Barrier PDHG (BPDHG), a nested algorithm which incorporates a logarithmic barrier function into the PDHG framework to alleviate prolonged plateaus in the KKT residual on selected instances is proposed.
Abstract
Primal Dual Hybrid Gradient (PDHG) method has been verified to exhibit a two stage convergence behavior, in which a prolonged active set identification phase may be a major issue of slow convergence. In this paper, we propose Barrier PDHG (BPDHG), a nested algorithm which incorporates a logarithmic barrier function into the PDHG framework to alleviate this problem. We first establish convergence of the inner iterations, derive an error bound for the corresponding inner problem. Then we prove that the outer sequence generated by BPDHG approaches the optimal solution set of the LP problem we considered. Furthermore, we integrate the barrier technique into the {Primal Dual Linear Programming} (PDLP) framework to develop the corresponding Barrier PDLP (BPDLP) method. Numerical experiments show that the barrier modification can alleviate prolonged plateaus in the KKT residual on selected instances. We also investigate an empirical instance-dependent indicator for identifying LP problems on which BPDLP is more likely to outperform PDLP.
Interior-point methods (IPMs) are among the most widely used algorithms for constrained optimization, yet their Newton-based search directions require costly second-order information and large linear-system solves. Learning to optimize offers cheaper updates learned from data, but the singular behavior of logarithmic b...
A. Madabhushi, Jia-Lin Liu, Min-Xin Zhang· 0 citations
This work considers a quadratic minmax problem with coupled inner constraints and proposes a method to compute a class of stationary points and shows in particular that the method is polynomial in the special case where the inner feasible set of the authors' constrained minmax problem is independent from outer variable...
Stefano Cipolla, O. Stein, Alain B. Zemkoho· 1 citation
It is shown that strict complementarity, together with a quadratic facial-violation property of the associated complementary faces, implies uniform quadratic growth of both the primal and dual augmented Lagrangians near a strictly complementary solution, and the local equivalence of three regularity conditions is prove...
We study a class of weakly convex optimization problems in which the objective is the sum of a smooth convex term and a weakly convex term that may be non-smooth. To exploit this structure, we develop a splitting technique based on the alternating direction method of multipliers (ADMM), which decouples the minimization...
Sheng-Han Mei, Cheng-Yu Ke, Yifei Lou et al.· Frontiers in Applied Mathema...· 0 citations
Projected reflected gradient (PRG) method proposed by Malitsky is efficient for solving monotone variational inequality (MVI), while the existing upper bound of step size is not tight due to the inequality scaling in the theoretical analysis. In this paper, we construct an averaged variant of PRG method for more genera...
This work proposes a smooth approximation of PBO through reformulation, penalization and regularization, and establishes convergence guarantees in terms of both minimizers and stationarity, and develops two single-loop algorithms for deterministic and stochastic PBOs, respectively.
Qi Cao, Bo Zeng, Shang-Zhi Zeng et al.· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
