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 p th order Taylor model and show that the algorithm needs at most $$O(\epsilon ^{-(p+1)/(p-q+1)})$$ O ( ϵ - ( p + 1 ) / ( p - q + 1 ) ) evaluations of the objective function and its first p derivatives (whenever they exist) to produce an $$(\epsilon ,\delta )$$ ( ϵ , δ ) -approximate q th-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 $$\ell _2$$ ℓ 2 – $$\ell _a$$ ℓ a norm with $$a\in (0,1)$$ 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.