An interior-point method for solving box-constrained underdetermined nonlinear systems

Numerical solution of KKT systems in PDE-constrained optimization problems via the affine scaling trust-region approach†

“…In particular, the paper [6] is devoted to the proposal of a subspace trust-region method tailored for large-scale problems. The affine scaling trust-region approach has also been extended to non-smooth constrained nonlinear systems by Kanzow and Klug [16] and underdetermined nonlinear systems by Francisco et al [13]. These methods show good theoretical and numerical properties, as it was enlighted in the above-mentioned papers.…”

“…Compute the matrix D k by Equation (17) (12), (13) and (16). else -solve the linear system (14) by a Sparse Direct Method for augmented system and compute (p v , p μ ) -compute p s , p z by (12), (13 (20) satisfies Equations (23) and (24). Form p( k ) by (20).…”

Trust-region quadratic methods for nonlinear systems of mixed equalities and inequalities

“…The implementation with μ = 0 is based on the Gauss-Newton model and will be denoted as TREI-GN. The core of this method is similar to the Gauss-Newton methods given in [10,14,31]. The other implementation is obtained setting μ > 0 and gives rise to a Levenberg-Marquardt method; therefore it will be denoted TREI-LM method.…”

“…Recently, the solution of this problem has been the subject of research. Francisco et al [14] designed a trust-region interior point method for underdetermined nonlinear systems which generalizes the method [1] for square systems while Kanzow et al [24,25] proposed global projected Levenberg-Marquardt methods. In particular, the numerical solution of general nonsquare systems of bound-constrained nonlinear equations was addressed in [25] while the paper [24] is focused on overdetermined nonsmooth systems of equations.…”

A new trust-region method for solving systems of equalities and inequalities

“…Systems of the form (1) has many applications in the real world problems. It is appeared in the formulation of the problems such as model formulation design, detection of feasibility of an optimization problem and in the restoration phase of filter methods; see, e.g., Fletcher and Leyffer (2003), Francisco et al (2005) and Gould and Toint (2004). To cover the variety of applications, throughout the paper, we allow any relation between the parameters n and m.…”