Global optimization by V‐dense curves in topological vector spaces

The purpose of this paper is to solve a global optimization problem raised from industry, named the M‐problem, posed in the space L₁ of Lebesgue measurable functions of integrable absolute value.

The paper introduces the new concept of V‐dense curve (VDC), a generalization of that of α‐dense curve, to densify subsets of topological vector spaces not necessarily metrisable. It is proved that the feasible set of the M‐problem, namely the subset D of L₁ of all probability functions, is densifiable by VDC provided that L₁ to be endowed with the weak topology.

It is proved that the M‐problem, consisting of finding a probability function f of D associated to the mean life of an electronic devise that minimizes the expectation defined by a certain functional on L₁, is not a well‐posed problem in D. Nevertheless, by virtue of the compactness, the M‐problem has solution on each weak VDC in D for arbitrary weak 0‐neighbourhood V, which allows to find an approximate probability function with arbitrary precision.

The paper has designed, by means of the VDC‐method, a convergent algorithm to find approximate solutions in ill‐posed global optimization problems when the feasible set is contained in a non‐metrisable topological vector space.