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An Inexact Boosted Difference of Convex Algorithm for Nondifferentiable Functions

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Pith's one-line read Inexact boosted DC algorithm with approximate subproblem solves and nonmonotone line search still converges to critical points for nondifferentiable difference-of-convex problems, at O(1/√N) step-norm complexity.

desk verdict A sound, incremental extension of nmBDCA to inexact subproblem solves, with correct proofs but an overstated abstract and a stopping rule that only certifies ε-criticality. read the letter →

arxiv 2412.05697 v1 pith:5K2AJ7QB submitted 2024-12-07 math.OC

Orizon P. Ferreira , Boris S. Mordukhovich , Wilkreffy M. S. Santos , João Carlos O. Souza This is my paper · ORCID
classification math.OC MSC 49J5390C2665K0565K10
keywords DCfunctionsBoosteddifferenceofconvexalgorithmNonmonotonelinesearchConvergenceanalysisComplexityInexactNondifferentiableoptimization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper presents an inexact version of the Boosted Difference of Convex Functions Algorithm (BDCA) for minimizing functions written as the difference of two convex functions, allowing both parts to be nondifferentiable. In the proposed algorithm, the convex subproblem that generates each search direction is solved only approximately, up to a controlled relative error, and the step size is chosen by an inexact nonmonotone Armijo line search. Under the assumptions that both DC components are strongly convex, the objective is bounded below, the error-tolerance sequence is summable, and the nonmonotonicity parameters follow one of three strategies, the algorithm is well-defined and every accumulation point of its iterates is a critical point of the DC program. The paper also gives an $O(1/\sqrt{N})$ iteration-complexity bound on the minimal step norm and reports numerical experiments where the inexact method outperforms the nonmonotone BDCA and the monotone DCA.

What carries the argument

The load-bearing object is the relative-error condition $\|w_k - \xi_k\| \le \theta \|y_k - x_k\|$ together with $w_k \in \partial_{\varepsilon_k} h(x_k)$ and $\xi_k \in \partial g(y_k)$. This condition makes the approximate subproblem solve 'exact enough': combined with the strong convexity of $g$ and $h$, it yields the descent estimate $\varphi(y_k) \le \varphi(x_k) - (\sigma/2 - \theta)\|d_k\|^2 + \varepsilon_k$ (Proposition 13), playing the role that the descent lemma plays in exact DCA and BDCA. The second mechanism is the nonmonotone Armijo line search with slack $\nu_k$; it keeps the line search well-defined even when $d_k$ is not a descent direction, and the summability/decay conditions on $\nu_k$ control the cumulative slack so that the quasi-monotone decrease still forces $\|d_k\| \to 0$. The $\varepsilon_k$-subdifferential of $h$, defined by $h(z) \ge h(x) + \langle w, z - x \rangle - \varepsilon_k$, is what allows inexact subgradient information at the current iterate.

What would settle it

Run exactly the paper's Algorithm 2 on a DC program with a known unique minimizer (such as Example 5, $\varphi(x,y) = x^2 + y^2 + x + y - |x| - |y|$), verifying at each iteration that $w_k \in \partial_{\varepsilon_k} h(x_k)$, $\xi_k \in \partial g(y_k)$, and $\|w_k - \xi_k\| \le \theta \|y_k - x_k\|$ with summable $\varepsilon_k$ and $\nu_k$ chosen by (A2). If the iterates accumulate at any point $\bar{x}$ with $\partial g(\bar{x}) \cap \partial h(\bar{x}) = \emptyset$, the main theorem is false; conversely, if the hypotheses hold and every accumulation point is critical, the claim is confirmed.

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Extended reading notes

Core claim

The central claim is that the acceleration strategy of BDCA can be made inexact without losing convergence, even when the first DC component is nondifferentiable and the subproblem cannot be solved exactly. Specifically, for the problem $\min \varphi = g - h$ with $g$ and $h$ strongly convex of modulus $\sigma > 0$ and $\varphi$ bounded below, the algorithm at each iteration chooses an $\varepsilon_k$-subgradient $w_k$ of $h$ at $x_k$, computes an approximate solution $y_k$ of the subproblem together with a subgradient $\xi_k$ of $g$ at $y_k$ satisfying the relative-error condition $\|w_k - \xi_k\| \le \theta \|y_k - x_k\|$ for fixed $\theta < \sigma/2$, and then performs a nonmonotone Armijo line search along $d_k = y_k - x_k$ with nonmonotonicity measure $\nu_k$. If the $\nu_k$ follow strategy (A2) (summable) or (A3) (eventually small relative to $\|d_k\|^2$) and the errors $\varepsilon_k$ are summable, then every accumulation point of $(x_k)$ is critical, meaning $\partial g(\bar{x}) \cap \partial h(\bar{x}) \ne \emptyset$ (Theorems 17, 18, and 21). The proof engine is the descent estimate $\varphi(y_k) \le \varphi(x_k) - (\sigma/2 - \theta)\|d_k\|^2 + \varepsilon_k$, which holds even though $d_k$ need not be a descent direction, and which turns the algorithm into a quasi-monotone process. The complexity bound is $\min_{0 \le k \le N-1} \|d_k\| \le C/\sqrt{N}$, with $C$ depending only on the initial gap, the sums of $\nu_k$ and $\varepsilon_k$, and $\sigma/2 - \theta$.

