Mathematical Programming

, Volume 165, Issue 2, pp 471–507 | Cite as

From error bounds to the complexity of first-order descent methods for convex functions

  • Jérôme Bolte
  • Trong Phong Nguyen
  • Juan Peypouquet
  • Bruce W. Suter
Full Length Paper Series A


This paper shows that error bounds can be used as effective tools for deriving complexity results for first-order descent methods in convex minimization. In a first stage, this objective led us to revisit the interplay between error bounds and the Kurdyka-Łojasiewicz (KL) inequality. One can show the equivalence between the two concepts for convex functions having a moderately flat profile near the set of minimizers (as those of functions with Hölderian growth). A counterexample shows that the equivalence is no longer true for extremely flat functions. This fact reveals the relevance of an approach based on KL inequality. In a second stage, we show how KL inequalities can in turn be employed to compute new complexity bounds for a wealth of descent methods for convex problems. Our approach is completely original and makes use of a one-dimensional worst-case proximal sequence in the spirit of the famous majorant method of Kantorovich. Our result applies to a very simple abstract scheme that covers a wide class of descent methods. As a byproduct of our study, we also provide new results for the globalization of KL inequalities in the convex framework. Our main results inaugurate a simple method: derive an error bound, compute the desingularizing function whenever possible, identify essential constants in the descent method and finally compute the complexity using the one-dimensional worst case proximal sequence. Our method is illustrated through projection methods for feasibility problems, and through the famous iterative shrinkage thresholding algorithm (ISTA), for which we show that the complexity bound is of the form \(O(q^{k})\) where the constituents of the bound only depend on error bound constants obtained for an arbitrary least squares objective with \(\ell ^1\) regularization.


Error bounds Convex minimization Forward-backward method KL inequality Complexity of first-order methods LASSO Compressed sensing 

Mathematics Subject Classification

90C06 90C25 90C60 65K05 



The authors would like to thank Amir Beck, Patrick Combettes, Édouard Pauwels, Marc Teboulle and the anonymous referee for very useful comments.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  • Jérôme Bolte
    • 1
    • 4
  • Trong Phong Nguyen
    • 1
  • Juan Peypouquet
    • 2
  • Bruce W. Suter
    • 3
  1. 1.Toulouse School of EconomicsUniversité Toulouse CapitoleToulouseFrance
  2. 2.Departamento de Matemática & AM2VUniversidad Técnica Federico Santa MaríaValparaísoChile
  3. 3.Air Force. Research Laboratory/RITBRomeUSA
  4. 4.The Research Institute of Time StudiesYamaguchi UniversityYamaguchiJapan

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