A Truncated Newton Algorithm for Large Scale Box Constrained Optimization

Abstract

A new method for the solution of minimization problems with simple bounds is presented. Global convergence of a general scheme requiring the approximate solution of a single linear system at each iteration is proved and a superlinear convergence rate is established without requiring the strict complementarity assumption. The algorithm proposed is based on a simple, smooth unconstrained reformulation of the bound constrained problem and may produce a sequence of points that are not feasible. Numerical results are reported.

On Affine-Scaling Interior-Point Newton Methods for Nonlinear Minimization with Bound Constraints

“…The optimization problem (P) has attracted quite a few researchers during the last 15 years, and a number of different methods for its solution may be found in [4,5,9,10,15,16,20,23,24,25,31]. The approach we follow in this work is typically called the affine-scaling interior-point Newton method.…”

An accurate active set newton algorithm for large scale bound constrained optimization

“…In recent years, a number of algorithms have been proposed to add and drop several constraints in an iteration (see [7], [10], [12], [13], [16], [20], [21], [22], [30]). …”

“…Many efficient methods such as Newton methods and trust region methods for unconstrained optimization have been successfully extended to handle the presence of bounds on the variables [4], [5], [19] and their local superlinear/quadratic convergence have been established [10], [12], [13], [20]. A trust region version of Newton's method for bound constrained problems is analyzed in [20].…”

“…However, in order to prove the local superlinear convergence, the method in [10] must resort to a particular-defined multiplier function, which involves solving a product form of the linear system. Later, Facchinei and Lucidi [13] employed a similar identification technique to estimate the active set. Unlike the algorithms for bound constrained problems that we have reviewed, it generates iterates that need not be feasible by employing a differentiable exact penalty function.…”

A convergent decomposition method for box-constrained optimization problems

“…This is a wellstudied problem and several methods have been proposed (see, e.g., [2,4,7,11,12] In many real applications, due to the particular structure of the problem and/or to the large sizes, the adoption of a decomposition approach may be the practicable way to efficiently solve the optimization problem (see, e.g., [5]). In a general decomposition framework, at any iteration, some variables are kept fixed to their current values, and the other variables are determined by solving the corresponding subproblem.…”