On local search in d.c. optimization problems

On the solution of convex bilevel optimization problems

“…Approaches using a local linearization of either the function f (·, ·) or the function ϕ(·), see e.g. Strekalovsky [35] and Tuy and Ghannadan [36] can be used in special cases.…”

On numerical solving the spherical separability problem

“…It is well-known [13][14][15]25,26] that in nonconvex problems, in particular, in D.C. programming problems, the classical convex optimization methods (as conjugate gradient, Newtonian, trust region methods etc.) turn out to be inefficient, in general, for reaching a global solution.…”

“…Since it is unknown in advance whether the sets may be separated with a sphere or not, the minimization of the classification error function arises here naturally [10]. This minimization problem turns out to be nonsmooth and nonconvex [12][13][14][15][16], that leads us to a nonconvex variational problem.…”

“…According to the theory developed on the base of global optimality conditions (GOC), the process of seeking for a global solution in nonconvex optimization problems consists of the two principal stages: (i) a special local search, which takes into account the structure of the problem under scrutiny [14], and (ii) the procedures (based on the global optimality conditions [13,15]) of escaping critical points (provided by the local search).…”

“…Further the spherical separability problem for two sets can be viewed as a problem of D.C. minimization [13][14][15]:…”