scholarly journals KKT conditions for weak ⁎ compact convex sets, theorems of the alternative, and optimality conditions

2014 ◽  
Vol 266 (2) ◽  
pp. 693-712
Author(s):  
Zili Wu
Keyword(s):  
Optimality Conditions ◽  
Convex Sets ◽  
Compact Convex ◽  
Kkt Conditions ◽  
Theorems Of The Alternative

scholarly journals Sufficient optimality conditions and duality for nonsmooth multiobjective optimization problems via higher order strong convexity

2017 ◽  
Vol 27 (2) ◽  
pp. 227-242
Author(s):  
Balendu Upadhyay ◽  
Ningthoujam Priyobarta ◽  
Yumnam Rohen
Keyword(s):  
Multiobjective Optimization ◽  
Optimality Conditions ◽  
Convex Sets ◽  
Optimization Problems ◽  
Compact Convex ◽  
Strong Duality ◽  
Sufficient Optimality Conditions ◽  
Dual Model ◽  
Support Functions ◽  
Nonsmooth Multiobjective Optimization

In this paper, we define some new generalizations of strongly convex functions of order m for locally Lipschitz functions using Clarke subdifferential. Suitable examples illustrating the non emptiness of the newly defined classes of functions and their relationships with classical notions of pseudoconvexity and quasiconvexity are provided. These generalizations are then employed to establish sufficient optimality conditions for a nonsmooth multiobjective optimization problem involving support functions of compact convex sets. Furthermore, we formulate a mixed type dual model for the primal problem and establish weak and strong duality theorems using the notion of strict efficiency of order m. The results presented in this paper extend and unify several known results from the literature to a more general class of functions as well as optimization problems.

scholarly journals An elementary renewal theorem for random compact convex sets

1995 ◽  
Vol 27 (4) ◽  
pp. 931-942 ◽  
Author(s):  
Ilya S. Molchanov ◽  
Edward Omey ◽  
Eugene Kozarovitzky
Keyword(s):  
Support Function ◽  
Convex Sets ◽  
Compact Convex ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Minkowski Sums ◽  
Image Position ◽  
Unit Vectors

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK(u) is the support function of K and is the set of all unit vectors u with EhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.

SYMMETRIES OF COMPACT CONVEX SETS

1974 ◽  
Vol 25 (1) ◽  
pp. 323-328 ◽  
Author(s):  
E. B. DAVIES
Keyword(s):  
Convex Sets ◽  
Compact Convex

PROJECTIVE COMPACT CONVEX SETS

1973 ◽  
Vol 24 (1) ◽  
pp. 301-306 ◽  
Author(s):  
A. W. WICKSTEAD
Keyword(s):  
Convex Sets ◽  
Compact Convex
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