Theory and Methods for Global Optimization — An Integral Approach

Let X be a Hausdorff topological space, S ⊂ X a closed set and f = X → R a real-valued function. The problem considered here is to find the infimum of f over S, $$\matrix{ {{\rm{\bar c}} = \inf {\rm{f}}\left( {\rm{x}} \right)} \cr {{\rm{x}} \in {\rm{s}}} \cr } $$ (1.1) and the set of all global minima: $${\rm{\bar H = }}\left\{ {{\rm{x}}\left| {{\rm{f}}\left( {\rm{x}} \right) = {\rm{\bar c,x}} \in {\rm{s}}} \right.} \right\}$$ (1.2) We assume in this paper: (A1) f is continuous: (A2) There is α ∈ R’ such that the level set $$\matrix{ {{{\rm{H}}_{\rm{\alpha }}} = \left\{ {{\rm{x}}\left| {{\rm{f}}\left( {\rm{x}} \right) \le {\rm{\alpha }}} \right.} \right\}} \cr {{\rm{is}}\,{\rm{compact}}\,{\rm{and}}\,{{\rm{H}}_{\rm{\alpha }}} \cap {\rm{s}} \ne \phi {\rm{.}}} \cr } $$ (1.3) Thus the problem (1.1) becomes to find $$\matrix{ {{\rm{\bar c = min}}\,{\rm{f}}\left( {\rm{x}} \right) = \min {\rm{f}}\left( {\rm{x}} \right)} \cr {{\rm{x}} \in {\rm{s}}\,{\rm{x}} \in {{\rm{H}}_{\rm{\alpha }}} \cap {\rm{S}}} \cr } $$ (1.4) and the set of all global minima Ħ is non empty.