  This paper is concerned with solving nonconvex learning problems with folded
concave penalty. Despite that their global solutions entail desirable
statistical properties, they lack optimization techniques that guarantee global
optimality in a general setting. In this paper, we show that a class of
nonconvex learning problems are equivalent to general quadratic programs. This
equivalence facilitates us in developing mixed integer linear programming
reformulations, which admit finite algorithms that find a provably global
optimal solution. We refer to this reformulation-based technique as the mixed
integer programming-based global optimization (MIPGO). To our knowledge, this
is the first global optimization scheme with a theoretical guarantee for folded
concave penalized nonconvex learning with the SCAD penalty [J. Amer. Statist.
Assoc. 96 (2001) 1348-1360] and the MCP penalty [Ann. Statist. 38 (2001)
894-942]. Numerical results indicate a significant outperformance of MIPGO over
the state-of-the-art solution scheme, local linear approximation and other
alternative solution techniques in literature in terms of solution quality.
