Inexact methods for constrained optimization problems and for constrained monotone nonlinear equations - Menezes, Tiago da Costa (2021)
In this work, we propose and analyze some methods to solve constrained optimization problems and constrained monotone nonlinear systems of equations. Our first algorithm is an inexact variable metric method for solving convex-constrained optimization problems. At each iteration of the method, the search direction is obtained by inexactly minimizing a strictly convex quadratic function over the closed convex feasible set. Here, we propose a new inexactness criterion for the search direction subproblems. Under mild assumptions, we prove that any accumulation point of the sequence generated by the method is a stationary point of the problem under consideration. Our second method consists of a Gauss-Newton algorithm with approximate projections for solving constrained nonlinear least squares problems. The local convergence of the method including results on its rate is discussed by using a general majorant condition. By combining the latter method and a nonmonotone line search strategy, we also propose a global version of this algorithm and analyze its convergence results. Our third approach corresponds to a framework, which is obtained by combining a safeguard strategy on the search directions with a notion of approximate projections, to solve constrained monotone nonlinear systems of equations. The global convergence of our framework is obtained under appropriate assumptions and some examples of methods which fall into this framework are presented. Numerical experiments illustrating the practical behaviors of the methods are reported and comparisons with existing algorithms are also presented.