Nonlinear error bounds for quasiconvex inequality systems

The error bound is an inequality that restricts the distance from a vector to a given set by a residual function. The error bound has so many useful applications, for example in variational analysis, in convergence analysis of algorithms, in sensitivity analysis, and so on. For convex inequality systems, Lipschitzian error bounds are studied mainly. If an inequality system is not convex, it is difficult to show the existence of a Lipschitzian global error bound in general. Hence for nonconvex inequality systems, Holderian error bounds and nonlinear error bounds have been investigated. For quasiconvex inequality systems, there are so many examples such that systems do not have Lipschitzian and Holderian error bounds. However, the research of nonlinear error bounds for quasiconvex inequality systems have not been investigated yet as far as we know. In this paper, we study nonlinear error bounds for quasiconvex inequality systems. We show the existence of a global nonlinear error bound by a generator of a quasiconvex function and a constraint qualification. We show well-posedness of a quasiconvex function by the error bound.