Interior Quasi-Subgradient Method with Non-Euclidean Distances for Constrained Quasi-Convex Optimization Problems in Hilbert Spaces

An interior quasi-subgradient method is proposed based on the proximal distance to solve constrained nondifferentiable quasi-convex optimization problems in Hilbert spaces. It is shown that a newly introduced generalized Gâteaux subdifferential is a subset of a quasi-subdifferential. The convergence properties, including the global convergence and iteration complexity, are investigated under the assumption of the Hölder condition of order p, when using the constant/diminishing/dynamic stepsize rules. Convergence rate results are obtained by assuming a Hölder-type weak sharp minimum condition relative to an induced proximal distance.