Jorge J. Moré

Benchmarking optimization software with performance profiles.

Abstract.We propose performance profiles — distribution functions for a performance metric — as a tool for benchmarking and comparing optimization software. We show that performance profiles combine the best features of other tools for performance evaluation.

Testing Unconstrained Optimization Software

Much of the testing of optimization software is inadequate because the number of test functmns is small or the starting points are close to the solution. In addition, there has been too much emphasm on measurmg the efficmncy of the software and not enough on testing reliability and robustness. To address this need, we have produced a relatwely large but easy-to-use collection of test functions and designed gmdelines for testing the reliability and robustness of unconstrained optimization software.

Computing a Trust Region Step

We propose an algorithm for the problem of minimizing a quadratic function subject to an ellipsoidal constraint and show that this algorithm is guaranteed to produce a nearly optimal solution in a finite number of iterations. We also consider the use of this algorithm in a trust region Newtons method. In particular, we prove that under reasonable assumptions the sequence generated by Newtons method has a limit point which satisfies the first and second order necessary conditions for a minimizer of the objective function. Numerical results for GQTPAR, which is a Fortran implementaton of our algorithm, show that GQTPAR is quite successful in a trust region method. In our tests a call to GQTPAR only required 1.6 iterations on the average.

A Characterization of Superlinear Convergence and its Application to Quasi-Newton Methods

Let F be a mapping from real n-dimensional Euclidean space into itself. Most practical algorithms for finding a zero of F are of the form

The NEOS Server

x_{k+1} = x_{k} B_{k}^{-1_{Fx_{k}}}

Line search algorithms with guaranteed sufficient decrease

where

Recent Developments in Algorithms and Software for Trust Region Methods

\{B_{k}\}

ON THE SOLUTION OF LARGE QUADRATIC PROGRAMMING PROBLEMS WITH BOUND CONSTRAINTS

is a sequence of non-singular matrices. The main result of this paper is a characterization theorem for the superlinear convergence to a zero of F of sequences of the above form. This result is then used to give a unified treatment of the results on the superlinear convergence of the Davidon-Fletcher-Powell method obtained by Powell for the case in which exact line searches are used, and by Broyden, Dennis, and More for the case without line searches. As a by-product, several results on the asymptotic behavior of the sequence

Benchmarking Derivative-Free Optimization Algorithms

\{B_{k}\}

Optimization Software Guide

are obtained. An interesting aspect of these results is that superlinear convergence is obtained without any consistency conditions; i.e. without requiring that the sequence