R. Tyrrell Rockafellar

Ralph Tyrrell Rockafellar is an American mathematician and one of the leading scholars in optimization theory and related fields of analysis and combinatorics. He is professor emeritus at the departments of mathematics and applied mathematics at the University of Washington, Seattle, he was born in Wisconsin. Rockafellar received the John von Neumann Theory Prize from the Institute for Operations Research and Management Science and delivered the 1993 John von Neumann Lecture for the Society for Industrial and Applied Mathematics. Rockafellar and his coauthor Roger J-B Wets were awarded the Frederick W. Lanchester Prize for 1997 by Institute for Operations Research and the Management Sciences; the Institute for Scientific Information lists Rockafellar as a cited researcher. Rockafellar, R. Tyrrell. Convex analysis. Princeton landmarks in mathematics. Princeton, NJ: Princeton University Press. Pp. xviii+451. ISBN 978-0-691-01586-6. MR 1451876. Rockafellar, R. Tyrrell. Conjugate duality and optimization.

Lectures given at Baltimore, Md.. June, 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16. Society for Industrial and Applied Mathematics, Philadelphia, Pa. 1974. Vi+74 pp. Rockafellar, Ralph T; the theory of subgradients and its applications to problems of optimization. Convex and nonconvex functions. Heldermann Verlag, Berlin, 1981. Vii+107 pp. ISBN 3-88538-201-6 R. T. Rockafellar. 1984. Network Flows and Monotropic Optimization. Wiley. Rockafellar, R. Tyrrell. Variational analysis. Grundlehren der mathematischen Wissenschaften. 317. Berlin: Springer-Verlag. Pp. xiv+733. Doi:10.1007/978-3-642-02431-3. ISBN 978-3-540-62772-2. MR 1491362. Retrieved 12 March 2012. Dontchev, Asen L. and Rockafellar, R. Tyrrell. Implicit functions and solution mappings. A view from variational analysis. Springer Monographs in Mathematics. Springer, Dordrecht, 2009. Xii+375 pp. ISBN 978-0-387-87820-1. Rockafellar, R. T.. "On the maximal monotonicity of subdifferential mappings".

Pacific J. Math. 33: 209–216. Doi:10.2140/pjm.1970.33.209. Rockafellar, R. T.. "The multiplier method of Hestenes and Powell applied to convex programming". J. Optimization Theory Appl. 12: 555–562. Doi:10.1007/bf00934777. Rockafellar, R. T.. "Augmented Lagrange multiplier functions and duality in nonconvex programming". SIAM J. Control. 12: 268–285. Doi:10.1137/0312021. Rockafellar, R. T.. "Augmented Lagrangians and applications of the proximal point algorithm in convex programming". Math. Oper. Res. 1: 97–116. Doi:10.1287/moor.1.2.97. Rockafellar, R. T.. "Lagrange multipliers and optimality". SIAM Rev. 35: 183–238. Doi:10.1137/1035044. Rockafellar, R. T.. "The Elementary Vectors of a Subspace of R N "". In R. C. Bose and T. A. Dowling. Combinatorial Mathematics and its Applications; the University of North Carolina Monograph Series in Probability and Statistics. Chapel Hill, North Carolina: University of North Carolina Press. Pp. 104–127. MR 0278972. Rockafellar, R. T.. "Scenarios and policy aggregation in optimization under uncertainty".

