We present a new method for the large-scale trust-region subproblem. The is matrix-free in sense that only matrix-vector products are required. recast subproblem as parameterized eigenvalue problem and compute an optimal value parameter. then find solution of from eigenvectors associated with two smallest eigenvalues corresponding to algorithm uses different interpolating scheme than existing methods introduces unified iteration naturally includes so-called hard case. show well defined...

A MATLAB 6.0 implementation of the LSTRS method is presented. was described in Rojas et al. [2000]. designed for large-scale quadratic problems with one norm constraint. The based on a reformulation trust-region subproblem as parameterized eigenvalue problem, and consists an iterative procedure that finds optimal value parameter. adjustment parameter requires solution problem at each step. relies matrix-vector products only has low fixed storage requirements, features make it suitable...

The evaluation complexity of general nonlinear, possibly nonconvex, constrained optimization is analyzed. It shown that, under suitable smoothness conditions, an $\epsilon$-approximate first-order critical point the problem can be computed in order $O(\epsilon^{1-2(p+1)/p})$ evaluations problem's functions and their first $p$ derivatives. This achieved by using a two-phase algorithm inspired Cartis, Gould, Toint [SIAM J. Optim., 21 (2011), pp. 1721--1739; SIAM 23 (2013), 1553--1574]. also...

In this paper we discuss a specialization of the augmented Lagrangian-type algorithm Conn, Gould, and Toint to solution strictly convex quadratic programming problems with simple bounds equality constraints. The new feature presented is adaptive precision control auxiliary in inner loop basic which yields rate convergence that does not have any term accounts for inexact problems. Moreover, boundedness penalty parameter achieved used. Numerical experiments illustrate efficiency encourage its usage.

Minimization of a differentiable function subject to box constraints is proposed as strategy solve the generalized nonlinear complementarity problem (GNCP) defined on polyhedral cone. It not necessary calculate projections that complicate and sometimes even disable implementation algorithms for solving these kinds problems. Theoretical results relate stationary points minimized solutions GNCP are presented. Perturbations also considered, obtained related resolution GNCPs with very general...

A nonmonotone strategy for solving nonlinear systems of equations is introduced. The idea consists combining efficient local methods with an algorithm that reduces monotonically the squared norm system in a proper way. used are Newton's method and two quasi-Newton algorithms. Global iterations based on recently introduced box-constrained minimization Numerical experiments presented

This work takes advantage of the spectral projected gradient direction within inexact restoration framework to address nonlinear optimization problems with nonconvex constraints. The proposed strategy includes a convenient handling constraints, together nonmonotonic features speed up convergence. numerical performance is assessed by experiments hard-spheres problems, pointing out that provides an adequate environment for extension method general nonlinearly constrained optimization.

In this paper, we present a brief survey of methods for solving nonlinear least-squares problems. We pay specific attention to that take into account the special structure Most discussed belong quasi-Newton family (i.e. structured (SQN)). Our comprises some traditional and modern developed At end, suggest few topics further research.

We introduce a sequential optimality condition for locally Lipschitz constrained nonsmooth optimization, verifiable just using derivative information, and which holds even in the absence of any constraint qualification. present practical algorithm that generates iterates either fulfilling new necessary or converging to stationary points infeasibility measure. A main feature devised is allow stronger control over than usually obtained by exact penalty strategies, ensuring theoretical...