On Solving Optimization Problems with Hidden Nonconvex Structures

On numerical solving the spherical separability problem

“…So, it seems to be quite natural (but hardly ever grounded) the reaction of the specialists propagating methods of direct selection-such as the method of branches and bounds (and cuts methods), which, as known, suffer the curse of dimension, when the volume of computations grows exponentially side by side with the growth of the problems dimension [13,15,16]. We are sure, there exists also another way of solving nonconvex problems of high dimension.…”

“…As well-known [13,15,16], the A.D.Alexandov's functions (or the (D.C.) functions, which may be represented as a difference of two convex functions), form a linear space, which is dense in the space of continuous functions (in the topology of homogeneous convergence on compacts). So, problems of D.C. programming represent a rather large and, besides, very attractive class of optimization problems, for which we developed the theory of global search [13,15].…”

“…At the same time, in nonconvex problems, the direct application of standard methods may have unpredictable consequences [13,15,16,20,21], and sometimes may even distract one from the desired solution. So, it seems to be quite natural (but hardly ever grounded) the reaction of the specialists propagating methods of direct selection-such as the method of branches and bounds (and cuts methods), which, as known, suffer the curse of dimension, when the volume of computations grows exponentially side by side with the growth of the problems dimension [13,15,16].…”

“…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.…”

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