On the pervasiveness of difference-convexity in optimization and statistics
Jong-Shi Pang
University of Southern California
Abstract:
With the increasing interest in applying the methodology of difference-of-convex
(dc) optimization to diverse problems in engineering and statistics, we show
that many well-known functions arising therein can be represented as the
difference of two convex functions. These include a univariate folded concave
function commonly employed in statistical learning, the value function of a
copositive recourse function in two-stage stochastic programming, and many
composite statistical functions in risk analysis, such as the value-at-risk
(VaR), conditional value-at-risk (CVaR), expectation-based, VaR-based, and
CVaR-based random deviation functionals. We also discuss decomposition methods
for computing directional stationary points of a class of nonsmooth, nonconvex
dc programs that combined the Gauss-Seidel idea, the alternating direction
method of multipliers, and a special technique to handle the negative of a
pointwise max function.