Math. Oper. Res. 16: 119–147. Doi:10.1287/moor.16.1.119. Rockafellar, R. Tyrrell Monotone processes of concave type. Memoirs of the American Mathematical Society, No. 77 American Mathematical Society, Providence, R. I. 1967 i+74 pp. Aardal, Karen. "Optima interview Roger J.-B. Wets". Optima: Mathematical Programming Society Newsletter: 3–5. "An Interview with R. Tyrrell Rockafellar". SIAG/Opt News and Views. 15. 2004. Wets, Roger J-B, Roger J-B, ed. "Foreword", Special Issue on Variational Analysis and their Applications, Mathematical Programming and Heidelberg: Springer Verlag, 104: 203–204, doi:10.1007/s10107-005-0612-5, ISSN 0025-5610 Homepage of R. Tyrrell Rockafellar at the University of Washington. R. Tyrrell Rockafellar at the Mathematics Genealogy Project Biography of R. Tyrrell Rockafellar from the Institute of Operations Research and the Management Sciences

Banach space

In mathematics, more in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit, within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Banach spaces grew out of the study of function spaces by Hilbert, Fréchet, Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are Banach spaces. A Banach space is a vector space X over the field R of real numbers, or over the field C of complex numbers, equipped with a norm ‖ ⋅ ‖ X and, complete with respect to the distance function induced by the norm, to say, for every Cauchy sequence in X, there exists an element x in X such that lim n → ∞ x n = x, or equivalently: lim n → ∞ ‖ x n − x ‖ X = 0.

The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space X is a Banach space if and only if each convergent series in X converges, ∑ n = 1 ∞ ‖ v n ‖ X < ∞ implies that ∑ n = 1 ∞ v n converges in X. Completeness of a normed space is preserved if the given norm is replaced by an equivalent one. All norms on a finite-dimensional vector space are equivalent; every finite-dimensional normed space over R or C is a Banach space. If X and Y are normed spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B. In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space X to another normed space is continuous if and only if it is bounded on the closed unit ball of X. Thus, the vector space B can be given the operator norm ‖ T ‖ = sup. For Y a Banach space, the space B is a Banach space with respect to this norm. If X is a Banach space, the space B = B forms a unital Banach algebra.

If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T: X → Y such that T and its inverse T −1 are continuous. If one of the two spaces X or Y is complete so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry, i.e. ||T|| = ||x|| for every x in X. The Banach–Mazur distance d between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and Y differ; every normed space X can be isometrically embedded in a Banach space. More for every normed space X, there exist a Banach space Y and a mapping T: X → Y such that T is an isometric mapping and T is dense in Y. If Z is another Banach space such that there is an isometric isomorphism from X onto a dense subset of Z Z is isometrically isomorphic to Y; this Banach space Y is the completion of the normed space X. The underlying metric space for Y is the same as the metric completion of X, with the vector space operations extended from X to Y.

The completion of X is denoted by X ^. The cartesian product X × Y of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are used, such as ‖ ‖ 1 = ‖ x ‖ + ‖ y ‖, ‖ ‖ ∞ = max and give rise to isomorphic normed spaces. In this sense, the product X × Y is only if the two factors are complete. If M is a closed linear subspace of a normed space X, there is a natural norm on the quotient space X / M, ‖ x + M ‖ = inf m ∈ M ‖ x + m ‖; the quotient X / M is a Banach space

Werner Fenchel

Moritz Werner Fenchel was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory which would, in time, serve as the foundation for nonlinear programming. A German-born Jew and early refugee from Nazi suppression of intellectuals, Fenchel lived most of his life in Denmark. Fenchel's monographs and lecture notes are considered influential. Fenchel was born on 3 May 1905 in Berlin, his younger brother was the Israeli architect Heinz Fenchel. Fenchel studied mathematics and physics at the University of Berlin between 1923 and 1928, he wrote his doctorate thesis in geometry under Ludwig Bieberbach. From 1928 to 1933, Fenchel was Professor E. Landau's Assistant at the University of Göttingen. During a one-year leave between 1930 and 1931, Fenchel spent time in Rome with Levi-Civita, as well as in Copenhagen with Harald Bohr and Tommy Bonnesen, he visited Denmark again in 1932. Fenchel taught at Göttingen until 1933, when the Nazi discrimination laws led to mass-firings of Jews.

