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A Note on Complexity for Two Classes of Structured Non-Smooth Non-Convex Compositional Optimization

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Pith's one-line read This note proves that for nonsmooth nonconvex compositional optimization with an easy-prox outer function, a smoothing gradient method finds a stochastic $(\delta,\epsilon)$-stationary point in $O(1/(\delta\epsilon^2))$ iterations, and…

desk verdict Useful new rates for two structured non-smooth compositional problems, but Theorem 4.7's weak-convexity constant needs correction before the stated complexity is supported. read the letter →

arxiv 2411.14342 v1 pith:CPI4LZG6 submitted 2024-11-21 math.OC

Yao Yao , Qihang Lin , Tianbao Yang This is my paper · ORCID
classification math.OC MSC 90C2690C3065K05
keywords compositionaloptimizationnon-smoothnon-convexsmoothingmethodprox-lineardifference-of-convexGoldsteinstationarypointiterationcomplexityMoreauenvelope
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

This note establishes iteration-complexity guarantees for first-order methods on compositional optimization problems $\min_x h(g(x))$ where the inner map $g$ is smooth and the outer function $h$ is Lipschitz, nonsmooth, and nonconvex but has one of two exploitable structures. For an outer function with an easily solvable proximal mapping, the smoothing compositional gradient method (SCGM) returns a stochastic $(\delta,\epsilon)$-stationary point defined through the chain rule and a Goldstein-type enlargement of $\partial h$ in $O(1/(\delta\epsilon^2))$ iterations, improving the known $O(1/(\delta\epsilon^3))$ bound for generic $(\delta,\epsilon)$-Goldstein stationarity by a factor of $1/\epsilon$. For an outer function of the difference-of-convex form $h_2(g_2(x))-h_1(g_1(x))$ with easy prox-linear subproblems, the prox-linear approximate gradient method (PAGM) finds a stochastic nearly $2\epsilon$-critical point in $O(1/\epsilon^2)$ iterations. These results show that structure-aware stationarity notions let nonsmooth nonconvex compositions be solved at complexities comparable to smooth nonconvex optimization.

What carries the argument

The Moreau envelope is the machinery common to both algorithms. In the first setting, $h_\mu(g(x)) = \frac{1}{2\mu}\|g(x)\|^2 - G(x)$ with $G(x)=\max_y\{ \frac{1}{\mu}y^\top g(x)-\frac{1}{2\mu}\|y\|^2 - h(y)\}$, a DC decomposition; the proximal point $v(g(x_k))\in\mathrm{prox}_{\mu h}(g(x_k))$ gives the subgradient $\frac{1}{\mu}\nabla g(x_k)^\top(g(x_k)-v(g(x_k)))$, and the smoothness and weak-convexity of the two pieces produce the descent inequality. In the second setting, the Moreau envelopes $f_{i,\mu}(z)$ smooth each DC component, and Proposition 5.7 shows the prox-linear subproblem (20) contracts the error $\|x_i^{k+1}-x_i^*(z_k)\|$ by a factor $1-tc$, which drives the $O(1/\epsilon^2)$ bound once the gradient step on $z_k$ is chosen suitably.

What would settle it

Take a scalar example with $h(y)=|y|$ and $g(x)=\sin(10x)$ on a bounded interval, and compute the finite-difference second quotient of $G(x)=\max_y\{ y g(x)-\frac{1}{2}y^2-|y|\}$; the lower bound will scale with $\beta=100$ rather than $L_g=10$, showing inequality (11) cannot hold with the claimed $C_v L_g/\mu$ modulus.

