Complexity of partially separable convexly constrained optimization with non-Lipschitzian singularities

An adaptive regularization algorithm using high-order models is proposed for solving partially separable convexly constrained nonlinear optimization problems whose objective function contains non-Lipschitzian 'q-norm regularization terms for q ϵ (0; 1). It is shown that the algorithm using a pth-order Taylor model for p odd needs in general at most O(ϵ ^-(p+1)=p) evaluations of the objective function and its derivatives (at points where they are defined) to produce an ϵ-approximate first-order critical point. This result is obtained either with Taylor models, at the price of requiring the feasible set to be kernel centered" (which includes bound constraints and many other cases of interest), or with non-Lipschitz models, at the price of passing the difficulty to the computation of the step. Since this complexity bound is identical in order to that already known for purely Lipschitzian minimization subject to convex constraints [C. Cartis, N. I. M. Gould, and Ph. L. Toint, IMA J. Numer. Anal., 32 (2012), pp. 1662-1695], the new result shows that introducing non-Lipschitzian singularities in the objective function may not affect the worst-case evaluation complexity order. The result also shows that using the problem's partially separable structure (if present) does not affect the complexity order either. A final (worse) complexity bound is derived for the case where Taylor models are used with a general convex feasible set.