Two novel augmented Lagrangian algorithms with exact multipliers are developed, designed respectively for the full row-rank case and the general matrix case, where all subproblems are globally optimized via closed-form solutions.
Abstract
This paper studies a non-separable composite $\ell_0$-$\ell_2$ regularization model that simultaneously enforces sparsity and smoothness for inverse problems. The $\ell_0$ norm induces inherent nonconvexity and nonsmoothness, while linear transformations further introduce nonseparability, making the problem computationally challenging to solve. The existing inexact augmented Lagrangian method suffers from high computational complexity and unstable convergence. To overcome these difficulties, we develop two novel augmented Lagrangian algorithms with exact multipliers, designed respectively for the full row-rank case and the general matrix case, where all subproblems are globally optimized via closed-form solutions. Furthermore, we prove linear convergence of the proposed method when the transformation matrix is full row rank. In the general setting, all accumulation points of the generated sequence are KKT points for the original problem. Numerical experiments on synthetic data, trend filtering, and image smoothing demonstrate the superior efficiency and accuracy of the proposed methods over the existing method, confirming our theoretical analysis.
This work proposes a nonlinear-residual linearized augmented Lagrangian method (NR-LALM) that replaces this subproblem by a regularized Gauss-Newton-type step while retaining the classical multiplier update based on the nonlinear constraint residual.
Benqi Liu, Kangkang Deng, Zichen Wang et al.· 0 citations
The main contribution is a finite-time mechanism for converting stationarity of the truncated minimax problem into a KKT certificate for the original constrained problem and establishing explicit convergence rates for the proposed method in terms of the KKT residual.
In this paper, we propose a balanced augmented Lagrangian method based on accelerated stochastic ADMM (b-ASADMM) to efficiently solve structured separable nonconvex optimization problems subject to linear constraints. The objective function in this problem comprises potentially nonsmooth and smooth functions, where the smooth function is an average of multiple nonconvex smooth functions. The involved smooth subproblem is tackled by an accelerated stochastic gradient method based on weighting of stochastic item and pre-variable. The involved nonsmooth subproblem is solved under incorporation of Bregman distance to avoid the case that subproblem does not have a closed-form solution due to the complicated quadratic term or other hindering. The involved balanced augmented Lagrangian method advances the original ALM by balancing its subproblems and improving its implementation. In contrast to most deterministic and stochastic ADMMs, our dual variable allows a more flexible and larger step-size region. By standard smoothness assumption, we establish the global convergence and iteration complexity of the generated sequence. Furthermore, we provide a linear convergence rate of b-ASADMM under a local error bound condition and the weakly convex property of the nonsmooth component. Numerical experiments on the graph-guided fused Lasso problem and the smooth clipped absolute deviation penalty problem are conducted to verify the effectiveness of b-ASADMM.
In this work, we consider the problem of sparse signal recovery known as compressed sensing using $\ell_1$-minimization.
We show how the $\ell_1$-minimization problem (also known as basis pursuit) can be transformed into an equivalent linear programming (LP) problem, and provide a proof of the equivalence of these two problems.
We conduct an experimental comparison of modern solvers (Gurobi, HiGHS, CPLEX, and Clarabel) for solving the LP problem on test data generated according to theoretical recovery guarantees for matrices with normally distributed elements. The results show that the open-source solver Clarabel is a competitive alternative to proprietary solvers in terms of speed.
We also propose a method for verifying the uniqueness of the obtained solution using an auxiliary quadratic programming problem with a strictly convex objective function.
A geometric interpretation of the uniqueness conditions is provided, and the application of the method is demonstrated on an example of a matrix with integer elements.
Anastasiia O. Storozhenko, P. Stetsyuk· KyivAcademUs2026· 0 citations
We develop a non-asymptotic approximation, sampling, and finite-iteration optimization theory for variational physics-informed approximation of uniformly monotone nonlinear multiscale elliptic equations. For boundary-compatible neural feature classes, the population error splits into approximation, empirical quadrature, and projected-gradient terms, with all non-approximation constants uniform in the microscopic scale \(\varepsilon\). Assuming a quantitative corrected \(H^1\)-estimate, a two-scale state class yields \[ \mathcal A_m^\varepsilon \le C\bigl(\varepsilon+\Phi_{0,m_0}^2+\Phi_{1,m_1}^2\bigr) \] in arbitrary dimension. We further introduce a convex primal-dual physics loss whose population value is a computable upper certificate for the state error. With additional flux-corrector regularity, a divergence-compatible two-scale flux class gives a certified state-flux bound combining \(O(\varepsilon)\) approximation, state and flux feature errors, empirical sampling error, and an \(O(K^{-1})\) optimization term. In contrast, for general periodic nonlinear fluxes satisfying a natural nondegeneracy condition, the empirical Rademacher complexities of strong-residual and squared-residual classes are bounded below by constant multiples of \((\varepsilon\sqrt N)^{-1}\) and \((\varepsilon^2\sqrt N)^{-1}\), respectively. These optimizer-independent lower bounds hold in every spatial dimension. Numerical experiments confirm the predicted \(\varepsilon\)- and \(N\)-scalings for nonlinear fluxes in \(d=1,2,3\), validate every computed primal-dual certificate, and show that corrector-enriched classes substantially reduce energy and \(H^1\) errors as the microscopic scale is refined.
We consider generalized Nash equilibrium problems among $N$ players with convex quadratic costs and shared affine constraints, assuming only that the game's pseudogradient is merely monotone. We show that computing a variational generalized Nash equilibrium (v-GNE) is equivalent to solving a single convex quadratic program (QP) derived from the players'joint Karush--Kuhn--Tucker conditions. Building on this, we show that the regularization of such a QP yields an $\varepsilon$-approximated v-GNE with suboptimality vanishing linearly in the regularization parameter. Next, we propose an accelerated proximal-point scheme and an accelerated projected-gradient method, both attaining an $\mathcal O(1/k^2)$-approximated v-GNE at the $k$-th iteration. We also demonstrate that an invertible Jacobian of the game allows for reduction to a lower-dimensional QP. Theoretical analysis and numerical experiments show the proposed methods substantially outperform the existing approaches to solve monotone linear-quadratic v-GNE problems.