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Extended reading notes

Core claim

The central claim is that the compositional structure of (1), rather than an obstacle, is an asset for nonsmooth nonconvex optimization. Theorem 4.7 shows that under Assumption 4.1 (Lipschitz outer function with easy prox, smooth inner map with bounded range), SCGM—a subgradient descent on the Moreau envelope $h_\mu(g(x))$—finds a stochastic chain-rule $(\delta,\epsilon)$-stationary point in $O(1/(\delta\epsilon^2))$ iterations. Theorem 5.8 shows that under Assumption 5.1 (DC outer function with easy prox-linear subproblems), PAGM finds a stochastic nearly $2\epsilon$-critical point of (2) in $O(1/\epsilon^2)$ iterations. The proofs hinge on smoothing via Moreau envelopes and on measuring stationarity through the chain-rule subdifferential $\nabla g(x)^\top \partial h(g(x)+\delta B)$, a weaker target than the full Goldstein subdifferential yet still meaningful as an approximate KKT condition for the constrained reformulation.

Load-bearing premise

The proof of Theorem 4.7 assumes that the term $G(x)$ in the Moreau-envelope decomposition (3) is $C_v L_g/\mu$-weakly convex; the natural uniform weak-convexity modulus of this max function is $\beta C_v/\mu$, so the descent inequality (11) is only justified if $\beta \le L_g$, a condition not stated in Assumption 4.1.

Editorial extensions

If this is right

  • For the first structure, SCGM finds a chain-rule $(\delta,\epsilon)$-stationary point in $O(1/(\delta\epsilon^2))$ iterations, a factor $1/\epsilon$ better than the $O(1/(\delta\epsilon^3))$ rate known for generic $(\delta,\epsilon)$-Goldstein stationarity.
  • For compositional DC problems, PAGM finds a nearly $2\epsilon$-critical point in $O(1/\epsilon^2)$ iterations, improving on the $O(1/\epsilon^4)$ rates of Moreau-envelope smoothing methods that solve subproblems inexactly.
  • Because each iteration uses only one proximal evaluation of the outer function and one gradient evaluation of the inner map, the iteration counts in Theorems 4.7 and 5.8 translate directly into first-order oracle complexity.
  • The stationarity targets carry algorithmic meaning: a chain-rule $(\delta,\epsilon)$-stationary point is an $\epsilon$-KKT point of the constrained reformulation $h(v)$ subject to $g(x)=v$, and a nearly $\epsilon$-critical point is a nearly $2\epsilon$-KKT point of the consensus reformulation of (2).

Reading between the lines

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

  • A natural extension is stochastic or variance-reduced versions of both algorithms; the contraction structure in Proposition 5.7 suggests sample-average or SVRG-style updates could preserve the $O(1/\epsilon^2)$ rate with finite-sample oracles under bounded-variance assumptions.
  • The chain-rule stationarity notion used in Theorem 4.7 is not the Goldstein condition; if the downstream task only needs KKT conditions of the lifted problem (9), then the cheaper $O(1/(\delta\epsilon^2))$ rate is the right measure, and optimal rates for structured nonsmooth problems should be defined relative to the structure, not the worst-case function class.
  • The DC structure visible in (3) connects the two settings: the same Moreau-envelope machinery that smooths an easy-prox outer function also underlies the DC prox-linear analysis, so one could expect a unified treatment or algorithms that interpolate between the two assumptions.
  • For DC compositional problems with strongly concave conjugates, the PAGM rate suggests that the bottleneck is the subproblem solver rather than the outer geometry; using inexact or stochastic prox-linear solves would likely trade the $O(1/\epsilon^2)$ rate for higher complexity, so implementing exact solves for structured $h_i$ is the practical key.
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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

2 major / 4 minor

Summary. The paper studies the compositional optimization problem min_x h(g(x)) with Lipschitz, nonsmooth, and generally nonconvex h, and smooth g. In Section 4, assuming the proximal mapping of h is easy to compute, the authors propose a smoothing compositional gradient method (Algorithm 1) and prove that it finds a stochastic chain-rule (δ, ε)-stationary point in O(1/(δε^2)) iterations (Theorem 4.7). In Section 5, assuming h is a difference of two convex Lipschitz functions composed with smooth maps, the authors propose a prox-linear approximate gradient method (Algorithm 2) and prove that it finds a stochastic nearly 2ε-critical point in O(1/ε^2) iterations (Theorem 5.8). The analysis is based on Moreau-envelope smoothing, weak convexity of the smoothed objectives, and standard descent arguments.

