335 research outputs found
On affine scaling inexact dogleg methods for bound-constrained nonlinear systems
Within the framework of affine scaling trust-region methods for bound constrained problems, we discuss the use of a inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Focusing on bound-constrained systems of nonlinear equations, an inexact affine scaling method for large scale problems, employing the inexact dogleg procedure, is described. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given
Constrained dogleg methods for nonlinear systems with simple bounds
We focus on the numerical solution of medium scale bound-constrained systems of nonlinear equations. In this context, we consider an affine-scaling trust region approach that allows a great flexibility in choosing the scaling matrix used to handle the bounds. The method is based on a dogleg procedure tailored for constrained problems and so, it is named Constrained Dogleg method. It generates only strictly feasible iterates. Global and locally fast convergence is ensured under standard assumptions. The method has been implemented in the Matlab solver CoDoSol that supports several diagonal scalings in both spherical and elliptical trust region frameworks. We give a brief account of CoDoSol and report on the computational experience performed on a number of representative test problem
Implementing a smooth exact penalty function for equality-constrained nonlinear optimization
We develop a general equality-constrained nonlinear optimization algorithm
based on a smooth penalty function proposed by Fletcher (1970). Although it was
historically considered to be computationally prohibitive in practice, we
demonstrate that the computational kernels required are no more expensive than
other widely accepted methods for nonlinear optimization. The main kernel
required to evaluate the penalty function and its derivatives is solving a
structured linear system. We show how to solve this system efficiently by
storing a single factorization each iteration when the matrices are available
explicitly. We further show how to adapt the penalty function to the class of
factorization-free algorithms by solving the linear system iteratively. The
penalty function therefore has promise when the linear system can be solved
efficiently, e.g., for PDE-constrained optimization problems where efficient
preconditioners exist. We discuss extensions including handling simple
constraints explicitly, regularizing the penalty function, and inexact
evaluation of the penalty function and its gradients. We demonstrate the merits
of the approach and its various features on some nonlinear programs from a
standard test set, and some PDE-constrained optimization problems
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