becoming too abstract, subjects which may be considered 'unpractical' are not touched upon. The author gives plausible reasons, without forsaking rigor, to show how the subject develops 'naturally'. Professor Ponstein has provided a concise account of optimization which should be readily accessible to anyone with a basic understanding of topology and functional analysis. Advanced students and professionals concerned with operations research, optimal control and mathematical programming will welcome this useful and interesting book.
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assume assumptions Banach space bifunction F Chebyshev approximation closed compact conjugate consider constraint qualification Conversely convex function convex set convex topological vector Corollary defined dual pair dual problem dual solution element equality constraints equation Euclidean space example exists feasible Fenchel duality finding inf finite finite-dimensional fixed point follows Frechet differentiable function h given hence holds hyperplane implies inequality constraints infimum L(dom Lagrangian duality Lemma linear programming linear space linear subspace locally convex topological mapping mathematical programming means neighbourhood nonnegative norm normal obtained oo otherwise open mapping theorem optimal solution optimization problem perturbation function positive cone primal problem problem of finding proof of Theorem regularity conditions replace require saddle-point satisfied separation solvable solved space with dual subspace supK supremum topological vector space variables variational inequality wi wi wi
Page 196 - Necessary optimality criteria in mathematical programming in the presence of differentiability.
Page 200 - RT ROCKAFELLAR, Monotone operators and the proximal point algorithm, SIAM J. Control Opt., 14 (1976), pp.
Page 196 - Zlobec. S. (1976). Characterization of optimality in convex programming without a constraint qualification. JOTA 20, 417-37.