Fenchel emigrated to Denmark somewhere between April and September 1933 obtaining a position at the University of Copenhagen. In December 1933, Fenchel married fellow German refugee mathematician Käte Sperling; when Germany occupied Denmark and eight-thousand other Danish Jews received refuge in Sweden, where he taught at the Danish School in Lund. After the Allied powers' liberation of Denmark, Fenchel returned to Copenhagen. In 1946, Fenchel was elected a member of the Royal Danish Academy of Letters. On leave between 1949 and 1951, Fenchel taught in the U. S. at the University of Southern California, Stanford University, Princeton University. From 1952 to 1956 Fenchel was the professor in mechanics at the Polytechnic in Copenhagen. From 1956 to 1974 he was the professor in mathematics at the University of Copenhagen. Professor Fenchel died on 24 January 1988. Fenchel lectured on "Convex Sets and Functions" at Princeton University in the early 1950s, his lecture notes shaped the field of convex analysis, according to the monograph Convex Analysis of R. T. Rockafellar.

Werner Fenchel at the Mathematics Genealogy Project Werner Fenchel website – contains CV, links to archive, etc

Infimum and supremum

In mathematics, the infimum of a subset S of a ordered set T is the greatest element in T, less than or equal to all elements of S, if such an element exists. The term greatest lower bound is commonly used; the supremum of a subset S of a ordered set T is the least element in T, greater than or equal to all elements of S, if such an element exists. The supremum is referred to as the least upper bound; the infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary ordered sets are considered; the concepts of infimum and supremum are similar to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum or maximum. For instance, the positive real numbers ℝ+ does not have a minimum, because any given element of ℝ+ could be divided in half resulting in a smaller number, still in ℝ+.

There is, however one infimum of the positive real numbers: 0, smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. A lower bound of a subset S of a ordered set is an element a of P such that a ≤ x for all x in S. A lower bound a of S is called an infimum of S. An upper bound of a subset S of a ordered set is an element b of P such that b ≥ x for all x in S. An upper bound b of S is called a supremum of S if for all upper bounds z of S in P, z ≥ b. Infima and suprema do not exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique. Ordered sets for which certain infima are known to exist become interesting. For instance, a lattice is a ordered set in which all nonempty finite subsets have both a supremum and an infimum, a complete lattice is a ordered set in which all subsets have both a supremum and an infimum.

More information on the various classes of ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset S exists, it is unique. If S contains a greatest element that element is the supremum. If S contains a least element that element is the infimum; the infimum of a subset S of a ordered set P, assuming it exists, does not belong to S. If it does, it is a minimum or least element of S. Similarly, if the supremum of S belongs to S, it is a maximum or greatest element of S. For example, consider the set of negative real numbers; this set has no greatest element, since for every element of the set, there is another, element. For instance, for any negative real number x, there is another negative real number x 2, greater. On the other hand, every real number greater than or equal to zero is an upper bound on this set. Hence, 0 is the least upper bound of the negative reals, so the supremum is 0; this set has a supremum but no greatest element.

However, the definition of maximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique. Whereas maxima and minima must be members of the subset, under consideration, the infimum and supremum of a subset need not be members of that subset themselves. A ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no smaller element, an upper bound; this does not say that each minimal upper bound is smaller than all other upper bounds, it is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a total one. In a ordered set, like the real numbers, the concepts are the same; as an example, let S be the set of all finite subsets of natural numbers and consider the ordered set obtained by taking all sets from S together with the set of integers ℤ and the set of positive real numbers ℝ+, ordered by subset inclusion as above.

Both ℤ and ℝ+ are greater than all finite sets of natural numbers. Yet, neither is ℝ+ smaller than ℤ nor is the converse true: both sets are minimal upper bounds but none is a supremum; the least-upper-bound property is an example of the aforementioned completeness properties, typical for the set of real numbers. This property is sometimes called Dedekind completeness. If an ordered set S has the property that every nonempty subset of S having an upper bound has a least upper bound S is said to have the least-upper-bound property; as noted above, the set ℝ of all real numbers has the least-upper-bound property. The set ℤ of integers has the least-upper-bound property.