A Necessary and Sufficient Optimality Condition for a Class of Nonconvex Scalar Variational Problems

This article studies the minimization of the functional $$u\mapsto\int_{0}^{1}f(\dot{u})$$ among all convex functions $u$ that satisfy the additional obstacle constraint $u\geq \ovu$, $u(0)=\ovu(0)$, $u(1)=\ovu(1)$ where $\ovu$ 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.