Alternating direction method of multipliers (ADMM) has received tremendous interest for solving numerous problems in machine learning, statistics and signal processing. However, it is known that the performance of ADMM and many of its variants is very sensitive to the penalty parameter of a quadratic penalty applied to the equality constraints. Although several approaches have been proposed for dynamically changing this parameter during the course of optimization, they do not yield theoretical improvement in the convergence rate and are not directly applicable to stochastic ADMM. In this paper, we develop a new ADMM and its linearized variant with a new adaptive scheme to update the penalty parameter. Our methods can be applied under both deterministic and stochastic optimization settings for structured non-smooth objective function. The novelty of the proposed scheme lies at that it is adaptive to a local sharpness property of the objective function, which marks the key difference from previous adaptive scheme that adjusts the penalty parameter per-iteration based on certain conditions on iterates. On theoretical side, given the local sharpness characterized by an exponent $\theta\in(0, 1]$, we show that the proposed ADMM enjoys an improved iteration complexity of $\widetilde O(1/\epsilon^{1-\theta})$\footnote{$\widetilde O()$ suppresses a logarithmic factor.} in the deterministic setting and an iteration complexity of $\widetilde O(1/\epsilon^{2(1-\theta)})$ in the stochastic setting without smoothness and strong convexity assumptions. The complexity in either setting improves that of the standard ADMM which only uses a fixed penalty parameter. On the practical side, we demonstrate that the proposed algorithms converge comparably to, if not much faster than, ADMM with a fine-tuned fixed penalty parameter.

In particular, it remains an open question how to quantify the improvement in ADMM's theoretical convergence by using adaptive penalty parameters. Of course, the answer to this question depends on the adaptive scheme being used.

Swarm trajectory optimization problems are a well-recognized class of multi-agent optimal control problems with strong nonlinearity. However, the heuristic nature of needing to set the final time for agents beforehand and the time-consuming limitation of the significant number of iterations prohibit the application of existing methods to large-scale swarm of Unmanned Aerial Vehicles (UAVs) in practice. In this paper, we propose a spatial-temporal trajectory optimization framework that accomplishes multi-UAV consensus based on the Alternating Direction Multiplier Method (ADMM) and uses Differential Dynamic Programming (DDP) for fast local planning of individual UAVs. The introduced framework is a two-level architecture that employs Parameterized DDP (PDDP) as the trajectory optimizer for each UAV, and ADMM to satisfy the local constraints and accomplish the spatial-temporal parameter consensus among all UAVs. This results in a fully distributed algorithm called Distributed Parameterized DDP (D-PDDP). In addition, an adaptive tuning criterion based on the spectral gradient method for the penalty parameter is proposed to reduce the number of algorithmic iterations. Several simulation examples are presented to verify the effectiveness of the proposed algorithm.

Abstract--The Minimum Bisection Problem is a well-known NP-hard problem in combinatorial optimization, with practical applications in areas such as parallel computing, network design, and machine learning. In this paper, we examine the potential of using D-Wave Systems' quantum annealing solvers to solve the Minimum Bisection Problem, which we formulate as a Quadratic Unconstrained Binary Optimization model. A key challenge in this formulation lies in choosing an appropriate penalty parameter, as it plays a crucial role in ensuring both the quality of the solution and the satisfaction of the problem's constraints. T o address this, we introduce a novel machine learning-based approach for adaptive tuning of the penalty parameter . Specifically, we use a Gradient Boosting Regressor model trained to predict suitable penalty parameter values based on structural properties of the input graph, the number of nodes and the graph's density. This method enables the penalty parameter to be adjusted dynamically for each specific problem instance, improving the solver's ability to balance the competing goals of minimizing the cut size and maintaining equally sized partitions. We test our approach on a large dataset of randomly generated Erd os-R enyi graphs with up to 4000 nodes, and we compare the results with classical partitioning algorithms, Metis and Kernighan-Lin. Experimental findings demonstrate that our adaptive tuning strategy significantly improves the performance of quantum annealing hybrid solver and consistently outperforms the classical methods used, indicating its potential as an alternative for graph partitioning problem. RAPH partitioning is a fundamental problem in combinatorial optimization with applications in task scheduling for multiprocessor computers with focus on parallel computation, partitioning the circuit with applications in microchips design, social network analysis e.g. The problem involves dividing a given graph G into two or more subsets while optimizing certain objective, such as minimizing the number of inter-edges and/or assigned costs between them and producing balanced partitions. While general Graph Partitioning Problem (GPP) allow for flexible partition sizes, a significant special case is the Minimum Bisection Problem (MBP), where the graph is partitioned into two equal-sized subsets while minimizing the number of inter-edges [1].

