Positivity 6: 261–274, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
Subdifferentiability and the Duality Gap
N. E. GRETSKY
Department of Mathematics, University of California, Riverside, CA, U.S.A. (E-mail:
Department of Economics, University of California, Los Angeles, CA, U.S.A.
(E-mail: firstname.lastname@example.org email@example.com)
(Received 16 February 2001; accepted 1 June 2001)
Abstract. We point out a connection between sensitivity analysis and the fundamental theorem of
linear programming by characterizing when a linear programming problem has no duality gap. The
main result is that the value function is subdifferentiable at the primal constraint if and only if there
exists an optimal dual solution and there is no duality gap. To illustrate the subtlety of the condition,
we extend Kretschmer’s gap example to construct (as the value function of a linear programming
problem) a convex function which is subdifferentiable at a point but is not continuous there. We also
apply the theorem to the continuum version of the assignment model.
AMS Classiﬁcation: 90C48, 46N10
Key words: duality gap, value function, subdifferentiability, assignment model
The purpose of this note is to point out a connection between sensitivity analysis
and the fundamental theorem of linear programming. The subject has received
considerable attention and the connection we ﬁnd is remarkably simple. In fact,
our observation in the context of convex programming follows as an application
of conjugate duality [11, Theorem 16]. Nevertheless, it is useful to give a separate
proof since the conclusion is more readily established and its import for linear
programming is more clearly seen.
The main result (Theorem 1) is that in a linear programming problem there
exists an optimal dual solution and there is no duality gap if and only if the value
function is subdifferentiable at the primal constraint.
The result is useful because the value function is convex, so there are simple
sufﬁcient conditions that it be subdifferentiable at a given (constraint) point. In
particular, if the value function of the linear programming problem is a proper con-
vex function, then the value function is lower semicontinuous and standard results
from convex analysis guarantee that the value function is locally bounded, locally
Lipschitz, and subdifferentiable at every interior point of its domain. When ap-
plied to inﬁnite-dimensional linear programming, Theorem 1 contains the Dufﬁn–
Karlovitz no-gap theorem for constraint spaces whose positive cone has interior
, and the Charnes–Cooper–Kortanek no-gap theorem for semi-inﬁnite programs