Trust-region methods for rectangular systems of nonlinear equations

Here Θ : X → IR is a continuously differentiable mapping, X ⊆ IR is an open set containing the feasible region Ω and Ω is an n-dimensional box, Ω = {x ∈ IR : l ≤ x ≤ u}. These inequalities are meant component-wise and l ∈ (IR ∪−∞), u ∈ (IR ∪∞). Taking into account the variety of applications yielding the problem (1), we allow any relationship between m and n. The relevance of this problem is well known. It arises in equality-constrained optimization, restoration feasibility for nonlinear programming, parameter identification problems, see [1, 2, 3]. Moreover, the general class of problems given by nonlinear system of equalities and inequalities can be cast as in (1). We propose two trust-region methods for bound-constrained nonlinear systems. The first method is based on a Gauss-Newton model, the second one is based on a regularized Gauss-Newton model and results to be a LevenbergMarquardt method. The globalization strategy uses affine scaling matrices arising in bound-constrained optimization. Under reasonable assumptions, these methods globally converge to a solution of (1) or to a first-order stationary point for the bound-constrained least-squares problem