DC optimization approach to robust controls: the optimal scaling value problem

The optimal scaling problem (OSP) for constant scaling in output feedback control is an inherently difficult nonconvex problem for which existing local search algorithms can at best locate a local solution. Because of the presence of additional nonconvex constraints, OSP is a harder problem than the feasibility problem (FP) studied in Tuan et al. However, like FP, it can be restated as a problem of globally minimizing a convex function under DC constraints, i.e. constraints that can be expressed in terms of differences of convex functions. A particular structure of this DC optimization problem is that it becomes convex when a relatively small number of "complicating" variables are held fixed. We propose alternative branch and bound algorithms for OSP, which exploit this structure by branching upon the complicating variables and use adaptive subdivision strategies to speed up the convergence to the global solution.