Convergence analysis of linearized $\ell_q$ penalty methods for nonconvex optimization with nonlinear equality constraints

Abstract: 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 $\ell_q$ 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 $\mathcal{O}(1/{\epsilon^{2+ (q-1)/q}})$ 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.