Load-bearing premise

The load-bearing premise is that at every iteration a solver can certify a pair $(y_k, \xi_k)$ satisfying the relative-error bound $\|w_k - \xi_k\| \le \theta \|y_k - x_k\|$ for the fixed tolerance $\theta < \sigma/2$; if no such certificate can be produced, the descent estimate and all downstream convergence and complexity claims no longer have ground to stand on.

Editorial extensions

If this is right

  • When both DC components are nondifferentiable and only approximate subproblem solves are available, the algorithm still provably converges to critical points, so inexact inner solvers can be used without losing the convergence guarantee.
  • The complexity estimate means that after about $O(1/\varepsilon^2)$ iterations, some step $\|d_k\|$ is below $\varepsilon$, giving a concrete stopping criterion for practical implementations.
  • Setting $\varepsilon_k = 0$ and $\theta = 0$ recovers the exact nonmonotone BDCA, so the analysis contains the existing nmBDCA theory as a special case.
  • The nonmonotonicity strategies (A1)–(A3) provide concrete, checkable rules for choosing $\nu_k$, including the cost-update scheme of [44], which makes the slack an algorithmic parameter rather than an abstract assumption.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • In practice the true bottleneck is not the line search but the relative-error certificate: for nonsmooth convex $h$, producing $w_k \in \partial_{\varepsilon_k} h(x_k)$ and then bounding its distance to some $\xi_k \in \partial g(y_k)$ requires solving a nearby convex problem, so an adaptive $\theta$ or a duality-gap surrogate for (12) would be a natural follow-up.
  • The convergence target is criticality, which is strictly weaker than the stationarity notion based on the limiting subdifferential (whose calculus rules would require plus-minus symmetry that $\partial \varphi$ lacks); the paper notes in Remark 3 that the symmetric subdifferential construction could be used to get stronger stationarity.
  • The same quasi-monotone accounting (descent estimate plus slack plus summable errors) could be transplanted to other DC variants such as proximal or inertial algorithms with approximate oracles, simply by swapping the descent estimate while keeping the slack/error bookkeeping.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The manuscript proposes an inexact nonmonotone boosted difference-of-convex algorithm (InmBDCA) for minimizing g−h where both g and h may be nondifferentiable but are strongly convex. At each iteration, the method chooses w_k in the ε_k-subdifferential of h at x_k and computes a pair (y_k, ξ_k) satisfying ξ_k ∈ ∂g(y_k) and ||w_k − ξ_k|| ≤ θ||y_k − x_k||; then it sets d_k = y_k − x_k and performs a nonmonotone Armijo-type linesearch. Under assumptions (H1)–(H2), with summable error parameters and with either summable or asymptotically controlled nonmonotonicity terms, the authors prove that every accumulation point of the infinite iterate sequence is a critical point in the sense ∂g ∩ ∂h ≠ ∅. They also provide O(1/sqrt(N)) iteration-complexity bounds on the minimal step norm. The numerical section reports two two-dimensional experiments verifying that approximate subproblem solutions obtained with fminsearch satisfy the relative-error inequalities used in the algorithm.

Significance. If the theoretical results are taken at face value, the paper gives a useful extension of BDCA/nmBDCA to an inexact setting with fully nonsmooth DC decompositions. The proof architecture is coherent: Proposition 13 provides a clean descent estimate, Proposition 16 correctly transfers d_k→0 into criticality via subdifferential closure, and Theorems 17, 18, and 21 establish the stated accumulation-point and complexity results under explicit assumptions. The authors also provide explicit bounds rather than only qualitative convergence, and the numerical section makes a step toward showing that the relative-error condition can actually be satisfied by black-box solvers. However, the paper as written has several mismatches between the algorithm's stated parameter ranges and the theorem hypotheses, an oracle formulation of the key inner step, a stopping-rule guarantee that is weaker than the abstract claims, and an unsupported numerical performance claim. These issues affect the precision and scope of the main claims, though the central derivation appears fixable.

