Differential Stability Properties of Convex Optimization and Optimal Control Problems

This paper studies the solution stability of convex optimization and discrete convex optimal control problems in Banach spaces, where the solution set may be empty. For both the optimization problem and the optimal control problem, formulas for the \(\varepsilon \) $\epsilon$ -subdifferential of the optimal value function are derived without qualification conditions. We first calculate the \(\varepsilon \) $\epsilon$ -subdifferential of the optimal value function to a parametric optimization problem with geometrical and functional constraints. We then use the obtained results to derive a formula for computing the \(\varepsilon \) $\epsilon$ -subdifferential of the optimal value function to a discrete convex optimal control problem with linear state equations, control constraints and initial, terminal conditions.