Iteration-complexity of a Rockafellar's proximal method of multipliers for convex programming based on second-order approximations

ABSTRACT

This paper studies the iteration-complexity of a new primal-dual algorithm based on Rockafellar's proximal method of multipliers (PMM) for solving smooth convex programming problems with inequality constraints. In each step, either a step of Rockafellar's PMM for a second-order model of the problem is computed or a relaxed extragradient step is performed. The resulting algorithm is a (large-step) relaxed hybrid proximal extragradient (r-HPE) method of multipliers, which combines Rockafellar's PMM with the r-HPE method. We obtain pointwise $\mathcal{O}\left(1/k\right)$ and ergodic $\mathcal{O}\left(1/k^{3/2}\right)$ global convergence rates at the price of solving, at each iteration, quadratic quadratically constrained convex programming problems. These convergence rates are superior to the corresponding pointwise $\mathcal{O}\left(1/\sqrt{k}\right)$ and ergodic $\mathcal{O}\left(1/k\right)$ currently known for standard proximal-point methods, thanks to the incorporation of second-order information. To the best of our knowledge, this is the first time that the above mentioned rates and results are obtained for solving the smooth convex programming problems with inequality constraints.

KEYWORDS:

• Convex programming
• proximal method of multipliers
• second-order methods
• hybrid extragradient
• complexity

2000 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 90C25
• 90C30
• 47H05

Acknowledgments

This work was done while M. Marques Alves was a postdoc at the School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, United States.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of M. Marques Alves was partially supported by CNPq [grants numbers 406250/2013-8, 306317/2014-1 and 405214/2016-2]. The work of R.D.C. Monteiro was partially supported by CNPq [grant number 406250/2013-8] and NSF [grant number CMMI-1300221]. The work of Benar F. Svaiter was partially supported by CNPq [grants numbers 474996/2013-1, 302962/2011-5], FAPERJ [grant number E-26/102.940/2011], and PRONEX-Optimization.