1,721,038 research outputs found
THE MASTER SPACE OF N=1 GAUGE THEORIES
No full textTalk given at the Fine Theoretical Physics Institute (FTPI) workshop, Continuous Advances in QCD (CAQCD-08), held May 15-18, 2008.Zaffaroni, Alberto. (2008). THE MASTER SPACE OF N=1 GAUGE THEORIES. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/42212
Codifferentiable mapings with applications to vector optimization.
No full textCodifferentiable mappings are defined as the ones which can be locally approximated by a particular type of difference convex mappings, adapting an analogous notion recently introduced for scalar functions. Some calculus rules are proved and some applications to vector optimization problems described by codifferentiable criteria and constraints are given
Superlinear separation for radiant and coradiant sets
No full textThe paper studies radiant and coradiant sets of some normed space X from the point of view of separation properties between a set A of X and a point x not included in A; indeed they show striking similarities with the ones holding for convex sets and can be obtained by simply changing halfspaces (level sets of linear continuous functions), with level sets of continuous superlinear functions. In a geometric perspective we can say that radiant sets are separated by means of convex coradiant sets and coradiant sets are separated by means of convex radiant sets. The identification between the geometric and the analytic approach passes through the well-known Minkoski gauge and the study of concave continuous gauges of convex coradiant sets. The results are then applied to the study of abstract convexity with respect to the family L of continuous superlinear functions, to the characterization of evenly coradiant convex sets and to the subdifferentiability of positively homogeneous functions
Convex coradiant sets with a continuous concave cogauge
No full textThe paper studies convex coradiant sets and their cogauges. While the concave gauge of a convex coradiant set is superlinear but discontinuous and its Minkowski cogauge is (possibly) continuous but is not concave, we are interested in those convex coradiant sets which admit a continuous concave cogauge. These sets are characterized in primal terms using their outer kernel and in dual terms using their reverse polar set. It is shown that a continuous concave cogauge, if it exists, is not unique; we prove that the class of continuous concave cogauges of some set C admits a greatest element and characterize its support set as the intersection of the reverse polar of C and the polar of its outer kernel
Superlinear separation and dual properties of radiant functions
No full textSuperlinear functionals are used to separate points from a radiant set according to both a strict and a weak version. Strict separation characterizes closed radiant sets; weak separation is used to define evenly radiant sets, which are characterized by means of a property of the tangent cone to the set at points of the boundary. The separation properties can be described via a polarity relation between a normed space X and the set L of continuous superlinear functionals defined on X. Radiant functions are the ones which are increasing along rays, i.e. the ones whose lower level sets are radiant and so they extend the class of quasiconvex functions with minimum at the origin. We study two particular subclasses: the one of l.s.c. radiant functions, whose lower level sets are closed and radiant and the one of evenly radiant functions, whose lower levels are evenly radiant. We introduce a conjugate function (defined on L), in two different versions, and prove the coincidence between a function and its second conjugate when the function belongs to one of the classes mentioned above. The conjugate function is then used to give global optimality conditions for problems described by radiant objective and constraints
IS EVERY RADIANT FUNCTION THE SUM OF QUASICONVEX FUNCTIONS?
No full textAn open question in the study of quasiconvex function is the characterization of the class of functions which are sum of quasiconvex functions. In this paper we restrict attention to quasiconvex radiant functions, i.e. those whose level sets are radiant as well as convex and deal with the claim that a function can be expressed as the sum of quasiconvex radiant functions if and only if it is radiant. Our study is carried out in the framework of Abstract Convex Analysis: the main tool is the description of a supremal generator of the set of radiant functions, i.e. a class of elementary functions whose sup-envelope gives radiant functions, and of the relation between the elementary generators of radiant functions and those of quasiconvex radiant functions. An important intermediate result is a nonlinear separation theorem in which a superlinear function is used to separate a point from a closed radiant set
Conically equivalent convex sets and applications
No full textGiven a normed space X and a cone K μ X, two closed, convex sets A and B in X¤ are said to be K-equivalent if the support functions of A and B coincide on K. We characterize the greatest set in an equivalence class, analyze the equivalence between two sets, find conditions for the existence and the uniqueness of a minimal set, extending previous results. We give some applications to the study of gauges of convex radiant sets and of cogauges of convex coradiant sets. Moreover we study the minimality of a second
order hypodifferential
Asymptotic conditions for weak and proper optimality in infinite dimensional convex vector optimization
No full textWe establish necessary and sufficient dual conditions for weak and proper minimality of infinite dimensional vector convex programming problems without any regularity conditions. The optimality conditions are given in asymptotic form using epigraphs of conjugate function and subdifferentials. It is shown how these asymptotic conditions yield standard Lagrangian conditions under appropriate regularity conditions.The main tool used to obtain these results is a new solvability result of Motzkin type for cone convex systems. We also provide local Lagrangian conditions for certain nonconvex problems using convex approximations
Conically equivalent convex sets and applications
No full textGiven a normed space X and a cone K in X, two closed, convex sets A and B in X* are said to be K-equivalent if the support functions of A and B coincide on K. We characterize the greatest set in an equivalence class, analyze the equivalence between two sets, find conditions for the existence and the uniqueness of a minimal set, extending previous results. We give some applications to the study of gauges of convex radiant sets and of cogauges of convex coradiant sets. Moreover we study the minimality of a second order hypodifferential
Convex Radiant Costarshaped Sets and the Least Sublinear Gauge
Full text linkThe paper studies convex radiant sets (i.e. containing the origin) of a linear normed space X and their representation by means of a gauge. By gauge of a convex radiant set C we mean a sublinear function p such that C=[p< 1]. We characterize the class of convex radiant sets which admit a gauge different from the Minkowski gauge in two different ways: they are contained in a translate of their recession cone or, equivalently, they are costarshaped, that is complement of a starshaped set. We prove that the family of all sublinear gauges of a convex radiant set admits a least element and characterize its support set in terms of polar sets. The key concept for this study is the outer kernel of C, that is the kernel (in the sense of Starshaped Analysis) of the complement of C. We also devote some attention to the relation between costarshaped and hyperbolic convex sets
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