Sensitivity analysis in convex optimization through the circatangent derivative (Q2346913)

The authors study the problem of maximizing of a vector function \(f\) under a general inequality constraint \(g(x) \leq b\) in which the partial orders of the outcome space \(Y\) for \(f\) and the value space for \(g\) are defined by closed cones with nonempty interiors and \(b\) is a parameter from an open set. They define an optimal solution \(x_0\) of the problem to be a feasible solution satisfying \(T\circ f(x_0)\leq T\circ f(x)\) for every feasible solution \(x\), where \(T\) is a linear, continuous, positive operator from \(Y\) to another Banach space equipped with a partial order in which every nonempty bounded from below set has an infimum. The main result of this work is concerned with the Clarke derivative of the optimal value mapping under some differentiability assumptions of the Lagrange multiplier mapping. A formula to compute the Clarke derivative of the optimal value mapping is also given.