# Adaptive smoothing algorithms for nonsmooth composite convex minimization

## Author

Listed:
• Quoc Tran-Dinh

() (University of North Carolina at Chapel Hill (UNC))

## Abstract

Abstract We propose an adaptive smoothing algorithm based on Nesterov’s smoothing technique in Nesterov (Math Prog 103(1):127–152, 2005) for solving “fully” nonsmooth composite convex optimization problems. Our method combines both Nesterov’s accelerated proximal gradient scheme and a new homotopy strategy for smoothness parameter. By an appropriate choice of smoothing functions, we develop a new algorithm that has the $$\mathcal {O}\left( \frac{1}{\varepsilon }\right)$$ O 1 ε -worst-case iteration-complexity while preserves the same complexity-per-iteration as in Nesterov’s method and allows one to automatically update the smoothness parameter at each iteration. Then, we customize our algorithm to solve four special cases that cover various applications. We also specify our algorithm to solve constrained convex optimization problems and show its convergence guarantee on a primal sequence of iterates. We demonstrate our algorithm through three numerical examples and compare it with other related algorithms.

## Suggested Citation

• Quoc Tran-Dinh, 2017. "Adaptive smoothing algorithms for nonsmooth composite convex minimization," Computational Optimization and Applications, Springer, vol. 66(3), pages 425-451, April.
• Handle: RePEc:spr:coopap:v:66:y:2017:i:3:d:10.1007_s10589-016-9873-6
DOI: 10.1007/s10589-016-9873-6
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## References listed on IDEAS

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1. Quoc Tran Dinh & Carlo Savorgnan & Moritz Diehl, 2013. "Combining Lagrangian decomposition and excessive gap smoothing technique for solving large-scale separable convex optimization problems," Computational Optimization and Applications, Springer, vol. 55(1), pages 75-111, May.
2. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
3. Radu Boţ & Christopher Hendrich, 2013. "A double smoothing technique for solving unconstrained nondifferentiable convex optimization problems," Computational Optimization and Applications, Springer, vol. 54(2), pages 239-262, March.
4. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2012. "Double smoothing technique for large-scale linearly constrained convex optimization," CORE Discussion Papers RP 2423, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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### Keywords

Nesterov’s smoothing technique; Accelerated proximal-gradient method; Adaptive algorithm; Composite convex minimization; Nonsmooth convex optimization;

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