Warren Hare

Heinz H. Bauschke, Warren L. Hare, Walaa M. Moursi

The problem of finding a minimizer of the sum of two convex functions - or, more generally, that of finding a zero of the sum of two maximally monotone operators - is of central importance in variational analysis. Perhaps the most popular method of solving this problem is the Douglas-Rachford splitting method. Surprisingly, little is known about the range of the Douglas-Rachford operator. In this paper, we set out to study this range systematically. We prove that for 3* monotone operators a very pleasing formula can be found that reveals the range to be nearly equal to a simple set involving the domains and ranges of the underlying operators. A similar formula holds for the range of the corresponding displacement mapping. We discuss applications to subdifferential operators, to the infimal displacement vector, and to firmly nonexpansive mappings. Various examples and counter-examples are presented, including some concerning the celebrated Brezis-Haraux theorem.

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, 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.