Earlier and Recent Results on Convex Mappings and Convex Optimization
31 Aug 2019-Vol. 3, pp 136-148
TL;DR: The main purpose of this review-paper is to recall and partially prove earlier, as well as recent results on convex optimization, published by the author in the last decades, and most of theorems have a clear geometric meaning.
Abstract: The main purpose of this review-paper is to recall and partially prove earlier, as well as recent results on convex optimization, published by the author in the last decades. Examples are given along the article. Some of these results have been published recently. Most of theorems have a clear geometric meaning. Minimum norm elements are characterized in normed vector spaces framework. Distanced convex subsets and related parallel hyperplanes preserving the distance are also discussed. The convex involved objective-mappings are real valued or take values in an order-complete vector lattice. On the other side, an optimization problem related to Markov moment problem is solved in the end.
Citations
More filters
18 Dec 2019
TL;DR: In this paper, a constrained optimization problem is solved, as an application of minimum principle for a sum of strictly concave continuous functions, subject to a linear constraint, firstly for finite sums of elementary such functions, and corresponding results for infinite sums of the such type of functions follow as a consequence, passing to the limit.
Abstract: A constrained optimization problem is solved, as an application of minimum principle for a sum of strictly concave continuous functions, subject to a linear constraint, firstly for finite sums of elementary such functions. The motivation of solving such problems is minimizing and evaluating the (unknown) mean of a random variable, in terms of the (known) mean of another related random variable. The corresponding result for infinite sums of the such type of functions follows as a consequence, passing to the limit. Note that in our statements and proofs the condition on the positive numbers is not essential for the interesting part of the results. So, our work refers not only to means of random variables, but to more general weighted means. A related example is given. A corresponding result for special concave mappings taking values into an order-complete Banach lattice of self-adjoint operators is also proved. Namely, one finds a lower bound for a sum of special concave mappings with ranges in the above mention order-complete Banach lattice, under a suitable linear constraint.
Cites background from "Earlier and Recent Results on Conve..."
...Other aspects of optimization theory have been recently recalled in [12]....
[...]
References
More filters
Book•
01 Jan 1966
TL;DR: In this paper, the Riesz representation theorem is used to describe the regularity properties of Borel measures and their relation to the Radon-Nikodym theorem of continuous functions.
Abstract: Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index
9,642 citations
Book•
01 Jan 1966
TL;DR: The Krein-Milman theorem as an integral representation theorem has been applied to the metrizable case of the Choquet boundary as mentioned in this paper, and it has been used to define a new set of integral representation theorems for monotonic functions.
Abstract: Preface.- Introduction. The Krein-Milman theorem as an integral representation theorem.- Application of the Krein-Milman theorem to completely monotonic functions.- Choquet's theorem: The metrizable case.- The Choquet-Bishop-de Leeuw existence theorem.- Applications to Rainwater's and Haydon's theorems.- A new setting: The Choquet boundary.- Applications of the Choquet boundary to resolvents.- The Choquet boundary for uniform algebras.- The Choquet boundary and approximation theory.- Uniqueness of representing measures.- Properties of the resultant map.- Application to invariant and ergodic measures.- A method for extending the representation theorems: Caps.- A different method for extending the representation theorems.- Orderings and dilations of measures.- Additional Topics.- References.- Index of symbols.- Index.
1,143 citations