Complexity of linearized quadratic penalty for optimization with nonlinear equality constraints

Abstract In this paper we consider a nonconvex optimization problem with nonlinear equality constraints. We assume that both, the objective function and the functional constraints, are locally smooth. For solving this problem, we propose a linearized quadratic penalty method, i.e., we linearize the objective function and the functional constraints in the penalty formulation at the current iterate and add a quadratic regularization, thus yielding a subproblem that is easy to solve, and whose solution is the next iterate. Under a new adaptive regularization parameter choice, we provide convergence guarantees for the iterates of this method to an $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> first-order optimal solution in $${\mathcal {O}}({\epsilon ^{-2.5}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2.5</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> iterations. Finally, we show that when the problem data satisfy Kurdyka–Lojasiewicz property, e.g., are semialgebraic, the whole sequence generated by the proposed algorithm converges and we derive improved local convergence rates depending on the KL parameter. We validate the theory and the performance of the proposed algorithm by numerically comparing it with some existing methods from the literature.