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| 2. |
- Niculescu, Constantin P., et al.
(författare)
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Convex Functions and Their Applications : A Contemporary Approach
- 2018
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Bok (refereegranskat)abstract
- This second edition provides a thorough introduction to contemporary convex function theory with many new results. A large variety of subjects are covered, from the one real variable case to some of the most advanced topics. The new edition includes considerably more material emphasizing the rich applicability of convex analysis to concrete examples. Chapters 4, 5, and 6 are entirely new, covering important topics such as the Hardy-Littlewood-Pólya-Schur theory of majorization, matrix convexity, and the Legendre-Fenchel-Moreau duality theory.This book can serve as a reference and source of inspiration to researchers in several branches of mathematics and engineering, and it can also be used as a reference text for graduate courses on convex functions and applications.
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| 3. |
- Niculescu, Constantin P., et al.
(författare)
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Convex Functions on a Normed Linear Space
- 2018
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Ingår i: Duality and Convex Optimization. - Cham : Springer. - 9783319783369 - 9783319783376 ; , s. 107-184
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Bokkapitel (refereegranskat)abstract
- Convex functions and their relatives are ubiquitous in a large variety of applications such as optimization theory, mass transportation, mathematical economics, and geometric inequalities related to isoperimetric problems. This chapter is devoted to a succinct presentation of their theory in the context of real normed linear spaces, but most of the illustrations will refer to the Euclidean space RN,">RN,RN, the matrix space MN(R)">MN(R)MN(R) of all N×N">N×NN×N-dimensional real matrices (endowed with the Hilbert–Schmidt norm or with the operator norm), and the Lebesgue spaces Lp(RN)">Lp(RN)Lp(RN) with p∈[1,∞]">p∈[1,∞]p∈[1,∞].
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| 4. |
- Niculescu, Constantin P., et al.
(författare)
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Convex Functions on Intervals
- 2018
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Ingår i: Duality and Convex Optimization. - Cham : Springer. - 9783319783369 - 9783319783376 ; , s. 1-70
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Bokkapitel (refereegranskat)abstract
- The study of convex functions of one real variable offers an excellent glimpse of the beauty and fascination of advanced mathematics. The reader will find here a large variety of results based on simple and intuitive arguments that have remarkable applications. At the same time they provide the starting point of deep generalizations in the setting of several variables, that will be discussed in the next chapters.
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| 5. |
- Niculescu, Constantin P., et al.
(författare)
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Convex Sets in Real Linear Spaces
- 2018
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Ingår i: Convex Functions and Their Applications. - Cham : Springer. - 9783319783369 - 9783319783376 ; , s. 71-106
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Bokkapitel (refereegranskat)abstract
- The natural domain for a convex function is a convex set. In this chapter we review some basic facts, necessary for a deep understanding of the concept of convexity in real linear spaces. For reader’s convenience, all results concerning the separation of convex sets in Banach spaces are stated in Section 2.2 with proofs covering only the particular (but important) case of Euclidean spaces. Full details in the general case are to be found in Appendix B
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| 6. |
- Niculescu, Constantin P., et al.
(författare)
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Convexity and Majorization
- 2018
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Ingår i: Convex Functions and Their Applications. - Cham : Springer. - 9783319783369 - 9783319783376 ; , s. 185-226
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Bokkapitel (refereegranskat)abstract
- This chapter is aimed to offer a glimpse on the majorization theory and the beautiful inequalities associated to it. Introduced by G. H. Hardy, J. E. Littlewood, and G. Pólya (Messenger Math. 58:145–152, (1929), [208]) in 1929, and popularized by their celebrated book on Inequalities (Hardy et al., Inequalities, Cambridge University Press, 1952, [209]), the relation of majorization has attracted along the time a big deal of attention not only from the mathematicians, but also from people working in various other fields such as statistics, economics, physics, signal processing, data mining, etc. Part of this research activity is summarized in the 900 pages of the recent book by A. W. Marshall, I. Olkin, and B. Arnold (Inequalities: theory of majorization and its applications. Springer, New York (2011), [305]).
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| 7. |
- Niculescu, Constantin P., et al.
(författare)
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Convexity in Spaces of Matrices
- 2018
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Ingår i: Convex Functions and Their Applications. - Cham : Springer. - 9783319783369 - 9783319783376 ; , s. 227-254
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Bokkapitel (refereegranskat)abstract
- In this chapter we investigate three subjects concerning the convexity of functions defined on a space of matrices (or just on a convex subset of it). The first one is devoted to the convex spectral functions, that is, to the convex functions F:Sym(n,R)→R">F:Sym(n,R)→RF:Sym(n,R)→R whose values F(A) depend only on the spectrum of A. The main result concerns their description as superpositions f∘Λ">f∘Λf∘Λ between convex functions f:Rn→R">f:Rn→Rf:Rn→R invariant under permutations, and the eigenvalues map Λ">ΛΛ.
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| 8. |
- Niculescu, Constantin P., et al.
(författare)
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Duality and Convex Optimization
- 2018
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Ingår i: Convex Functions and Their Applications. - Cham : Springer. - 9783319783369 - 9783319783376 ; , s. 255-300
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Bokkapitel (refereegranskat)abstract
- Convex optimization is one of the main applications of the theory of convexity and Legendre–Fenchel duality is a basic tool, making more flexible the approach of many concrete problems. The diet problem, the transportation problem, and the optimal assignment problem are among the many problems that during the Second World War and immediately after led L. Kantorovich, T. C. Koopmans, F. L. Hitchcock, and G. B. Danzig to develop the mathematical theory of linear programming. Soon it was realized that most results extend to the framework of convex functions, which marked the birth of convex programming. Later on, W. Fenchel, R. T. Rockafellar, and J. J. Moreau laid the foundations of convex analysis.
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| 9. |
- Niculescu, Constantin P., et al.
(författare)
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Old and new on the Hermite-Hadamard inequality
- 2003
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Ingår i: Real Analysis Exchange. - : Michigan State University Press. - 0147-1937 .- 1930-1219. ; 29:2, s. 663-685
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Tidskriftsartikel (refereegranskat)abstract
- The goal of this paper is to describe the panorama of Mathematics grown up from the celebrated inequality of Hermite and Hadamard. Both old and new results are presented, complemented and discussed within this framework.
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| 10. |
- Niculescu, Constantin P., et al.
(författare)
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Special Topics in Majorization Theory
- 2018
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Ingår i: Duality and Convex Optimization. - Cham : Springer. - 9783319783369 - 9783319783376 ; , s. 301-326
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Bokkapitel (refereegranskat)abstract
- The primary aim of this chapter is to discuss the connection between the Hermite–Hadamard double inequality and Choquet’s theory. Noticed first by Niculescu (Math Inequal Appl 5(3):479–489, 2002, [356]), Niculescu (Math Inequal Appl 5(4):619–623, 2002, [357]) (during the conference Inequalities 2001, in Timişoara), this connection led him to a partial extension of the majorization theory beyond the classical case of probability measures, using the so-called Steffensen–Popoviciu measures. Their main feature is to offer a large framework under which the Jensen–Steffensen inequality remains available. As a consequence, one obtains the extension of the left-hand side of Hermite–Hadamard double inequality to a context involving signed Borel measures on arbitrary compact convex sets. A similar extension of the right-hand side of this inequality is known only in dimension 1, the higher dimensional case being still open.
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