In the Real-Parameter Black-Box Optimization Benchmark (BBOB) multi-modal functions are divided into two sets: “Multi-modal functions with adequate global structure” and “Multi-modal functions with weak global structure”. What is the formal definition of “adequate” and “weak” global structure?
Article Real-Parameter Black-Box Optimization Benchmarking 2009: Noi...
Looking at the papers cited, especially those by Hansen et al (1009a,b), an "adequate" structure of the optimal solutions means that they are spread out in a near-symmetrical way, while "weak" ditto means that the optima are less regularly placed. But to be honest, I do not see a formal definition.
Definitions of local solutions is important in continuous minimization problem. That is, there exist a point such that a local minimum also a local maximum inner of flat region of a objective function. For example, the point 0 on convex function: max{x2-1,0} is a local minimum and also local maximum. Such a duplicated definition for one point is clear up using our new definition of local minimal value set. For an objective function, if the number of connected component of all local minimal value set is one, the function is weak unimodal, otherwise the function multimodal.
Our original paper of the abovel aricle have been written in japanese, its translated version is see :"H.Kanemitsu, M.Miyakosi and M.Shimbo: Properties of Unimodal and Multimodal Functions Defined by Local Minimal Values Set,Electronics and Communications(Part III: Fundamental Electronic Science), Vol.81, No.1, pp.42-51, 1998.1."
Moreover, the following aritcle is available on RearchGate: Hideo Kanemitsu et al.:
``Definitions and Properties of (Local) Minima and Multimodal Functions using Level Set for Continuous Optimization Problems,'' Proc. of 2013 International Symposium on Nonlinear Theory and its Applications (NOLTA'2013),
pp.94-97, 2013(Sep.), Santa Fe(U.S.A.).
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