On the pervasiveness of difference-convexity in optimization and statistics

Abstract

With the increasing interest in applying the methodology of difference-of-convex (dc) optimization to diverse problems in engineering and statistics, this paper establishes the dc property of many functions in various areas of applications not previously known to be of this class. Motivated by a quadratic programming based recourse function in two-stage stochastic programming, we show that the (optimal) value function of a copositive (thus not necessarily convex) quadratic program is dc on the domain of finiteness of the program when the matrix in the objective function's quadratic term and the constraint matrix are fixed. The proof of this result is based on a dc decomposition of a piecewise LC 1 function (i.e., functions with Lipschitz gradients). Armed with these new results and known properties of dc functions existed in the literature, we show that many composite statistical functions in risk analysis, including the value-at-risk (VaR), conditional value-at-risk (CVaR), Optimized Certainty Equivalent (OCE), and the expectation-based, VaR-based, and CVaR-based random deviation functionals are all dc. Adding the known class of dc surrogate sparsity functions that are employed as approximations of the 0 function in statistical learning, our work significantly expands the classes of dc functions and positions them for fruitful applications. * All authors are affiliated with the Daniel J. Epstein

Computation of the maximum likelihood estimator in low-rank factor analysis

“…In the machine learning community, these methods are also known as the convex concave procedure [36,37]. These algorithms have gained significant traction in the wider optimization community, especially recently, due to their pervasive use in practice -some excellent recent works on this topic include [1,13,25,26] (see also references therein). However, to our knowledge, ours is the first paper to use this approach in the context of ML factor analysis.…”

Correction to: On the pervasiveness of difference-convexity in optimization and statistics

“…Suppose that Q is copositive on D ∞ and that dom(Q, D) is convex. Then the value function qp opt (q, b) is dc on dom(Q, D), provided that qp opt is a quadratic function on each polyhedral member in the family F in the above corrected Part (c) of Proposition 9 in [4]. In particular, qp opt (q, b) is dc on dom(Q, D), if Q is positive semidefinite.…”

“…After the publication of the DOI version of [4], the authors were alerted to two examples in the literature that show the value function of, respectively, a constraint-only [2], or an objective-only [3] perturbed (nonconvex) quadratic program (QP) may not be piecewise linear-quadratic, which forms a subclass of the class of piecewise quadratic functions. The latter class of piecewise functions is the principal conclusion of Part (c) of Proposition 9 in the paper [4]. These examples do not contradict the proof of the proposition but invalidate the statement of this part of the proposition.…”

“…Below we present the correct statement of this part without repeating its proof. We use the same notations as in [4] throughout this erratum; in particular, "dc" stands for differenceof-convex. fixed (q, b).…”

“…Another oversight in [4] is in the statement of Theorem 12. To apply Proposition 11, we need the value function qp opt (q, b) to be quadratic on each polyhedral piece S F in the family F in Part (c) of Proposition 9.…”