# On the Pervasiveness of Difference-Convexity in Optimization and Statistics

arXiv: https://arxiv.org/abs/1704.03535 · HTML: https://mlsanity.com/paper/1704.03535/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Apr 2017.

## Built on by (most cited first, 4 of 4)

- [On Tilted Losses in Machine Learning: Theory and Applications](https://mlsanity.com/paper/2109.06141/) (2021)
- [Non-geodesically-convex optimization in the Wasserstein space](https://mlsanity.com/paper/2406.00502/) (2024)
- [DC-LA: Difference-of-Convex Langevin Algorithm](https://mlsanity.com/paper/2601.22932/) (2026)
- [Distributed Difference of Convex Optimization](https://mlsanity.com/paper/2407.16728/) (2024)

## Abstract (arXiv)

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 well-known functions 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 LC1 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), expectation-based, VaR-based, and CVaR-based random deviation functions are all dc. Adding the known class of dc surrogate sparsity functions that are employed as approximations of the l_0 function in statistical learning, our work significantly expands the family of dc functions and positions them for fruitful applications.