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A Perturbed DCA for Computing d-Stationary Points of Nonsmooth DC Programs

Zhangcheng Feng, Yancheng Yuan

TL;DR

This work tackles the computation of $d$-stationary points for nonsmooth DC programs of the form $\min_x \phi(x)-\psi(x)$ with $\psi(x)=\max_i\psi_i(x)$. It introduces the perturbed DCA (pDCA), which perturbs the concave part so the active gradient reduces to a singleton almost surely, enabling a single strongly convex subproblem per iteration and improving efficiency relative to prior multi-subproblem schemes. Theoretical results establish subsequential convergence to $d$-stationary points almost surely under standard assumptions, and extensive experiments on $K$-sparse regression and $K$-means/$K$-medians clustering demonstrate substantial practical gains. The method balances sharpness and computational efficiency, offering a scalable tool for large-scale nonsmooth DC optimization with strong convergence guarantees.

Abstract

This paper introduces an efficient perturbed difference-of-convex algorithm (pDCA) for computing d-stationary points of an important class of structured nonsmooth difference-of-convex problems. Compared to the principal algorithms introduced in [J.-S. Pang, M. Razaviyayn, and A. Alvarado, Math. Oper. Res. 42(1):95--118 (2017)], which may require solving several subproblems for a one-step update, pDCA only requires solving a single subproblem. Therefore, the computational cost of pDCA for one-step update is comparable to the widely used difference-of-convex algorithm (DCA) introduced in [D. T. Pham and H. A. Le Thi, Acta Math. Vietnam. 22(1):289--355 (1997)] for computing a critical point. Importantly, under practical assumptions, we prove that every accumulation point of the sequence generated by pDCA is a d-stationary point almost surely. Numerical experiment results on several important examples of nonsmooth DC programs demonstrate the efficiency of pDCA for computing d-stationary points.

A Perturbed DCA for Computing d-Stationary Points of Nonsmooth DC Programs

TL;DR

This work tackles the computation of -stationary points for nonsmooth DC programs of the form with . It introduces the perturbed DCA (pDCA), which perturbs the concave part so the active gradient reduces to a singleton almost surely, enabling a single strongly convex subproblem per iteration and improving efficiency relative to prior multi-subproblem schemes. Theoretical results establish subsequential convergence to -stationary points almost surely under standard assumptions, and extensive experiments on -sparse regression and -means/-medians clustering demonstrate substantial practical gains. The method balances sharpness and computational efficiency, offering a scalable tool for large-scale nonsmooth DC optimization with strong convergence guarantees.

Abstract

This paper introduces an efficient perturbed difference-of-convex algorithm (pDCA) for computing d-stationary points of an important class of structured nonsmooth difference-of-convex problems. Compared to the principal algorithms introduced in [J.-S. Pang, M. Razaviyayn, and A. Alvarado, Math. Oper. Res. 42(1):95--118 (2017)], which may require solving several subproblems for a one-step update, pDCA only requires solving a single subproblem. Therefore, the computational cost of pDCA for one-step update is comparable to the widely used difference-of-convex algorithm (DCA) introduced in [D. T. Pham and H. A. Le Thi, Acta Math. Vietnam. 22(1):289--355 (1997)] for computing a critical point. Importantly, under practical assumptions, we prove that every accumulation point of the sequence generated by pDCA is a d-stationary point almost surely. Numerical experiment results on several important examples of nonsmooth DC programs demonstrate the efficiency of pDCA for computing d-stationary points.
Paper Structure (22 sections, 5 theorems, 74 equations, 1 figure, 4 tables, 3 algorithms)

This paper contains 22 sections, 5 theorems, 74 equations, 1 figure, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notations.
  3. Preliminaries and Motivating Examples
  4. A pDCA for Computing d-stationary Points of the DC Program \ref{['eq:dc_general']}
  5. Example 1 ($K$-sparse regularization):
  6. Example 2 (Capped $\ell_1$ penalty):
  7. Example 3 ($K$-medians clustering):
  8. Numerical Experiments
  9. Stopping criteria of pDCA.
  10. A simple comparison with DCA
  11. Numerical results on $K$-sparse regularized linear regression problem
  12. Data generation.
  13. Active piece selection.
  14. A dual semismooth Newton method for solving the subproblems
  15. Numerical results
  16. ...and 7 more sections

Key Result

Proposition 1

A vector $x^*\in\operatorname{dom}(\zeta)$ is a d-stationary point of the problem eq:dc_general if and only if: $\forall i\in\mathcal{M}(x^*)$, where $\sigma>0$. Moreover, if the function $\psi$ is piecewise affine, then any such d-stationary point is a local minimizer of $\zeta$.

Figures (1)

  • Figure 1: Comparison of pDCA and standard DCA for solving \ref{['ex: pdcavsdca']}.

Theorems & Definitions (18)

  • Example 1: $K$-sparse regularized problems gotoh2018dcsparseahn2017dclearning
  • Example 2: Capped $\ell_1$ penalty zhang2010cappedl1
  • Example 3: DC reformulations of $K$-means and $K$-medians clustering models
  • Definition 1: Critical point tao1997convex
  • Definition 2: d-stationary point pang2017computing
  • Proposition 1: pang2017computing
  • Proposition 2
  • proof
  • Remark 1
  • Lemma 1: Sequence boundedness
  • ...and 8 more