Approximate norm descent methods for constrained nonlinear systems

We address the solution of convex-constrained nonlinear systems of equations where the Jacobian matrix is unavailable or its computation/storage is burdensome. In order to efficiently solve such problems, we propose a new class of algorithms which are “derivative-free” both in the computation of the search direction and in the selection of the steplength. Search directions … Read more

On the convergence of an inexact Gauss-Newton trust-region method for nonlinear least-squares problems with simple bounds

We introduce an inexact Gauss-Newton trust-region method for solving bound-constrained nonlinear least-squares problems where, at each iteration, a trust-region subproblem is approximately solved by the Conjugate Gradient method. Provided a suitable control on the accuracy to which we attempt to solve the subproblems, we prove that the method has global and asymptotic fast convergence properties. … Read more

On the convergence of trust region algorithms for unconstrained minimization without derivatives

We consider iterative trust region algorithms for the unconstrained minimization of an objective function F(x) of n variables, when F is differentiable but no derivatives are available, and when each model of F is a linear or quadratic polynomial. The models interpolate F at n+1 points, which defines them uniquely when they are linear polynomials. … Read more

A Retrospective Trust-Region Method for Unconstrained Optimization

We introduce a new trust-region method for unconstrained optimization where the radius update is computed using the model information at the current iterate rather than at the preceding one. The update is then performed according to how well the current model retrospectively predicts the value of the objective function at last iterate. Global convergence to … Read more

A recursive trust-region method in infinity norm for bound-constrained nonlinear optimization

A recursive trust-region method is introduced for the solution of bound-constrained nonlinear nonconvex optimization problems for which a hierarchy of descriptions exists. Typical cases are infinite-dimensional problems for which the levels of the hierarchy correspond to discretization levels, from coarse to fine. The new method uses the infinity norm to define the shape of the … Read more

Recursive Trust-Region Methods for Multilevel Nonlinear Optimization (Part I): Global Convergence and Complexity

A class of trust-region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations. The algorithms in this class make use of the discretization level as a mean of speeding up the computation of the step. This use is recursive, leading to … Read more