A deterministic approach to global box-constrained optimization

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

“…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. They approximate λ min (H(x)) using Gershgorin's Theorem, but note that other approximations to λ min (H(x)), for x in some domain, have been proposed in the literature.…”

“…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 (…”

“…This is usually achieved using techniques such as vertex enumeration, interval arithmetic, convexification or problem specific constructs (Neumaier, 2004) but these may not be flexible enough or suitably scalable for the purpose of generic problem solvers. The approach in Evtushenko and Posypkin (2012) and oBB however, allows global solution for both first and second order models by solving over Euclidean balls rather than boxes, which ensures that the subproblems can be solved in polynomial time. Evtushenko and Posypkin (2012) use non-uniform rectangular partitions and while their algorithm is already in a parallel framework it only uses first order models.…”

“…The approach in Evtushenko and Posypkin (2012) and oBB however, allows global solution for both first and second order models by solving over Euclidean balls rather than boxes, which ensures that the subproblems can be solved in polynomial time. Evtushenko and Posypkin (2012) use non-uniform rectangular partitions and while their algorithm is already in a parallel framework it only uses first order models. oBB uses overlapping balls leading to an overlapping covering (as opposed to the traditional hyper-rectangular partition) of the domain.…”