Balanced Augmented Lagrangian Method for Convex Programming

Bingsheng He; Xiaoming Yuan

Home/Archive/arXiv

arXiv · 2108.08554

Balanced Augmented Lagrangian Method for Convex Programming

Bingsheng He·Xiaoming Yuan

View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the convex minimization model with both linear equality and inequality constraints, and reshape the classic augmented Lagrangian method (ALM) by balancing its subproblems. As a result, one of its subproblems decouples the objective function and the coefficient matrix without any extra condition, and the other subproblem becomes a positive definite system of linear equations or a positive definite linear complementary problem. The balanced ALM advances the classic ALM by enlarging its applicable range, balancing its subproblems, and improving its implementation. We also extend our discussion to two-block and multiple-block separable convex programming models, and accordingly design various splitting versions of the balanced ALM for these separable models. Convergence analysis for the balanced ALM and its splitting versions is conducted in the context of variational inequalities through the lens of the classic proximal point algorithm.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Bingsheng He, Xiaoming Yuan. 2021-08-19. Balanced Augmented Lagrangian Method for Convex Programming. https://arxiv.org/abs/2108.08554

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

AdGT: Decentralized Gradient Tracking with Adaptive Per-Agent Stepsizes

In decentralized optimization, gradient-tracking methods typically rely on a single global stepsize. This choice can be conservative when agents have local objectives with different smoothness constants, since the stepsize must remain stable for the agent with the largest smoothness constant. This paper proposes AdGT, a decentralized gradient-tracking method in which each agent adapts its own stepsize using local gradient variation and a single global safety factor. The method reduces fixed-stepsize tuning effort and allows agents to exploit local smoothness information during the iterations. For smooth and strongly convex local objectives over undirected networks, we prove that the analyzed AdGT update converges linearly to the exact consensus optimizer. We also study two adaptive stepsize updates that use changes in the gradient-tracking direction. We characterize when the corresponding candidate determines the stepsize and prove conditional lower and upper stepsize bounds and linear convergence under an additional relative tracking-disagreement condition. Experiments on logistic regression, ridge regression, synthetic quadratic problems, and a linear-regression benchmark against state-of-the-art decentralized solvers show that AdGT often reaches a given accuracy in fewer iterations or gradient evaluations than tuned fixed-stepsize GT and the tested baselines, especially under heterogeneous local smoothness. In the topology experiments, each tested AdGT update uses one common safety factor across all graphs, whereas the fixed GT stepsize is tuned separately for each graph and seed.

math.OC↗

arXiv2026

Brockett cost function for symplectic eigenvalues

The sum of symplectic eigenvalues and corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson's theorem can be computed via minimization of a trace cost function under the symplecticity constraint. Optimal solutions to this problem only offer a symplectic basis for the symplectic eigenspace corresponding to the sought symplectic eigenvalues. In this note, we introduce a Brockett cost function and investigate its properties and the connection with the symplectic eigenvalues and eigenvectors of the considered matrix. Specifically, we prove that any stationary point consists of symplectic eigenvectors, characterize the saddle points and global minimizers based on which the trace minimization theorem for the symplectic eigenvalues is re-established, and the nonexistence of local nonglobal minimizers is justified.

math.OC↗

arXiv2026

Riemannian Bilevel Optimization with Gradient Aggregation

We study bilevel optimization on Riemannian manifolds when the lower-level solution set is a positive-dimensional submanifold, so that implicit differentiation fails. We propose Riemannian Bilevel Descent Aggregation (RBDA), which extends bilevel descent aggregation to manifolds. Its inner loop aggregates the lower-level descent direction with the upper-level gradient under a decaying multiplier, and its hypergradient is the reverse-mode derivative of the unrolled loop. Under geodesic convexity and quadratic growth of the lower level, the inner iterates converge to a point of the optimistic solution set at a polynomial rate. Approximate minimizers of the objective with a finite number of inner iterations converge to minimizers of the optimistic value. In the experiments RBDA selects the optimistic solution where the implicit and unrolled estimators remain at the initial point or stop at a larger query loss. While each of its outer steps costs more than that of the unrolled estimator, it attains the highest test accuracy in data hyper-cleaning.

math.OC↗