Composite Quasi-DC Optimization

• Composite quasi-DC programs are a class of nonconvex, nonsmooth optimization problems characterized by directional derivatives expressed as the difference of two positively homogeneous convex functions.
• The framework employs composite formulations with pointwise-max structures, enabling iterative convex subproblems and descent algorithms that converge to directional or weak stationary points.
• Applications span modern statistical estimation and deep learning, with demonstrated success in robust regression, matrix recovery, and phase retrieval through tailored majorization-minimization and smoothing techniques.

Searching arXiv for the cited work and closely related papers on composite quasi-DC / DC-composite optimization. Composite quasi-DC programs are a class of nonconvex, nonsmooth optimization problems in which compositional structure and generalized DC-type directional information are central. In the terminology proposed in "Quasi-difference-convexity: Modernization of Quasi-differentiable Optimization" (Pang et al., 16 Jul 2025), a quasi-differentiable function is renamed a quasi-difference-convex, or quasi-dc, function: its directional derivative at a reference point is a difference of two positively homogeneous convex functions. In a general composite formulation over a closed convex set $X$, one studies objectives of the form $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$ with $\theta_j(x)=\phi_j(P^j(x))$, where $\phi_j$ is Bouligand differentiable and each inner component has mixed convex, concave, and max-of-differentiable structure (Pang et al., 16 Jul 2025). Earlier work on modern statistical estimation formulated a related "composite quasi-DC" framework through empirical risk minimization, $$F(\theta)=R(\theta)+(1/N)\sum_{i=1}^N \ell_i(\theta)$$, where the regularizer and losses admit dc decompositions, but the algorithmic treatment exploits a special pointwise-max structure rather than an explicit global DC decomposition (Cui et al., 2018).

1. Terminology, differentiability, and quasi-dc structure

A function $f:O\subseteq \mathbb{R}^n\to\mathbb{R}$ is Bouligand differentiable at $x\in O$ if it is locally Lipschitz and the directional derivative

$$f'(x;v)\coloneqq \lim_{\tau\downarrow 0}\frac{f(x+\tau v)-f(x)}{\tau}$$

exists for all $v\in\mathbb{R}^n$ (Pang et al., 16 Jul 2025). The quasi-dc property is defined at a reference point $\bar x$ by requiring $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$0 to be a difference of two positively homogeneous convex functions. Equivalently,

$\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$1

where $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$2 and $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$3 are convex and positively homogeneous, and there exist compact convex sets $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$4 and $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$5 such that

$\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$6

The pair $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$7 is called a quasi-differential of $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$8 at $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$9 (Pang et al., 16 Jul 2025).

This terminology shift is historically motivated. Quasi-differentiable functions were introduced by Pshenichnyi in a 1969 monograph written in Russian and translated in an English version in 1971, and the 2025 modernization proposes the name quasi-difference-convexity to align the class more closely with contemporary DC programming (Pang et al., 16 Jul 2025). A common misconception is that quasi-dc functions are simply DC functions in disguise. The available formulations do not make that identification: the quasi-dc definition is stated in terms of directional derivatives, while DC formulations are typically stated through explicit difference-of-convex decompositions of the function itself (Pang et al., 16 Jul 2025).

2. Canonical formulations of composite quasi-DC programs

One general composite quasi-dc program is defined on a closed convex set $\theta_j(x)=\phi_j(P^j(x))$0 through

$\theta_j(x)=\phi_j(P^j(x))$1

where $\theta_j(x)=\phi_j(P^j(x))$2 is Bouligand differentiable and

$\theta_j(x)=\phi_j(P^j(x))$3

Here $\theta_j(x)=\phi_j(P^j(x))$4 is convex, $\theta_j(x)=\phi_j(P^j(x))$5 is concave, and $\theta_j(x)=\phi_j(P^j(x))$6 is $\theta_j(x)=\phi_j(P^j(x))$7, possibly nonconvex (Pang et al., 16 Jul 2025). The framework is organized through four standing assumption types: Type I for differentiable outer $\theta_j(x)=\phi_j(P^j(x))$8, Type II for convex outer $\theta_j(x)=\phi_j(P^j(x))$9, Type III for dd-convex outer functions, and Type IV for outer functions decomposed as $\phi_j$0 with $\phi_j$1 convex and $\phi_j$2 dd-convex (Pang et al., 16 Jul 2025).

