Convergence analysis of linearized ${\mathrm{\ell }}_q$ penalty methods for nonconvex optimization with nonlinear equality constraints

In this paper, we consider nonconvex optimization problems with nonlinear equality constraints. We assume that the objective function and the functional constraints are locally smooth. To solve this problem, we introduce a linearized ${\mathrm{\ell }}_q$ penalty based method, where $q\in (1,2]$ is the parameter defining the norm used in the construction of the penalty function. Our method involves linearizing the objective function and functional constraints in a Gauss-Newton fashion at the current iteration in the penalty formulation and introduces a quadratic regularization. This approach yields an easily solvable subproblem, whose solution becomes the next iterate. By using a novel dynamic rule for the choice of the regularization parameter, we establish that the iterates of our method converge to an $\epsilon$-first-order solution in $\mathcal{O}(1\mathrm{/}{\epsilon }^{2+(q-1)\mathrm{/}q})$ outer iterations. Finally, we put theory into practice and evaluate the performance of the proposed algorithm by making numerical comparisons with existing methods from literature.