High-Order Evaluation Complexity for Convexly-Constrained Optimization with Non-Lipschitzian Group Sparsity Terms

This paper studies high-order evaluation complexity for partially separable convexly-constrained optimization involving non-Lipschitzian group sparsity terms in a nonconvex objective function. We propose a partially separable adaptive regularization algorithm using a pth order Taylor model and show that the algorithm needs at most O(ϵ^{- ( p + 1 ) / ( p - q + 1 )}) evaluations of the objective function and its first p derivatives (whenever they exist) to produce an (ϵ, δ) -approximate qth-order stationary point. Our algorithm uses the underlying rotational symmetry of the Euclidean norm function to build a Lipschitzian approximation for the non-Lipschitzian group sparsity terms, which are defined by the group ℓ₂–ℓ_a norm with a∈ (0 , 1). The new result shows that the partially-separable structure and non-Lipschitzian group sparsity terms in the objective function do not affect the worst-case evaluation complexity order.