A second canonical formulation arises in statistical estimation: $\phi_j$3 with

$\phi_j$4

where all four component functions are convex, possibly nonsmooth (Cui et al., 2018). Although $\phi_j$5 is then a DC function in principle, the framework does not insist on an explicit DC decomposition of $\phi_j$6. Instead it exploits a special pointwise-max structure in each component (Cui et al., 2018).

That pointwise-max structure is central. Any convex, piecewise-affine or piecewise-quadratic function can be written as

$\phi_j$7

and the same representation is used for regularizers and losses. The data explicitly list $\phi_j$8, group-$\phi_j$9, SCAD, MCP, and truncated-$$F(\theta)=R(\theta)+(1/N)\sum_{i=1}^N \ell_i(\theta)$$0 as regularizers that can be written as a difference of two maxima of affine pieces; ReLU is a max of two affine forms; and a multi-layer ReLU network is a difference of two max-of-convex-PLQ functions (Cui et al., 2018). This suggests that the phrase "composite quasi-DC" is used in the literature in at least two closely related senses: one centered on quasi-differentials and Bouligand derivatives, and another centered on composite empirical-risk models with diff-max structure.

3. Majorization, convexification, and descent algorithms

The 2025 quasi-dc framework develops an iterative convex-programming based descent algorithm in a "single-$$F(\theta)=R(\theta)+(1/N)\sum_{i=1}^N \ell_i(\theta)$$1" version. Given $$F(\theta)=R(\theta)+(1/N)\sum_{i=1}^N \ell_i(\theta)$$2, one chooses $$F(\theta)=R(\theta)+(1/N)\sum_{i=1}^N \ell_i(\theta)$$3 from parameter sets $$F(\theta)=R(\theta)+(1/N)\sum_{i=1}^N \ell_i(\theta)$$4, then computes the unique minimizer $$F(\theta)=R(\theta)+(1/N)\sum_{i=1}^N \ell_i(\theta)$$5 of the strongly convex subproblem

$$F(\theta)=R(\theta)+(1/N)\sum_{i=1}^N \ell_i(\theta)$$6

where $$F(\theta)=R(\theta)+(1/N)\sum_{i=1}^N \ell_i(\theta)$$7. If $$F(\theta)=R(\theta)+(1/N)\sum_{i=1}^N \ell_i(\theta)$$8, the method stops and $$F(\theta)=R(\theta)+(1/N)\sum_{i=1}^N \ell_i(\theta)$$9 is a weak directional stationary point. Otherwise it sets $f:O\subseteq \mathbb{R}^n\to\mathbb{R}$0 and performs an Armijo line-search to find the smallest $f:O\subseteq \mathbb{R}^n\to\mathbb{R}$1 such that

$f:O\subseteq \mathbb{R}^n\to\mathbb{R}$2

then updates $f:O\subseteq \mathbb{R}^n\to\mathbb{R}$3 with $f:O\subseteq \mathbb{R}^n\to\mathbb{R}$4 (Pang et al., 16 Jul 2025). A "full-$f:O\subseteq \mathbb{R}^n\to\mathbb{R}$5" version also minimizes over all $f:O\subseteq \mathbb{R}^n\to\mathbb{R}$6; it is computationally heavier but yields true directional stationarity (Pang et al., 16 Jul 2025).

In the earlier composite quasi-DC framework for statistical estimation, the primary algorithm is a nonmonotone majorization-minimization method. For a representative summand $f:O\subseteq \mathbb{R}^n\to\mathbb{R}$7, the outer univariate convex $f:O\subseteq \mathbb{R}^n\to\mathbb{R}$8 is decomposed as

$f:O\subseteq \mathbb{R}^n\to\mathbb{R}$9

with $x\in O$0 convex non-decreasing and $x\in O$1 convex non-increasing. At the current iterate $x\in O$2, active-index sets are identified, convex majorants $x\in O$3 are formed for active pairs $x\in O$4, and a strongly convex subproblem in $x\in O$5 is solved after introducing slack variables $x\in O$6 and $x\in O$7 (Cui et al., 2018). The update selects the active pair yielding the smallest surrogate-plus-regularizer value. Because of the quadratic regularization, the method does not insist on $x\in O$8, but it does obtain decrease in the majorized sequence $x\in O$9 (Cui et al., 2018).

