Problems of the type: Minimize \[ I(x)=\ell(x(t_ 0),x(t_ 1))+\int^{t_ 1}_{t_ 0}L(t,x(t),\dot x(t))dt \] over a space of arcs \(x:[t_ 0,t_ 1]\to R^ n\) subject to a system of equations and inequality constraints on the endpoint pair \((x(t_ 0),x(t_ 1))\) and the triple \((t,x(t),\dot x(t))\) are known as the problems of Bolza type. In the paper, the authors deal with discrete-time analogues of these optimization problems considering first the deterministic case and further the stochastic one. They assume the functional to be convex but differentiability assumptions are omitted.
Necessary and sufficient conditions for the optimal solutions are given in the deterministic part. These results are used for the stochastic version.
In the stochastic part, it is assumed that decisions made at any time can only depend on the information collected about past random events, the future being known only in a probabilistic sense. Further, the necessary and sufficient conditions of optimal solutions are given, too. At the end the dual stochastic problem is discussed.
All assertions presented in the paper are very carefully proved using results of convex analysis.