Algorithms for the quasiconvex feasibility problem

We study the behavior of subgradient projections algorithms for the quasiconvex feasibility problem of finding a point x*∈Rnx*∈Rn that satisfies the inequalities f1(x*)⩽0,f2(x*)⩽0,…,fm(x*)⩽0f1(x*)⩽0,f2(x*)⩽0,…,fm(x*)⩽0, where all functions are continuous and quasiconvex. We consider the consistent case when the solution set is nonempty. Since the Fenchel–Moreau subdifferential might be empty we look at different notions of the subdifferential and determine their suitability for our problem. We also determine conditions on the functions, that are needed for convergence of our algorithms. The quasiconvex functions on the left-hand side of the inequalities need not be differentiable but have to satisfy a Lipschitz or a Hölder condition.