The same work derives a dual formulation for the MM subproblems and proposes a semismooth Newton method for their solution. Owing to strong convexity in $$f'(x;v)\coloneqq \lim_{\tau\downarrow 0}\frac{f(x+\tau v)-f(x)}{\tau}$$0, the inner minimizations are unique, the dual objective is $$f'(x;v)\coloneqq \lim_{\tau\downarrow 0}\frac{f(x+\tau v)-f(x)}{\tau}$$1, and generalized Jacobians are available in closed form because the prox mappings and constraint functions are piecewise-affine or smooth (Cui et al., 2018). This pairing of MM and semismooth Newton is one of the distinctive computational features of the pointwise-max composite quasi-DC framework.

4. Stationarity concepts and convergence theory

Directional stationarity is the principal optimality notion in the composite quasi-dc literature. In the diff-max framework, the directional derivative is

$$f'(x;v)\coloneqq \lim_{\tau\downarrow 0}\frac{f(x+\tau v)-f(x)}{\tau}$$2

and $$f'(x;v)\coloneqq \lim_{\tau\downarrow 0}\frac{f(x+\tau v)-f(x)}{\tau}$$3 is called d-stationary if $$f'(x;v)\coloneqq \lim_{\tau\downarrow 0}\frac{f(x+\tau v)-f(x)}{\tau}$$4 for all feasible directions $$f'(x;v)\coloneqq \lim_{\tau\downarrow 0}\frac{f(x+\tau v)-f(x)}{\tau}$$5 (Cui et al., 2018). The quasi-dc modernization distinguishes weak directional stationary points from directional stationary points, with the stronger conclusion attached to the full-$$f'(x;v)\coloneqq \lim_{\tau\downarrow 0}\frac{f(x+\tau v)-f(x)}{\tau}$$6 scheme and to certain regularity conditions (Pang et al., 16 Jul 2025).

For the nonmonotone MM scheme, the key convergence statement is that if $$f'(x;v)\coloneqq \lim_{\tau\downarrow 0}\frac{f(x+\tau v)-f(x)}{\tau}$$7 is bounded below on $$f'(x;v)\coloneqq \lim_{\tau\downarrow 0}\frac{f(x+\tau v)-f(x)}{\tau}$$8 and $$f'(x;v)\coloneqq \lim_{\tau\downarrow 0}\frac{f(x+\tau v)-f(x)}{\tau}$$9, then

$v\in\mathbb{R}^n$0

so $v\in\mathbb{R}^n$1, and any accumulation point $v\in\mathbb{R}^n$2 is d-stationary for $v\in\mathbb{R}^n$3 on $v\in\mathbb{R}^n$4 (Cui et al., 2018).

For the 2025 composite quasi-dc descent algorithm, subsequential convergence is established under bounded level sets. Any limit point $v\in\mathbb{R}^n$5 satisfies one of two conclusions: if $v\in\mathbb{R}^n$6, then $v\in\mathbb{R}^n$7 is a weak directional stationary solution; if $v\in\mathbb{R}^n$8 and uniform upper-approximation holds at $v\in\mathbb{R}^n$9, then $\bar x$0 is weak stationary. Moreover, if $\bar x$1 is a singleton and uniform upper-approximation holds for each $\bar x$2 at $\bar x$3, then $\bar x$4 is a directional stationary solution (Pang et al., 16 Jul 2025).

The same paper gives stronger sequential convergence under Lipschitz assumptions. If $\bar x$5 and $\bar x$6 are $\bar x$7 with Lipschitz gradients on the bounded level set, $\bar x$8 are Lipschitz there, and either "componentwise composite convexity" $\bar x$9 or "aggregate convexity" $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$00 holds for all $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$01, then for any $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$02 one may take unit steps, obtaining

$\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$03

uniformly, and any accumulation point is a directional stationary point (Pang et al., 16 Jul 2025). Under a uniform Kurdyka-\L{}ojasiewicz property with exponent $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$04, the convergence-rate trichotomy is explicit: finite convergence when $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$05, linear rate when $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$06, and sublinear rate $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$07 when $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$08 (Pang et al., 16 Jul 2025).

5. Relation to composite DC optimization and smoothing methods

Composite quasi-dc programs sit near a broader family of composite DC optimization models. A representative DC-composite class is

$\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$09

where $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$10 and $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$11 are $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$12, $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$13 and $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$14 are proper, lsc, convex, and $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$15 is proper, lsc, convex and simple (Tao et al., 2023). For this class, an inexact linearized proximal algorithm computes an inexact minimizer of a strongly convex majorization constructed with a partial linearization of the objective, and global convergence is derived under a KL property of a potential function; a verifiable condition is also given for KL exponent $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$16, which yields a local R-linear convergence rate (Tao et al., 2023).

