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An adaptive regularization algorithm using high-order models is proposed forpartially separable convexly constrained nonlinear optimization problems whoseobjective function contains non-Lipschitzian $\ell_q$-norm regularization termsfor $q\in (0,1)$. It is shown that the algorithm using an $p$-th order Taylormodel for $p$ odd needs in general at most $O(\epsilon^{-(p+1)/p})$ evaluationsof the objective function and its derivatives (at points where they aredefined) to produce an $\epsilon$-approximate first-order critical point. Thisresult is obtained either with Taylor models at the price of requiring thefeasible set to be 'kernel-centered' (which includes bound constraints and manyother cases of interest), or for non-Lipschitz models, at the price of passingthe difficulty to the computation of the step. Since this complexity bound isidentical in order to that already known for purely Lipschitzian minimizationsubject to convex constraints [CartGoulToin2016], the new result shows thatintroducing non-Lipschitzian singularities in the objective function may notaffect the worst-case evaluation complexity order. The result also shows thatusing the problem's partially separable structure (if present) does not affectcomplexity order either. A final (worse) complexity bound is derived for thecase where Taylor models are used with a general convex feasible set.

Partially separable convexly-constrained optimization with non-Lipschitzian singularities and its complexity