An Inexact Primal-Dual Smoothing Framework for Large-Scale Non-Bilinear Saddle Point Problems

Le Thi Khanh Hien (), Renbo Zhao () and William B. Haskell ()
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Le Thi Khanh Hien: University of Mons
Renbo Zhao: University of Iowa
William B. Haskell: Purdue University

Journal of Optimization Theory and Applications, 2024, vol. 200, issue 1, No 2, 34-67

Abstract: Abstract We develop an inexact primal-dual first-order smoothing framework to solve a class of non-bilinear saddle point problems with primal strong convexity. Compared with existing methods, our framework yields a significant improvement over the primal oracle complexity, while it has competitive dual oracle complexity. In addition, we consider the situation where the primal-dual coupling term has a large number of component functions. To efficiently handle this situation, we develop a randomized version of our smoothing framework, which allows the primal and dual sub-problems in each iteration to be inexactly solved by randomized algorithms in expectation. The convergence of this framework is analyzed both in expectation and with high probability. In terms of the primal and dual oracle complexities, this framework significantly improves over its deterministic counterpart. As an important application, we adapt both frameworks for solving convex optimization problems with many functional constraints. To obtain an $$\varepsilon $$ ε -optimal and $$\varepsilon $$ ε -feasible solution, both frameworks achieve the best-known oracle complexities.

Keywords: Non-bilinear saddle point problems; Inexact primal-dual smoothing; Convex optimization with functional constraints; Stochastic optimization; 90C06; 90C25; 49J52 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10957-023-02351-9

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