DOI: 10.68381/jca11025 ISSN: 0944-6532
A Necessary and Sufficient Optimality Condition for a Class of Nonconvex Scalar Variational Problems
Guillaume Carlier
This article studies the minimization of the functional
u\mapsto\int_{0}^{1}f(\dot{u})
u
↦
∫
0
1
f
(
u
˙
)
among all convex functions
u
u
that satisfy the additional obstacle constraint
u\geq {\underline{u}}
u
≥
u
‾
,
u(0)=\underline{u}(0)
u
(
0
)
=
u
‾
(
0
)
,
u(1)=\underline{u}(1)
u
(
1
)
=
u
‾
(
1
)
where
\underline{u}
u
‾
is a given convex function. We first show that this nonconvex problem is in fact equivalent to a linear programming problem. This enables us to establish a necessary and sufficient optimality condition.