New quasi-Newton method for solving systems of nonlinear equations

In this report, we propose the new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but it is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR decompositions of nonsymmetric approximations of the Jacobian matrix, so it requires $O(n^2)$ … Read more

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

An Augmented Lagrangian based Algorithm for Distributed Non-Convex Optimization

This paper is about distributed derivative-based algorithms for solving optimization problems with a separable (potentially nonconvex) objective function and coupled affine constraints. A parallelizable method is proposed that combines ideas from the fields of sequential quadratic programming and augmented Lagrangian algorithms. The method negotiates shared dual variables that may be interpreted as prices, a concept … Read more

An adaptive cubic regularisation algorithm for nonconvex optimization with convex constraints and its function-evaluation complexity

The adaptive cubic overestimation algorithm described in Cartis, Gould and Toint (2007) is adapted to the problem of minimizing a nonlinear, possibly nonconvex, smooth objective function over a convex domain. Convergence to first-order critical points is shown under standard assumptions, but without any Lipschitz continuity requirement on the objective’s Hessian. A worst-case complexity analysis in … Read more

Nonlinear programming without a penalty function or a filter

A new method is introduced for solving equality constrained nonlinear optimization problems. This method does not use a penalty function, nor a barrier or a filter, and yet can be proved to be globally convergent to first-order stationary points. It uses different trust-regions to cope with the nonlinearities of the objective function and the constraints, … Read more