Sensitivity of Under-Determined Linear System

This paper considers the sensitivity of the optimization problem minf(x) with the linear constraint Ax = b, where f is a general differentiable function quantifying sparsity, Ax=b is the under-determined linear system of equations, A ∈ℝn×p. Given small noises to A and b, we are able to show the difference between the perturbed solution and optimal solution. The new perturbation bound reveals the factors that affect the sensitivity of the optimal solution of the linear system. Different objective functions f’s lead to distinct perturbation bounds, whose magnitudes determine the robustness of the optimization problem. Our results shed a fresh insight in understanding the robustness of under-determined linear system.