major comments (4)
  1. [Abstract and Section 8] The abstract states that 'Numerical experiments show that InmBDCA outperforms both the nonsmooth BDCA (nmBDCA) and the monotone version of DCA in practical scenarios,' but Section 8 explicitly says 'we are not focused on analyzing the computational performance of the method compared to others' and only verifies that the relative-error inequalities (11)–(12) hold for two examples. The numerical performance claim is therefore unsupported by the reported experiments and must be corrected or replaced with an actual comparison.
  2. [Algorithm 2, Step 2, equations (11)–(12)] The entire convergence analysis depends on the existence, at every iteration, of a pair (y_k, ξ_k) satisfying the relative-error condition ||w_k − ξ_k|| ≤ θ||y_k − x_k|| with ξ_k ∈ ∂g(y_k). The paper proves existence via the exact solution of subproblem (7), but no implementable procedure or finite-time certificate is given for an inexact solver to obtain such a pair; the numerical section checks (11)–(12) only after the computation. Since Proposition 13 and Theorem 14 use (12) to derive the descent estimate, a finite-precision implementation that cannot certify (12) falls outside the stated guarantees. This should be stated explicitly as an oracle assumption, or an implementable inner-line-search/stopping rule with a verifiable certificate should be provided.
  3. [Algorithm 2, Step 3, and Theorems 14–18] The algorithm allows ν_k ≥ 0, and Theorems 17, 18, and 21 state assumptions only in terms of (A2) or (A3), which do not exclude ν_k = 0. However, Theorem 14 and Proposition 15, which justify well-definiteness of the linesearch and the descent estimate, explicitly assume ν_k > 0. If ν_k = 0 and d_k is not a descent direction for the nondifferentiable DC objective, the while loop in Step 3 need not terminate. The convergence and complexity theorems therefore do not cover the stated parameter range; either ν_k > 0 should be made a standing requirement, or the ν_k = 0 case must be analyzed separately.
  4. [Algorithm 2, Step 3, stopping rule] The stopping rule 'If d_k = 0 then STOP and return x_k' does not guarantee that x_k is critical for problem (5) when ε_k > 0; Proposition 13 only yields ε_k-criticality. A concrete instance is g(x) = x^2/2 + x^4/4, h(x) = x^2/2 + 2x, ε_0 = 2, x_0 = 0, for which all assumptions hold and Algorithm 2 returns the noncritical point 0. Moreover, the abstract's claim about accumulation points is an infinite-sequence statement and does not apply to a finite terminated run. The authors should either require ε_k = 0 at termination, or explicitly state that the finite output is an ε_K-critical point, and align the abstract and conclusion with that guarantee.
minor comments (6)
  1. [Proposition 13] The term 'ε_k-critical point' is used in the statement and proof but never defined in the paper; please add a formal definition and use it consistently throughout the stopping-rule discussion.
  2. [Remark 2] The remark claims that 'strict convexity of the objective function guarantees the uniqueness of minimizers; see Proposition 5,' but Proposition 5 applies to a differentiable strongly convex function, while the DC objective φ = g − h is not necessarily convex; the justification should be rephrased or removed.
  3. [Theorem 22] The statement of Theorem 22 writes 'ε_k ≤ ξ(ρ/2 − θ)||d_k||^2' while the proof uses σ/2 − θ; this appears to be a typo and should be corrected.
  4. [Theorem 22] The theorem statement has a factor (1 − ξ) in the denominator of the complexity bound, whereas the proof derives the bound with (1 − 2ξ); the displayed statement and the proof should be harmonized.
  5. [Algorithm 2 and Theorem 14] The initialization of the linesearch permits ¯λ_k = 0; with ¯λ_k = 0 the while loop never changes x_{k+1}, and the algorithm would not make progress. Either require ¯λ_k > 0 or explain why zero initial steps are admissible.
  6. [Section 8, Figures 1 and 2] The y-axis label '|| k−wk||' should presumably be '||ξ_k − w_k||', and the phrase 'MATLAB solves Example 5 inaccurately' is unclear; please clarify what 'inaccurately' means in that context.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: convergence results are proved from stated assumptions; the only self-citations are contextual and not load-bearing.

