Slater's condition
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In mathematics, Slater's condition (or Slater
condition) is a sufficient condition for strong
duality to hold for a convex optimization problem. This is a specific example of a constraint
qualification. In particular, if Slater's condition holds for the primal problem,
then the duality gap is 0, and if the dual value is finite then it is attained.[1]
[edit]Mathematics
Given the problem
with 
convex (and
therefore a convex optimization problem). Then strong duality holds if there exists an 
(where
relint is the relative interior and 
)
such that
-
and -
[2]
If the first 
constraints, 
are linear
functions, then strong duality holds if there exists an such
that
-
-
and -
[2]
[edit]Generalized
Inequalities
Given the problem
where 
is
convex and 
is 
-convex
for each 
.
Then Slater's condition says that if there exists an such
that
-
and -
then strong duality holds.[2]
[edit]References
- ^ Borwein,
Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 978-0-387-29570-1. - ^ a b c Boyd,
Stephen; Vandenberghe, Lieven (2004) (pdf). Convex Optimization.
Cambridge University Press. ISBN 978-0-521-83378-3.
Retrieved October 3, 2011.