Nonmonotone higher-order Taylor approximation methods for composite problems

Yassine Nabou

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Nonmonotone higher-order Taylor approximation methods for composite problems

Yassine Nabou

Optimization and Control · Submitted 3 Mar 2025

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Our method achieves the same nice global and rate of convergence properties as traditional higher-order methods while eliminating the need for global Lipschitz continuity assumptions, strict descent condition, or explicit boundedness of the iterates.

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We study composite optimization problems in which the smooth part of the objective function is $ p $-times continuously differentiable, where $ p \geq 1 $ is an integer. Higher-order methods are known to be effective for solving such problems, as they speed up convergence rates. These methods often require, or implicitly ensure, a monotonic decrease in the objective function across iterations. Maintaining this monotonicity typically requires that the $ p $-th derivative of the smooth part of the objective function is globally Lipschitz or that the generated iterates remain bounded. In this paper, we propose nonmonotone higher-order Taylor approximation (NHOTA) method for composite problems. Our method achieves the same nice global and rate of convergence properties as traditional higher-order methods while eliminating the need for global Lipschitz continuity assumptions, strict descent condition, or explicit boundedness of the iterates. Specifically, for nonconvex composite problems, we derive global convergence rate to a stationary point of order $ \mathcal{O}(k^{-\frac{p}{p+1}}) $, where $ k $ is the iteration counter. Moreover, when the objective function satisfies the Kurdyka-{\L}ojasiewicz (KL) property, we obtain improved rates that depend on the KL parameter. Furthermore, for convex composite problems, our method achieves sublinear convergence rate of order $ \mathcal{O}(k^{-p}) $ in function values. Finally, preliminary numerical experiments on nonconvex phase retrieval problems highlight the promising performance of the proposed approach.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study composite optimization problems in which the smooth part of the objective function is $ p $-times continuously differentiable, where $ p \geq 1 $ is an integer.	background 0.98	0.26	0.02	asserted 0.97
s1	Higher-order methods are known to be effective for solving such problems, as they speed up convergence rates.	background 1.00	0.04	0.03	asserted 0.91
s2	These methods often require, or implicitly ensure, a monotonic decrease in the objective function across iterations.	background 0.98	0.05	0.03	hedged possibility 0.53
s3	Maintaining this monotonicity typically requires that the $ p $-th derivative of the smooth part of the objective function is globally Lipschitz or that the generated iterates remain bounded.	background 0.90	0.06	0.03	asserted 0.80
s4	In this paper, we propose nonmonotone higher-order Taylor approximation (NHOTA) method for composite problems.	contribution 0.91	0.92	0.16	asserted 1.00
s5	Our method achieves the same nice global and rate of convergence properties as traditional higher-order methods while eliminating the need for global Lipschitz continuity assumptions, strict descent condition, or explicit boundedness of the iterates.	result 0.56	0.93	0.82	asserted 1.00
s6	Specifically, for nonconvex composite problems, we derive global convergence rate to a stationary point of order $ \mathcal{O}(k^{-\frac{p}{p+1}}) $, where $ k $ is the iteration counter.	result 1.00	0.96	0.20	asserted 1.00
s7	Moreover, when the objective function satisfies the Kurdyka-{\L}ojasiewicz (KL) property, we obtain improved rates that depend on the KL parameter.	result 0.99	0.93	0.08	asserted 1.00
s8	Furthermore, for convex composite problems, our method achieves sublinear convergence rate of order $ \mathcal{O}(k^{-p}) $ in function values.	result 1.00	0.94	0.12	asserted 1.00
s9	Finally, preliminary numerical experiments on nonconvex phase retrieval problems highlight the promising performance of the proposed approach.	result 0.90	0.95	0.06	asserted 0.56

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Original abstract

We study composite optimization problems in which the smooth part of the objective function is $ p $-times continuously differentiable, where $ p \geq 1 $ is an integer. Higher-order methods are known to be effective for solving such problems, as they speed up convergence rates. These methods often require, or implicitly ensure, a monotonic decrease in the objective function across iterations. Maintaining this monotonicity typically requires that the $ p $-th derivative of the smooth part of the objective function is globally Lipschitz or that the generated iterates remain bounded. In this paper, we propose nonmonotone higher-order Taylor approximation (NHOTA) method for composite problems. Our method achieves the same nice global and rate of convergence properties as traditional higher-order methods while eliminating the need for global Lipschitz continuity assumptions, strict descent condition, or explicit boundedness of the iterates. Specifically, for nonconvex composite problems, we derive global convergence rate to a stationary point of order $ \mathcal{O}(k^{-\frac{p}{p+1}}) $, where $ k $ is the iteration counter. Moreover, when the objective function satisfies the Kurdyka-{\L}ojasiewicz (KL) property, we obtain improved rates that depend on the KL parameter. Furthermore, for convex composite problems, our method achieves sublinear convergence rate of order $ \mathcal{O}(k^{-p}) $ in function values. Finally, preliminary numerical experiments on nonconvex phase retrieval problems highlight the promising performance of the proposed approach.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Builds on 2 papers

Building on the works of [ref] , our method is designed to enforce the following condition, which plays a crucial role in ensuring convergence [equation]

Convergence of ZH-type nonmonotone descent method for Kurdyka-{\L}ojasiewicz optimization problems · Nonmonotone higher-order Tensor method · 0.83
Building on the works of [ref] , our method is designed to enforce the following condition, which plays a crucial role in ensuring convergence [equation]

Convergence of Nonmonotone Proximal Gradient Methods under the Kurdyka-Lojasiewicz Property without a Global Lipschitz Assumption · Nonmonotone higher-order Tensor method · 0.83

Uses the method or tool of 1 paper

Following [ref] , $R(x_k)$ is computed as a convex combination of the previous reference value $ℛ_k-1$ and the new function value $f(x_k)$.

Convergence of Nonmonotone Proximal Gradient Methods under the Kurdyka-Lojasiewicz Property without a Global Lipschitz Assumption · Nonmonotone higher-order Tensor method · 0.98

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