Significance. If fully established, the claimed rates would improve on the O(1/(δε^3)) typical for Goldstein stationarity in nonsmooth nonconvex optimization and on the O(1/ε^4) of smoothing DC methods under the stated compositional structure. The paper is largely self-contained: the chain-rule stationarity notion is clearly motivated by the KKT conditions of the lifted constrained problem, Lemma 4.3 and Proposition 5.7 supply the needed technical bounds, and the proof skeletons are standard. The two structural settings are distinct and the algorithms are simple and practically motivated. However, two proof points are not currently supported as written; both are local and fixable without changing the claimed rates.

major comments (2)
  1. [§4, Theorem 4.7, Eq. (11)] The proof asserts that G(x) in (3) is C_v L_g/µ-weakly convex. Under Assumption 4.1 the natural uniform modulus for each active function is β‖y‖/µ, because the gradient of y^T g(x) with respect to x is ∇g(x)^T y, whose Lipschitz constant is at most β‖y‖ and not L_g‖y‖. The constant β is not controlled by L_g (for example, g(x)=sin(kx)/k in one dimension has L_g=1 and β=k), so inequality (11) and the definition C := L_g^2 + βC_g + C_v L_g in Theorem 4.7 are not justified unless Assumption 4.1 is augmented with β ≤ L_g, which the paper does not state. The O(1/(δε^2)) rate survives if C is corrected to L_g^2 + βC_g + βC_v (or a max{L_g, β} variant), but the theorem as stated is not proven.
  2. [§5, Theorem 5.8, Eq. (31)] Equation (31) does not follow from the preceding combined inequality. Summing that inequality after multiplying by 32µ^2/γ gives Σ_{k=1}^K(Δ_k+δ_k) + 7Δ_K - 7Δ_0 + (49 - 147/θ^2)δ_0 + (147/θ^2 - 48)δ_K ≤ (32µ^2/γ)(f_µ(z_1)-f_µ(z_{K+1})). Since θ=tc<1, the terms -7Δ_K and -(147/θ^2-48)δ_K are nonpositive and can be dropped, leaving an upper bound with 7Δ_0 + (147/θ^2-49)δ_0; there is no 49Δ_K term. Consequently the subsequent bound Δ_K ≤ 8µC_1 does not enter the estimate, and the displayed K containing the 1568µC_1 term is not supported by the derivation. The proof and the constants in K should be corrected; the O(1/ε^2) rate remains plausible after this correction.
minor comments (4)
  1. [§5, after Eq. (2)] The sentence defining f1 and f2 reads "We define f1(x) := h1(g1(x)) and f1(x) := h2(g2(x))", but the second definition should be f2(x) := h2(g2(x)).
  2. [§5, proof of Theorem 5.8] The constants C1 and C are introduced in the proof by "We assume that there exists..." without being stated in Assumption 5.1 or in the theorem statement; they are finite under the assumptions because g_i is bounded, h_i is Lipschitz, and f_µ is bounded below, but this should be stated explicitly so that the explicit K in the theorem is well defined.
  3. [§5, Proposition 5.7 and Theorem 5.8] The condition t^{-1} ≥ µ^{-1}+ρ implies tc < 1, which is used to ensure 1-t^2c^2 > 0 in the definition of γ and to justify sign choices in the proof of Theorem 5.8; this restriction should be stated explicitly where the constants are introduced.
  4. [§1 and references] In the introduction, "similar to but different form that in [44,19]" should be "similar to but different from that in [44,19]"; also reference [29] is incomplete, as it lacks venue and year information.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorems 4.7 and 5.8 are proved from self-contained descent arguments; self-citations are contextual and not load-bearing.