Another adjacent framework minimizes

$\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$17

with $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$18 smooth, $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$19 smooth, and $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$20 a DC decomposition with weakly convex, prox-friendly components (Yazawa et al., 18 Mar 2025). The proposed variable smoothing algorithm replaces each weakly convex component by its Moreau envelope, forms the surrogate

$\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$21

and performs gradient descent updates on $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$22 while driving $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$23. The method is inner-loop-free, and any cluster point is a DC-critical point of the original problem (Yazawa et al., 18 Mar 2025).

For linearly constrained DC programs, difference-of-Moreau-envelopes smoothing leads to the composite LCDC-ALM. In the formulation

$\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$24

with $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$25, the method combines smoothing with the augmented Lagrangian and yields an $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$26-approximate stationary solution in $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$27 outer iterations (Sun et al., 2021). These DC results are not stated as quasi-dc theorems. A plausible implication is that composite quasi-dc programs belong to a wider methodological continuum in which convexification, Moreau-envelope smoothing, and proximal-augmented descent are recurring design principles.

A second misconception is that composite quasi-dc methods are merely reformulations of classical DCA. The data support a more nuanced view. The 2025 quasi-dc framework is explicitly presented as a unified treatment of iterative convex-programming based descent algorithms for a broad class of composite quasi-dc programs (Pang et al., 16 Jul 2025), whereas the 2018 diff-max framework relies on nonmonotone MM with semismooth Newton subsolvers (Cui et al., 2018). Classical DCA remains relevant, but it is one algorithmic strand among several.

6. Modeling domains, applications, and examples

The modeling range described in the literature is broad. In the statistical estimation framework, the data explicitly cite continuous piecewise affine regression and deep learning as examples. A continuous piecewise-affine model is written as

$\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$28

and the same framework covers deep networks with piecewise-affine activations, including two-layer ReLU networks that can be written in diff-max form (Cui et al., 2018). Numerical results in that study state that piecewise-affine regression outperforms linear regression on synthetic and UCI data, converges reliably to d-stationary points, and that the semismooth Newton subsolver typically needs fewer than 10 Newton steps per MM subproblem (Cui et al., 2018).

The quasi-dc modernization enumerates a different, but overlapping, set of examples: single-ratio fractional programming with $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$29, products with $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$30, norm-based objectives with $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$31, robust deviations composed with Huber or SCAD loss, and piecewise affine DC models $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$32, for which Algorithm 1 specializes to proximal DCA (Pang et al., 16 Jul 2025). It also mentions Heaviside composites as ongoing work (Pang et al., 16 Jul 2025).

Composite DC methods closely related to the quasi-dc setting have been applied to robust low-rank recovery and robust phase retrieval. For matrix completion with outliers and non-uniform sampling, the model

$\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$33

is treated by iLPA with a dPPASN subsolver, and the reported numerical highlights include relative error $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$34 or better on synthetic data, 20–100 iterations with wall-clock times 1–50 s for $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$35 up to 3000, Jester NMAE $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$36–$\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$37 with iLPA about $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$38 faster than PAM, and MovieLens-1M and Netflix NMAE $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$39–$\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$40 in tens of seconds (Tao et al., 2023). In robust phase retrieval, a variable smoothing algorithm for a DC loss composed with a smooth mapping is proposed for quadratic measurements corrupted by outliers, and the numerical experiment is reported to show that DC loss functions are more robust against outliers than the $\Theta_{\max}(x)=\max_{1\le j\le J}\theta_j(x)$41 loss (Yazawa et al., 9 Apr 2026).

A further application of DC programming with a composite objective appears in dynamic panels with group-specific heterogeneity and spatially dependent errors. The Composite Quasi-Likelihood estimator is formulated as a mixed-integer d.c. program and solved by DCA; the estimator is stated to remain unbiased under misspecification of unobserved fixed effects and to achieve an oracle property for the group-specific slope parameters (Chu, 2017). This is not presented as a composite quasi-dc program in the quasi-differential sense, but it shows how composite modeling, latent grouping, and DC programming intersect in econometric practice.

Taken together, these works portray composite quasi-dc programming not as a single algorithm or a single normal form, but as a technically structured optimization viewpoint. Its defining features are compositional modeling, pointwise-max or directional-difference structure, strongly convex local surrogates, and stationarity notions based on directional derivatives rather than classical smooth first-order conditions (Pang et al., 16 Jul 2025).