A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions

Branching and bounding improvements for global optimization algorithms with Lipschitz continuity properties

“…To the best of our knowledge, none of these have been used before in place of a Lipschitz constant in Lipschitz based branch and bound algorithms. In this section, we show that some of these estimates are more accurate than the Lipschitz constant estimates considered in Fowkes et al (2012) and estimates using Gershgorin's Theorem in Evtushenko and Posypkin (2012).…”

“…The approach taken to estimate the gradient Lipschitz constant in Fowkes et al (2012) was to bound the norm of the Hessian over a suitable domain using interval arithmetic. Evtushenko and Posypkin (2012) suggest replacing the negative Lipschitz constant by a lower bound on the spectrum of the Hessian, λ min (H(x)), for x in some interval, which they claim yields a more accurate estimate.…”

“…The case where the lower bound is based on a Lipschitz constant L f of the objective function f has been well studied in the literature (see Pinter, 1996;Pardalos, Horst andThoai, 1995 andNeumaier, 2004 and references therein) and has the immediate form f (x) ≥ f (x B ) − L f (B) x − x B for some point x B in a subregion B and a Lipschitz constant L f (B) for f over B. A more accurate lower bound using a Lipschitz constant L g (B) of the gradient of the objective function g = ∇ x f can be derived using Taylor's theorem to first order (Evtushenko and Posypkin, 2012;Fowkes, Gould and Farmer, 2012) …”

“…Evtushenko and Posypkin (2012) have used (1.2) and refinements where −L g (B) is replaced by a lower bound on the spectrum of the Hessian in a special type of branch and bound algorithm using non-uniform subregions. The case where one goes a step further and uses second order Taylor's theorem to obtain a lower bound using a Lipschitz constant L H (B) for the Hessian H = ∇ xx f was considered in Fowkes et al (2012) f (x) ≥ c B (…”

“…Additionally, we extend one of these approaches to the second order bound (1.3) by replacing the Lipschitz constant estimate with a novel lower bound on the spectrum of the third order derivative tensor. We test the new proposals in the overlapping branch and bound (oBB) framework proposed in Fowkes et al (2012) where l B (x) is either q B (x) in (1.2) or c B (x) in (1.3), over each subdomain B. As one would expect, due to additional problem information being employed in the model, we find that in general second order models yield better lower bounds on the objective function compared to first order models and hence can potentially lead to better branch and bound algorithms for Lipschitz optimization.…”