full rationale

The claimed complexity results are not equivalent to their inputs by construction. For Theorem 4.7, the proof builds a descent inequality for the smoothed objective h_mu(g(x)) from smoothness of (1/(2 mu))||g||^2 in inequality (10) and from weak convexity of G(x) in inequality (11); steps (12)-(15) are a standard descent argument, and the conversion to a stochastic chain-rule (delta,epsilon)-stationary point uses only Lemma 4.2, which identifies (g-v)/mu as an element of partial h(v). For Theorem 5.8, the analysis uses independent, externally sourced ingredients (Lemmas 5.2, 5.5, and 5.6 from [13] and [43], and Proposition 13.37 of [34]) together with a self-contained contraction estimate in Proposition 5.7; no fitted quantity is later renamed as a prediction. The self-citations [17], [40], and [42] are used to motivate the Moreau-envelope smoothing framework and to compare iteration bounds, not as load-bearing proof steps. I therefore find no circular step. I do flag a non-circular correctness limitation: in equation (11), the natural uniform weak-convexity modulus of G(x)=max_y((1/mu)y^T g(x)-...) is beta C_v/mu, not C_v L_g/mu, because the y in prox_h(g(x)) is controlled by C_v and the Jacobian of g is beta-Lipschitz; as written, the stated constant C requires beta <= L_g or a corrected modulus, though the O(1/(delta epsilon^2)) order is unaffected. This is a gap in the stated constants, not a circular dependence.

Assumptions & free parameters 6 free parameters · 8 assumptions · 0 invented entities

The paper's central results depend on standard variational analysis results and the structural assumptions stated in Assumptions 4.1 and 5.1. The algorithm parameters (µ, γ, t, α) are chosen by the authors to make the proofs work, not fitted to data. No new entities are postulated.

free parameters (6)
  • µ (smoothing parameter, Section 4) = µ = δ/(2 L_h)
    Chosen to balance approximation error and gradient norm; appears in iteration count K = 16 L_h C ∆/(δ ε²).
  • γ (step size, Section 4) = γ = 2C/µ
    Set to cancel the quadratic term in the descent inequality (15).
  • µ (smoothing parameter, Section 5) = µ^{-1} > max{1, ρ}
    Chosen so the Moreau envelopes are smooth and the contraction factor in Proposition 5.7 is positive.
  • t (prox-linear step size, Section 5) = t^{-1} ≥ µ^{-1} + ρ
    Required for the contraction property in Proposition 5.7.
  • γ (step size, Algorithm 2) = γ = min{1/(4 L_µ), sqrt(t^3 c^4 µ^3 / (48(1 - t^2 c^2)))}
    Chosen to ensure the Lyapunov inequality (27) and the cross-term bounds in the proof of Theorem 5.8.
  • α (auxiliary constant in proof) = 7
    Hand-picked in the proof of Theorem 5.8 to simplify the Young's inequality coefficients.
assumptions (8)
  • standard math Chain rule for Fréchet and limiting subdifferentials (Lemma 4.4)
    Used to define the chain-rule stationarity target; follows from Rockafellar-Wets Theorem 10.6.
  • standard math Properties of Moreau envelope and proximal mapping for possibly nonconvex h (Lemma 4.2)
    Gives (g(x)-v)/µ ∈ ∂h(v) for v ∈ prox_µh(g(x)); standard variational analysis.
  • standard math Weak convexity of composite f_i = h_i∘g_i with modulus ρ = Lβ (Lemma 5.2)
    Quoted from Drusvyatskiy-Paquette [13]; used to construct smooth Moreau envelopes of f1 and f2.
  • standard math Lipschitz continuity of Moreau envelope proximal points (Lemma 5.5)
    From Zhang-Luo [43]; bounds the drift of x*_i(z) between z_k and z_{k-1} in Theorem 5.8.
  • domain assumption Assumption 4.1: h Lipschitz, lower bounded, easy prox; g C1 with β-Lipschitz Jacobian and bounded image
    Defines the problem class for the first result.
  • domain assumption Assumption 5.1: h_i convex Lipschitz with easy prox-linear subproblems; g_i smooth
    Defines the DC compositional class for the second result.
  • ad hoc to paper Existence of bounded constants C1 and C (introduced in proof of Theorem 5.8)
    Used to bound ∆_K and the objective decrease; these are not stated in Assumption 5.1 but follow from boundedness of g_i and Lipschitz h_i.
  • ad hoc to paper Implicit condition β ≤ L_g (or corrected constant) for weak convexity of G in (11)
    The proof of Theorem 4.7 uses C_v L_g/µ as the weak convexity modulus of G; the natural modulus is β C_v/µ, so the inequality is valid as written only if β ≤ L_g.

