Duality and Stability in Convex Optimization (Extended Results for the Saddle Case)

Due to the first chapter we realize that convex programs of the type (5.4) with random parameters (η,ξ) in the objective and in the right-hand side play an essential role in stochastic two-stage programs. We are aware of the fact that we have to integrate (at least approximately) the optimal value function (recourse function) of (5.4), given — for a fixed first-stage decision x — only implicitly by the infimum of (5.4). Although the optimal value function changes with x, we take into account that it remains a saddle function. This fact motivates to derive suitable approximates for the optimal value function, that are computable and allow conclusions about the goodness of the first-stage decision x. Such approximates will be derived in chapter III. At this point it is only of importance to stress that the existence of these approximates essentially depend on the existence of subgradients of the optimal value function at distinguished points. Hence, at first, we are confronted with investigating under which assumptions subgradients for the optimal value function at a certain distinguished point, say (η0,ξ0),exist. In other words we have to ensure dual solvability of the convex optimization problem — of the type (5.4) — with respect to (η,ξ) at (η0,ξ0).