On the Slater condition for the SDP relaxations of nonconvex sets

2001 ◽  
Vol 29 (4) ◽  
pp. 181-186 ◽  
Author(s):  
Levent Tunçel
Keyword(s):  
Slater Condition ◽  
Nonconvex Sets

scholarly journals Cutting-Planes for Optimization of Convex Functions over Nonconvex Sets

2014 ◽  
Vol 24 (2) ◽  
pp. 643-677 ◽  
Author(s):  
Daniel Bienstock ◽  
Alexander Michalka
Keyword(s):  
Convex Functions ◽  
Cutting Planes ◽  
Nonconvex Sets

scholarly journals Generalized Projections on Closed Nonconvex Sets in Uniformly Convex and Uniformly Smooth Banach Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Messaoud Bounkhel
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Set ◽  
Variational Problems ◽  
Generalized Projection ◽  
Uniformly Convex ◽  
Uniformly Smooth ◽  
Nonconvex Sets ◽  
Generalized Projections ◽  
Uniformly Smooth Banach Spaces

The present paper is devoted to the study of the generalized projectionπK:X∗→K, whereXis a uniformly convex and uniformly smooth Banach space andKis a nonempty closed (not necessarily convex) set inX. Our main result is the density of the pointsx∗∈X∗having unique generalized projection over nonempty close sets inX. Some minimisation principles are also established. An application to variational problems with nonconvex sets is presented.

scholarly journals Fixed point theorems for nonexpansive mappings on nonconvex sets in UCED Banach spaces

2002 ◽  
Vol 31 (4) ◽  
pp. 251-257 ◽  
Author(s):  
Wei-Shih Du ◽  
Young-Ye Huang ◽  
Chi-Lin Yen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Finite Union ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Nonconvex Sets ◽  
Firmly Nonexpansive ◽  
Asymptotically Regular

It is shown that every asymptotically regular orλ-firmly nonexpansive mappingT:C→Chas a fixed point wheneverCis a finite union of nonempty weakly compact convex subsets of a Banach spaceXwhich is uniformly convex in every direction. Furthermore, if{T i}i∈Iis any compatible family of strongly nonexpansive self-mappings on such aCand the graphs ofT i,i∈I, have a nonempty intersection, thenT i,i∈I, have a common fixed point inC.

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