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Pith. "Pith review of A Note on Complexity for Two Classes of Structured Non-Smooth Non-Convex Compositional Optimization." pith.science (2026). https://pith.science/paper/CPI4LZG6

@misc{pith2026241114342,
  author       = {Pith},
  title        = {Pith review of: A Note on Complexity for Two Classes of Structured Non-Smooth Non-Convex Compositional Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CPI4LZG6}},
  note         = {Machine review of arXiv:2411.14342}
}
abstract

This note studies numerical methods for solving compositional optimization problems, where the inner function is smooth, and the outer function is Lipschitz continuous, non-smooth, and non-convex but exhibits one of two special structures that enable the design of efficient first-order methods. In the first structure, the outer function allows for an easily solvable proximal mapping. We demonstrate that, in this case, a smoothing compositional gradient method can find a $(\delta,\epsilon)$-stationary point--specifically defined for compositional optimization--in $O(1/(\delta \epsilon^2))$ iterations. In the second structure, the outer function is expressed as a difference-of-convex function, where each convex component is simple enough to allow an efficiently solvable proximal linear subproblem. In this case, we show that a prox-linear method can find a nearly ${\epsilon}$-critical point in $O(1/\epsilon^2)$ iterations.

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Reference graph

Works this paper leans on

46 extracted references · 45 canonical work pages

  1. [1]

    O n the rate of convergence of the difference-of-convex algorithm (dca)

    Hadi Abbaszadehpeivasti, Etienne de Klerk, and Moslem Zamani. O n the rate of convergence of the difference-of-convex algorithm (dca). Journal of Optimization Theory and Applications , 202(1):475–496, 2024

    work page 2024

  2. [2]

    Difference-of-convex le arning: directional stationarity, optimality, and sparsity

    Miju Ahn, Jong-Shi Pang, and Jack Xin. Difference-of-convex le arning: directional stationarity, optimality, and sparsity. SIAM Journal on Optimization , 27(3):1637–1665, 2017

    work page 2017

  3. [3]

    A four-o perator splitting algorithm for nonconvex and nonsmooth optimization, 2024

    Jan Harold Alcantara, Ching pei Lee, and Akiko Takeda. A four-o perator splitting algorithm for nonconvex and nonsmooth optimization, 2024

    work page 2024

  4. [4]

    A new d ecomposition method for multiuser dc-programming and its applications

    Alberth Alvarado, Gesualdo Scutari, and Jong-Shi Pang. A new d ecomposition method for multiuser dc-programming and its applications. IEEE Transactions on Signal Processing , 62(11):2984–2998, 2014

    work page 2014

  5. [5]

    Stochastic a pproximations and differential inclusions

    Michel Bena¨ ım, Josef Hofbauer, and Sylvain Sorin. Stochastic a pproximations and differential inclusions. SIAM Journal on Control and Optimization , 44(1):328–348, 2005

    work page 2005

  6. [6]

    Conservative set valued fi elds, automatic differentiation, stochastic gradient method and deep learning, 2020

    J´ erˆ ome Bolte and Edouard Pauwels. Conservative set valued fi elds, automatic differentiation, stochastic gradient method and deep learning, 2020

    work page 2020

  7. [7]