full rationale

The paper's central claims are derived from the standing assumptions (H1)-(H2), the inexactness conditions (11)-(12), the nonmonotonicity strategies (A2)/(A3), and summability of the error sequence (epsilon_k). Theorems 17, 18, and 21 use the descent estimate (25) from Proposition 15 to show sum ||d_k||^2 < infinity, then Proposition 16 converts lim ||d_k|| = 0 and lim epsilon_k = 0 into criticality via the closedness properties of subdifferentials (Propositions 2 and 7). No target conclusion is assumed at the outset. The relative-error condition (12) is an implementability oracle about the inner solver, not a restatement of the convergence conclusion. The stopping-rule issue raised in the skeptic note is a correctness nuance about termination: when d_k = 0 with epsilon_k > 0, Proposition 13 only certifies an epsilon_k-critical point, and this is explicitly stated in the paper rather than disguised as full criticality. That is a precision problem in the abstract's wording, not circularity. Self-citations to [18] for the nonmonotone strategies are present, but the paper reproves the relevant estimates directly (e.g., Remark 7 and Theorems 17-22), so the self-citation is not load-bearing. Numerical examples are taken from prior work [3,5] as benchmarks, which is normal external validation rather than circular reasoning.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central convergence theorem rests on standard convex analysis facts, the problem assumptions (H1)-(H2), and an implementability oracle for the inexact subproblem. No parameters are fitted to data and no new entities are introduced.

assumptions (4)
  • domain assumption (H1) g and h are strongly convex with common modulus sigma > 0
    Invoked at the start of Section 3 and used throughout, e.g., Propositions 13 and Theorem 14, to obtain strong monotonicity and descent estimates. The authors argue it is nonrestrictive by adding the same quadratic to both components.
  • domain assumption (H2) phi = g - h is bounded below
    Used to sum descent inequalities and derive complexity bounds, e.g., Theorems 17 and 21.
  • ad hoc to paper The inner solver can produce (y_k, xi_k) satisfying (11)-(12) at every iteration
    Algorithm 2 Step 2 assumes this oracle; it is not proven to be satisfiable by any given black-box solver, only that the exact solution of (7) with xi_k = w_k satisfies it.
  • standard math Standard convex analysis facts: nonempty compact subdifferentials, epsilon-subdifferential closure, and strong convexity characterizations (Propositions 1-7)
    Background results cited from textbooks [9,22,33,34].

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Cite this review

Pith. "Pith review of An Inexact Boosted Difference of Convex Algorithm for Nondifferentiable Functions." pith.science (2026). https://pith.science/paper/5K2AJ7QB

@misc{pith2026241205697,
  author       = {Pith},
  title        = {Pith review of: An Inexact Boosted Difference of Convex Algorithm for Nondifferentiable Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5K2AJ7QB}},
  note         = {Machine review of arXiv:2412.05697}
}
read the original abstract

In this paper, we introduce an inexact approach to the Boosted Difference of Convex Functions Algorithm (BDCA) for solving nonconvex and nondifferentiable problems involving the difference of two convex functions (DC functions). Specifically, when the first DC component is differentiable and the second may be nondifferentiable, BDCA utilizes the solution from the subproblem of the DC Algorithm (DCA) to define a descent direction for the objective function. A monotone linesearch is then performed to find a new point that improves the objective function relative to the subproblem solution. This approach enhances the performance of DCA. However, if the first DC component is nondifferentiable, the BDCA direction may become an ascent direction, rendering the monotone linesearch ineffective. To address this, we propose an Inexact nonmonotone Boosted Difference of Convex Algorithm (InmBDCA). This algorithm incorporates two main features of inexactness: First, the subproblem therein is solved approximately allowing us for a controlled relative error tolerance in defining the linesearch direction. Second, an inexact nonmonotone linesearch scheme is used to determine the step size for the next iteration. Under suitable assumptions, we demonstrate that InmBDCA is well-defined, with any accumulation point of the sequence generated by InmBDCA being a critical point of the problem. We also provide iteration-complexity bounds for the algorithm. Numerical experiments show that InmBDCA outperforms both the nonsmooth BDCA (nmBDCA) and the monotone version of DCA in practical scenarios.

Figures

Figures reproduced from arXiv: 2412.05697 by the authors.

Figure 1
Figure 1. Example 5 starting from x 0 = (6.2945, 8.1158). In Example 6, the constants in the design of Algorithm 2 are set as ρ = 0.6, β = 0.1, λ¯ = 1, and θ = 0.2. The sequence of parameters (νk)k∈N are chosen as νk = 0.01 ∥d k∥ 2 k+1 for all k ∈ N. Example 6 ([5, Example 3.4]). Let ϕ : R 2 → R given by ϕ(x, y) := 1 2 (x 2 + y 2 ) + |x| + |y| − 5 2 x. We have the following DC decomposition of the cost function: ϕ(x, y) = g(x… view at source ↗
Figure 2
Figure 2. Example 6 starting from x 0 = (−4.4615, −9.0766). 20 [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.