On Some Generalized Polyhedral Convex Constructions

ABSTRACT Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal cone, conjugate function, subdifferential are studied thoroughly in this paper. Among other things, we show how a generalized polyhedral convex set can be characterized through the finiteness of the number of its faces. In addition, it is proved that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function. The obtained results can be applied to scalar optimization problems described by generalized polyhedral convex sets and generalized polyhedral convex functions.

[1]  Xi Yin Zheng.  Pareto Solutions of Polyhedral-valued Vector Optimization Problems in Banach Spaces , 2009 .

[2]  H. Weyl.  Elementare Theorie der konvexen Polyeder , 1934 .

[3]  R. Durier,et al.  Polyhedral Norms in an Infinite Dimensional Space , 1993 .

[4]  R. Rockafellar.  Convex Analysis: (pms-28) , 1970 .

[5]  Branko Grünbaum.  Diameters of Polytopes , 2003 .

[6]  Helmut Gfrerer,et al.  JOHANNES KEPLER UNIVERSITY LINZ Institute of Computational Mathematics On Directional Metric Subregularity and Second-Order Optimality Conditions for a Class of Nonsmooth Mathematical Programs , 2012 .

[7]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.

[8]  C. Tammer,et al.  Theory of Vector Optimization , 2003 .

[9]  E. Anderson.  Linear Programming In Infinite Dimensional Spaces , 1970 .

[10]  Nguyen Ngoc Luan.  Efficient solutions in generalized linear vector optimization , 2017 .

[11]  N. M. Nam,et al.  Generalized Differentiation and Characterizations for Differentiability of Infimal Convolutions , 2014, 1404.0787.

[12]  A. W. Tucker,et al.  1. THE ELEMENTARY THEORY OF CONVEX POLYHEDRA , 1951 .

[13]  A. Banerjee.  Convex Analysis and Optimization , 2006 .

[14]  Infinite-Dimensional Polyhedrality , 2004, Canadian Journal of Mathematics.

[15]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[16]  Xiaoqi Yang,et al.  The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces , 2008 .

[17]  Boris S. Mordukhovich,et al.  Lipschitzian stability of parametric variational inequalities over generalized polyhedra in Banach s , 2011 .

[18]  D. Blackwell,et al.  5. Admissible Points of Convex Sets , 1953 .

[19]  C. B. Luis.  Une caractérisation complète des minima locaux en programmation quadratique , 1980 .

[20]  Nguyen Dong Yen,et al.  A representation of generalized convex polyhedra and applications , 2019, Optimization.

[21]  H. Brezis.  Functional Analysis, Sobolev Spaces and Partial Differential Equations , 2010 .

[22]  Klaudia Beich,et al.  Theory Of Vector Optimization , 2016 .

[23]  S. Semmes.  Topological Vector Spaces , 2003 .

[24]  M. Planitz,et al.  Analysis, algebra, and computers in mathematical research , 1995 .

[25]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[26]  Wen Song,et al.  Linearly perturbed generalized polyhedral normal cone mappings and applications , 2016 .

[27]  H. Minkowski,et al.  Geometrie der Zahlen , 1896 .

[28]  Benjamin Pfaff,et al.  Perturbation Analysis Of Optimization Problems , 2016 .

[29]  Thomas Strömberg.  The operation of infimal convolution , 1996 .

[31]  A. Ioffe,et al.  Theory of extremal problems , 1979 .

[32]  Helmut Gfrerer,et al.  On Metric Pseudo-(sub)Regularity of Multifunctions and Optimality Conditions for Degenerated Mathematical Programs , 2014 .

[33]  W. Fenchel.  Convex cones, sets, and functions , 1953 .

[34]  Zdzisław Denkowski,et al.  Set-Valued Analysis , 2021 .

[35]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[36]  V. Klee.  Some characterizations of convex polyhedra , 1959 .

[37]  Nguyen Dong Yen,et al.  Quadratic Programming and Affine Variational Inequalities: A Qualitative Study , 2005 .

[38]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.