# Introduction to Optimization Theory in a Hilbert Space

Applied Mathematics and Optimization. We study a class of stochastic optimization problems in which the state as well as the observation spaces are permitted to be Hilbert spaces of non-finite dimension. Although there have been previous attempts in the Hilbert space setting, our results, techniques, as well as applications, are totally different.

We initiate the use of Gauss measure on a Hilbert space even though it is only finitely additive; and an associated theory of white noise, in contrast to the Wiener process theory, which is novel even in the finite dimensional case. We only treat time-invariant systems, but no strong ellipticity or coercivity conditions are used; we exploit the theory of semigroups of operators in contrast to the Lions-Magenes theory.

A key result involves a far-reaching generalization of the Factorization theorem of Krein. We apply the results to the problem of boundary observation and control for partial differential equations. By the creation of a special state space, we can apply the theory to problems in which the state equations are finitedimensional but the noise does not have a rational spectrum. In a final section, we present a stochastic theory for inverse problems System Identification in the Hilbert space setting.

The corresponding convergence conditions are satisfied for a large class of practically relevant problems in calculus of variations and optimal control. In particular, the complexity analysis in this paper implies that branch-and-lift can be applied to solve potentially non-convex and infinite-dimensional optimization problems without needing a-priori knowledge about the existence or regularity of minimizers, as the run-time bounds solely depend on the structural and regularity properties of the cost functional as well as the underlying Hilbert space and the geometry of the constraint set.

In order to demonstrate that these algorithmic ideas and complexity analysis are not of pure theoretical interest only, the practical applicability of branch-and-lift has been illustrated with a numerical case study for a problem of calculus of variations. The case study of an optimal control problem in [ 25 ] provides another illustration.

The authors would like to thank Co-Editor Dr. Sven Leyffer for his constructive comments about minimality of assumptions for the convergence of branch-and-lift. National Center for Biotechnology Information , U. Mathematical Programming. Math Program. Published online Dec Author information Article notes Copyright and License information Disclaimer. Corresponding author. Received Oct 28; Accepted Nov Abstract We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality.

Keywords: Infinite-dimensional optimization, Complete search, Branch-and-lift, Convergence analysis, Complexity analysis.

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Outline and contributions The paper starts by discussing several regularity conditions for sets and functionals defined in a Hilbert space in Sect. Some regularity conditions for sets and functionals in Hilbert space This section builds upon basic concepts in infinite-dimensional Hilbert spaces in order to arrive at certain regularity conditions for sets and functionals defined in such spaces. Proof Let G be a bounded and regular subset of H on C such that the condition 7 is satisfied.

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## Introduction to Optimization Theory in a Hilbert Space (lecture by A.v. VG

Remark 7 With regularity of the set G alone, i. Remark 8 In general, Lipschitz-continuity does not imply strong Lipschitz-continuity in an infinite-dimensional Hilbert space. Remark 10 Many recent optimization techniques for global optimization are based on the theory of positive polynomials and their associated linear matrix inequality LMI approximations [ 30 , 45 ], which are also originally inspired by moment problems.

Strategies for upper and lower bounding of functionals Besides partitioning, the efficient construction of tight upper and lower bounds on the global solution value of 1 for given subregions of H is key in a practical implementation of branch-and-lift. Open in a separate window. Conclusions This paper has presented a complete-search algorithm, called branch-and-lift, for global optimization of problems with a non-convex cost functional and a bounded and convex constraint sets defined on a Hilbert space.

References 1. Akhiezer NI. Translated by N. New York: Hafner Publishing Co. Albersmeyer J, Diehl M.

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The lifted Newton method and its application in optimization. SIAM J. Anderson EJ, Nash P. Linear Programming in Infinite-Dimensional Spaces. Hoboken: Wiley; Bampou D, Kuhn D. Polynomial approximations for continuous linear programs. Topology Optimization: Theory, Methods, and Applications. Berlin: Springer; Betts JT. Philadelphia: SIAM; Biegler LT. Solution of dynamic optimization problems by successive quadratic programming and orthogonal collocation. Bock, H. Pergamon Press, Oxford Bompadre A, Mitsos A.

Convergence rate of McCormick relaxations.

Convergence analysis of Taylor and McCormick-Taylor models. Boyd S, Vandenberghe L. Convex Optimization. Cambridge: University Press; Bryson AE, Ho Y. Applied Optimal Control. Washington: Hemisphere; Buie R, Abrham J.

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October 20, - Published on Amazon. The first eleven chapters are an excellent introduction to functional analysis. Both Hilbert and Banach spaces are introduced carefully. Then there are two short chapters on orthogonal expansions and classical fourier series and then linear operators are studied. From the point of view of a person who is interested in applications to physics and engineering one can say that the book is well motivated mainly because is so compact and because of the many notes on applications.

Chapters nine , ten and eleven on Green's functions and eigenfunctions expansions are extremely good. Chapters twelve and thirteen are poorly motivated from the point of view of applications.

## Introduction to Optimization Theory in a Hilbert Space | A.V. Balakrishnan | Springer

Finally chapters fourteen to sixteen try to exhibit the applications to complex analysis of operator theory and be helpfull to eletrical engineers. I think the book fails in this. So the ten first chapters of the book are excellent. The remaining less so. February 27, - Published on Amazon. Young has done an admirable job at presenting some really beautiful and useful aspects of Hilbert spaces in a manner comprehendable for advanced undergraduates. After reading the book and reflecting on the experience, I'm somewhat amazed at the amount of nice ideas that were presented in such a compact text.

The book cannot be compared with more rigorous and comprehensive texts such as Rudin, but you still get all the fundamentals of Hilbert space plus some wonderful applications. I must strongly disagree with the reader from Sao Paolo who says that chapters 12 and 13 are poorly motivated. These chapters are crucial for the final theorem of the book in chapter Parrott's Theorem in chapter 12 is the key to the foundational Nehari's theorem of chapter