Warren Hare

Warren Hare

Derivative-free optimization (DFO) is the mathematical study of the optimization algorithms that do not use derivatives. One branch of DFO focuses on model-based DFO methods, where an approximation of the objective function is used to guide the optimization algorithm. Proving convergence of such methods often applies an assumption that the approximations form {\em fully linear models} -- an assumption that requires the true objective function to be smooth. However, some recent methods have loosened this assumption and instead worked with functions that are compositions of smooth functions with simple convex functions (the max-function or the $\ell_1$ norm). In this paper, we examine the error bounds resulting from the composition of a convex lower semi-continuous function with a smooth vector-valued function when it is possible to provide fully linear models for each component of the vector-valued function. We derive error bounds for the resulting function values and subgradient vectors.

Yiwen Chen, Warren Hare, Lindon Roberts

We study three quadratic models in model-based derivative-free optimization: the minimum norm (MN), minimum Frobenius norm (MFN), and quadratic generalized simplex derivative (QS) models. Despite their widespread use, their approximation accuracy and relationships have not been systematically explored. We establish fully linear error bounds for all three models, removing the uniformly bounded model Hessian assumption required in existing MN analyses and deriving the first such results for the QS model. We further analyze Hessian approximation accuracy via directional error bounds, showing that all three models achieve fully quadratic accuracy along sample directions under a mild condition on the sample set. This reveals a form of directional fully quadratic accuracy not captured by existing theory. Finally, we characterize the relationships among these models, identifying conditions under which they coincide and clarifying their structural connections.

Yiwen Chen, Warren Hare, Amy Wiebe

Gradient approximations are a class of numerical approximation techniques that are of central importance in numerical optimization. In derivative-free optimization, most of the gradient approximations, including the simplex gradient, centred simplex gradient, and adapted centred simplex gradient, are in the form of simplex derivatives. Owing to machine precision, the approximation accuracy of any numerical approximation technique is subject to the influence of floating point errors. In this paper, we provide a general framework for floating point error analysis of simplex derivatives. Our framework is independent of the choice of the simplex derivative as long as it satisfies a general form. We review the definition and approximation accuracy of the generalized simplex gradient and generalized centred simplex gradient. We define and analyze the accuracy of a generalized version of the adapted centred simplex gradient. As examples, we apply our framework to the generalized simplex gradient, generalized centred simplex gradient, and generalized adapted centred simplex gradient. Based on the results, we give suggestions on the minimal choice of approximate diameter of the sample set.

Yasha Pushak, Warren Hare, Yves Lucet

This paper addresses the problem of finding multiple near-optimal, spatially-dissimilar paths that can be considered as alternatives in the decision making process, for finding optimal corridors in which to construct a new road. We further consider combinations of techniques for reducing the costs associated with the computation and increasing the accuracy of the cost formulation. Numerical results for five algorithms to solve the dissimilar multipath problem show that a "bidirectional approach" yields the fastest running times and the most robust algorithm. Further modifications of the algorithms to reduce the running time were tested and it is shown that running time can be reduced by an average of 56 percent without compromising the quality of the results.