    Nonconvex lagrangian-based optimization: Monitoring schemes and global convergence.Mathematics of operations research, 43(4):1210–1232, 2018

    J´ erˆ ome Bolte, Shoham Sabach, and Marc Teboulle. Nonconvex lagrangian-based optimization: Monitoring schemes and global convergence.Mathematics of operations research, 43(4):1210–1232, 2018

    work page 2018

  8. [8]

    A unifying fram ework of high-dimensional sparse estimation with difference-of-convex (dc) regularizations

    Shanshan Cao, Xiaoming Huo, and Jong-Shi Pang. A unifying fram ework of high-dimensional sparse estimation with difference-of-convex (dc) regularizations. Statistical Science, 37(3):411–424, 2022

    work page 2022

Show all 46 references
  1. [9]

    Composite differe nce-max programs for modern statistical estimation problems

    Ying Cui, Jong-Shi Pang, and Bodhisattva Sen. Composite differe nce-max programs for modern statistical estimation problems. SIAM Journal on Optimization , 28(4):3344–3374, 2018

    2018

  2. [10]

    Stochastic model-based minimization of weakly convex functions

    Damek Davis and Dmitriy Drusvyatskiy. Stochastic model-based minimization of weakly convex functions. SIAM Journal on Optimization , 29(1):207–239, 2019

    2019

  3. [11]

    Damek Davis, Dmitriy Drusvyatskiy, Sham Kakade, and Jason D. Lee. Stochastic subgradient method converges on tame functions. Foundations of Computational Mathematics , 20(1):119–154, Feb 2020

    2020

  4. [12]

    Drusvyatskiy, A

    D. Drusvyatskiy, A. D. Ioffe, and A. S. Lewis. Nonsmooth optim ization using taylor-like models: error bounds, convergence, and termination criteria. Mathematical Programming, 185(1):357–383, Jan 2021. 13

    2021

  5. [13]

    Drusvyatskiy and C

    D. Drusvyatskiy and C. Paquette. Efficiency of minimizing compos itions of convex functions and smooth maps. Mathematical Programming, 178(1):503–558, Nov 2019

    2019

  6. [14]

    Dmitriy Drusvyatskiy and Adrian S. Lewis. Error bounds, quadr atic growth, and linear conver- gence of proximal methods. Math. Oper. Res. , 43(3):919–948, August 2018

    2018

  7. [15]

    Duchi and Feng Ruan

    John C. Duchi and Feng Ruan. Stochastic methods for compos ite and weakly convex optimization problems. SIAM Journal on Optimization , 28(4):3229–3259, 2018

    2018

  8. [16]

    A. A. Goldstein. Optimization of lipschitz continuous functions. Mathematical Programming, 13(1):14–22, Dec 1977

    1977

  9. [17]

    Single-loop st ochastic algorithms for differ- ence of max-structured weakly convex functions

    Quanqi Hu, Qi Qi, Zhaosong Lu, and Tianbao Yang. Single-loop st ochastic algorithms for differ- ence of max-structured weakly convex functions. In Advances in Neural Information Processing Systems 37: Annual Conference on Neural Information Proces sing Systems 2024, NeurIPS 2024 , 2024

    2024

  10. [18]

    Non-smooth weakly-c onvex finite-sum coupled compositional optimization

    Quanqi Hu, Dixian Zhu, and Tianbao Yang. Non-smooth weakly-c onvex finite-sum coupled compositional optimization. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine, editors, Advances in Neural Information Processing Systems , volume 36, pages 5348–

  11. [19]

    Determin- istic nonsmooth nonconvex optimization

    Michael Jordan, Guy Kornowski, Tianyi Lin, Ohad Shamir, and Ma nolis Zampetakis. Determin- istic nonsmooth nonconvex optimization. In The Thirty Sixth Annual Conference on Learning Theory, pages 4570–4597. PMLR, 2023

    2023

  12. [20]

