Global convergence and the Powell singular function

The Powell singular function was introduced 1962 by M.J.D. Powell as an unconstrained optimization problem. The function is also used as nonlinear least squares problem and system of nonlinear equations. The function is a classic test function included in collections of test problems in optimization as well as an example problem in text books. In … Read more

On large scale unconstrained optimization problems and higher order methods

Third order methods will in most cases use fewer iterations than a second order method to reach the same accuracy. However, the number of arithmetic operations per iteration is higher for third order methods than a second order method. Newton’s method is the most commonly used second order method and Halley’s method is the most … Read more

Using Partial Separability of Functions in Generating Set Search Methods for Unconstrained Optimisation

Generating set Search Methods (GSS), a class of derivative-free methods for unconstrained optimisation, are in general robust but converge slowly. It has been shown that the performance of these methods can be enhanced by utilising accumulated information about the objective function as well as a priori knowledge such as partial separability. This paper introduces a … Read more

A generating set search method exploiting curvature and sparsity

Generating Set Search method are one of the few alternatives for optimising high fidelity functions with numerical noise. These methods are usually only efficient when the number of variables is relatively small. This paper presents a modification to an existing Generating Set Search method, which makes it aware of the sparsity structure of the Hessian. … Read more

Graph Coloring in the Estimation of Sparse Derivative Matrices: Instances and Applications

We describe a graph coloring problem associated with the determination of mathematical derivatives. The coloring instances are obtained as intersection graphs of row partitioned sparse derivative matrices. The size of the graph is dependent on the partition and can be varied between the number of columns and the number of nonzero entries. If solved exactly … Read more

Optimal Direct Determination of Sparse Jacobian Matrices

It is well known that a sparse Jacobian matrix can be determined with fewer function evaluations or automatic differentiation \emph{passes} than the number of independent variables of the underlying function. In this paper we show that by grouping together rows into blocks one can reduce this number further. We propose a graph coloring technique for … Read more

The Use of Java Arrays for Matrix Computations

In the paper it is shown how to utilize the flexibility in native Java arrays for matrix computations. Suitable datastructures for symmetric and sparse matrices are introduced. A disadvantage of the native Java arrays is shown when used as two-dimensional array for dense matrix computation. Numerical results show that the efficiency is not lost using … Read more

Sparsity issues in the computation of Jacobian Matrices

The knowledge of sparsity information plays an important role in efficient determination of sparse Jacobian matrices. In a recent work, we have proposed sparsity-exploiting substitution techniques to determine Jacobian matrices. In this paper, we take a closer look at the underlying combinatorial problem. We propose a column ordering heuristic to augment the “usable sparsity” in … Read more

Reducing the number of AD passes for computing a sparse Jacobian matrix

A reduction in the computational work is possible if we do not require that the nonzeros of a Jacobian matrix be determined directly. If a column or row partition is available, the proposed substitution technique can be used to reduce the number of groups in the partition further. In this chapter, we present a substitution … Read more