Hadamard directional differentiability of the optimal value of a linear second-order conic programming problem
Qingsong Duan, Mengwei Xu, Liwei Zhang, Sainan Zhang · Journal of Industrial and Management Optimization · 2020
In this paper, we consider perturbation properties of a linear second-order conic optimization problem and its Lagrange dual in which all parameters in the problem are perturbed. We prove the upper semi-continuity of solution mappings for the pertured problem and its Lagrange dual problem. We demonstrate that the optimal value function can be expressed as a min-max optimization problem over two compact convex sets, and it is proven as a Lipschitz continuous function and Hadamard directionally differentiable.