    Krzysztof C. Kiwiel. Convergence of the gradient sampling algor ithm for nonsmooth nonconvex optimization. SIAM Journal on Optimization , 18(2):379–388, 2007

    2007

  13. [21]

    Oracle complexity in nonsmooth nonconvex optimization

    Guy Kornowski and Ohad Shamir. Oracle complexity in nonsmooth nonconvex optimization. Journal of Machine Learning Research , 23(314):1–44, 2022

    2022

  14. [22]

    Minimizing comp ositions of differences- of-convex functions with smooth mappings

    Hoai An Le Thi, Van Ngai Huynh, and Tao Pham Dinh. Minimizing comp ositions of differences- of-convex functions with smooth mappings. Math. Oper. Res. , 49(2):1140–1168, July 2023

    2023

  15. [23]

    Convergenc e analysis of difference- of-convex algorithm with subanalytic data

    Hoai An Le Thi, Van Ngai Huynh, and Tao Pham Dinh. Convergenc e analysis of difference- of-convex algorithm with subanalytic data. Journal of Optimization Theory and Applications , 179(1):103–126, 2018

    2018

  16. [24]

    A. S. Lewis and S. J. Wright. A proximal method for composite min imization. Mathematical Programming, 158(1):501–546, Jul 2016

    2016

  17. [25]

    Two-s tage stochastic programming with linearly bi-parameterized quadratic recourse

    Junyi Liu, Ying Cui, Jong-Shi Pang, and Suvrajeet Sen. Two-s tage stochastic programming with linearly bi-parameterized quadratic recourse. SIAM Journal on Optimization , 30(3):2530–2558, 2020

    2020

  18. [26]

    Risk-based robust statistical le arning by stochastic difference-of- convex value-function optimization

    Junyi Liu and Jong-Shi Pang. Risk-based robust statistical le arning by stochastic difference-of- convex value-function optimization. Operations Research, 71(2):397–414, 2023

    2023

  19. [27]

    A successive diffe rence-of-convex approxi- mation method for a class of nonconvex nonsmooth optimization pro blems

    Tianxiang Liu, Ting Kei Pong, and Akiko Takeda. A successive diffe rence-of-convex approxi- mation method for a class of nonconvex nonsmooth optimization pro blems. Math. Program., 176(1–2):339–367, July 2019

    2019

  20. [28]

    Analysis of nonsmooth stochastic approximation: the differential inclusion approach, 2018

    Szymon Majewski, B/suppress la˙ zej Miasojedow, and Eric Moulines. Analysis of nonsmooth stochastic approximation: the differential inclusion approach, 2018

    2018

  21. [29]

    A complete smooth regularization of dc optimiz ation problems

    Abdellatif Moudafi. A complete smooth regularization of dc optimiz ation problems

  22. [30]

    Lectures on Convex Optimization

    Yurii Nesterov. Lectures on Convex Optimization . Springer Publishing Company, Incorporated, 2nd edition, 2018

    2018

  23. [31]

    On th e pervasiveness of difference- convexity in optimization and statistics

    Maher Nouiehed, Jong-Shi Pang, and Meisam Razaviyayn. On th e pervasiveness of difference- convexity in optimization and statistics. Mathematical Programming, 174(1):195–222, 2019. 14

    2019

  24. [32]

    Com puting b-stationary points of nonsmooth dc programs

    Jong-Shi Pang, Meisam Razaviyayn, and Alberth Alvarado. Com puting b-stationary points of nonsmooth dc programs. Mathematics of Operations Research , 42(1):95–118, 2017

    2017

  25. [33]

    Decomposition methods for comput ing directional stationary solu- tions of a class of nonsmooth nonconvex optimization problems

    Jong-Shi Pang and Min Tao. Decomposition methods for comput ing directional stationary solu- tions of a class of nonsmooth nonconvex optimization problems. SIAM Journal on Optimization , 28(2):1640–1669, 2018

    2018

  26. [34]

