A mathematical programming approach to strong separation in normed spaces

This paper deals with an infinite-dimensional optimization approach to the strong separation of two bounded sets in a normed space. We present an approximation procedure, called Algorithm (A), such that a semi-infinite optimization problem must be solved at each step. Its global convergence is established under certain natural assumptions, and a stopping criterion is also provided. The particular case of strong separation in the space Lp(X, A, μ) is approached in detail. We also propose Algorithm (B), which is an implementable modification of Algorithm (A) for separating two bounded sets in Lp([a, b]), with [a, b] being an interval in R. Some illustative computational experience is reported, and a particular stopping criterion is provided for the case of functions of bounded variation in L2([a, b]).