    Tyrrell Rockafellar and Roger J.-B

    R. Tyrrell Rockafellar and Roger J.-B. Wets. Variational Analysis. Springer Verlag, Heidelberg, Berlin, New York, 1998

    1998

  27. [35]

    Algorithms for difference-of-con vex programs based on difference- of-moreau-envelopes smoothing

    Kaizhao Sun and Xu Andy Sun. Algorithms for difference-of-con vex programs based on difference- of-moreau-envelopes smoothing. INFORMS J. Optim. , 5:321–339, 2022

    2022

  28. [36]

    Convex analysis approach to dc pr ogramming: theory, algo- rithms and applications

    Pham Dinh Tao and LT Hoai An. Convex analysis approach to dc pr ogramming: theory, algo- rithms and applications. Acta mathematica vietnamica , 22(1):289–355, 1997

    1997

  29. [37]

    No dimension-free determinis tic algorithm computes approx- imate stationarities of lipschitzians

    Lai Tian and Anthony Man-Cho So. No dimension-free determinis tic algorithm computes approx- imate stationarities of lipschitzians. Mathematical Programming, pages 1–24, 2024

    2024

  30. [38]

    Alexr: An optimal single-loop algor ithm for convex finite-sum coupled compositional stochastic optimization, 2024

    Bokun Wang and Tianbao Yang. Alexr: An optimal single-loop algor ithm for convex finite-sum coupled compositional stochastic optimization, 2024

    2024

  31. [39]

    Complexity of finite-sum optimization with nonsmooth composite functions and non-lipschitz regularization

    Xiao Wang and Xiaojun Chen. Complexity of finite-sum optimization with nonsmooth composite functions and non-lipschitz regularization. SIAM Journal on Optimization , 34(3):2472–2502, 2024

    2024

  32. [40]

    Stochastic o ptimization for dc func- tions and non-smooth non-convex regularizers with non-asymptotic convergence

    Yi Xu, Qi Qi, Qihang Lin, Rong Jin, and Tianbao Yang. Stochastic o ptimization for dc func- tions and non-smooth non-convex regularizers with non-asymptotic convergence. In International Conference on Machine Learning , 2018

    2018

  33. [41]

    Proximal gradient method with extrapolation and line s earch for a class of non-convex and non-smooth problems

    Lei Yang. Proximal gradient method with extrapolation and line s earch for a class of non-convex and non-smooth problems. Journal of Optimization Theory and Applications , 200(1):68–103, Jan 2024

    2024

  34. [42]

    Large-scale optimization of partial AUC in a range of false positive rates

    Yao Yao, Qihang Lin, and Tianbao Yang. Large-scale optimization of partial AUC in a range of false positive rates. In Sanmi Koyejo, S. Mohamed, A. Agarwal, Da nielle Belgrave, K. Cho, and A. Oh, editors, Advances in Neural Information Processing Systems 35: Annu al Conference ...

    2022

  35. [43]

    A proximal alternating directionmethod of multiplier for linearly constrained nonconvex minimization

    Jiawei Zhang and Zhi-Quan Luo. A proximal alternating directionmethod of multiplier for linearly constrained nonconvex minimization. SIAM journal on optimization , 30(3):2272–2302, 2020

    2020

  36. [44]

    Complexity of finding stationary points of nonsmooth nonconvex functions

    Jingzhao Zhang, Hongzhou Lin, Stefanie Jegelka, Suvrit Sra, a nd Ali Jadbabaie. Complexity of finding stationary points of nonsmooth nonconvex functions. In Proceedings of the 37th Interna- tional Conference on Machine Learning , ICML’20. JMLR.org, 2020

    2020

  37. [45]

    Optimal algorithms for convex ne sted stochastic composite opti- mization

    Zhe Zhang and Guanghui Lan. Optimal algorithms for convex ne sted stochastic composite opti- mization. arXiv preprint arXiv:2011.10076 , 2020. 15

    2011 arXiv

  38. [5403]

    Curran Associates, Inc., 2023

    2023

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.