We propose an inexact proximal augmented Lagrangian method (P-ALM) for nonconvex structured optimization problems. The proposed method features an easily implementable rule not only for updating the penalty parameters, but also for adaptively tuning the proximal term. It allows the penalty parameter to grow rapidly in the early stages to speed up progress, while ameliorating the issue of ill-conditioning in later iterations, a well-known drawback of the traditional approach of linearly increasing the penalty parameters. A key element in our analysis lies in the observation that the augmented Lagrangian can be controlled effectively along the iterates, provided an initial feasible point is available. Our analysis, while simple, provides a new theoretical perspective about P-ALM and, as a by-product, results in similar convergence properties for its non-proximal variant, the classical augmented Lagrangian method (ALM). Numerical experiments, including convex and nonconvex problem instances, demonstrate the effectiveness of our approach.

Eq. 1 in the paper (re-stated in Eq. 2 below), and show that it is equal to the objective function in the Theorem in the paper (see Eq. 8 below) up to the factor In the following, we provide the detailed steps. We start by re-stating Eq. 1 in the paper 1 n null null nullA The details are outlined in Sections 2.2 and 2.3 below. See Eq. 1 above for the definitions of X, multiplied by the dropout-probability p, and q = 1 p. In line 6, we change the sum over the m columns back to matrix notation. Finally, in line 8, we used the substitutions from Eq. 1 as to obtain In lines 11 and 12, the squared loss is expanded into its four terms.

In this paper, we study a class of stochastic and finite-sum convex optimization problems with deterministic constraints. Existing methods typically aim to find an $ε$-$expectedly\ feasible\ stochastic\ optimal$ solution, in which the expected constraint violation and expected optimality gap are both within a prescribed tolerance $ε$. However, in many practical applications, constraints must be nearly satisfied with certainty, rendering such solutions potentially unsuitable due to the risk of substantial violations. To address this issue, we propose stochastic first-order methods for finding an $ε$-$surely\ feasible\ stochastic\ optimal$ ($ε$-SFSO) solution, where the constraint violation is deterministically bounded by $ε$ and the expected optimality gap is at most $ε$. Our methods apply an accelerated stochastic gradient (ASG) scheme or a modified variance-reduced ASG scheme $only\ once$ to a sequence of quadratic penalty subproblems with appropriately chosen penalty parameters. We establish first-order oracle complexity bounds for the proposed methods in computing an $ε$-SFSO solution. As a byproduct, we also derive first-order oracle complexity results for sample average approximation method in computing an $ε$-SFSO solution of the stochastic optimization problem using our proposed methods to solve the sample average problem.

Originality: I find the approach original and interesting, I find that other works have been cited and the section of related work is written clearly and detailed, it gives a nice overview. I think only that it is important to highlight more clearly the differences between [40] and the current work. In particular, it is unclear what is the penalty parameter, and how their method of adversarial training relates to this work - do they optimize a different bound or what quantities do they optimize, and do these quantities show up in the proposed bound? Quality: the work seems complete, and sound for as far as I could check. I could not check all the proofs in detail but I read the work in great detail.

Quadratic programming (QP) forms a crucial foundation in optimization, encompassing a broad spectrum of domains and serving as the basis for more advanced algorithms. Consequently, as the scale and complexity of modern applications continue to grow, the development of efficient and reliable QP algorithms is becoming increasingly vital. In this context, this paper introduces a novel deep learning-aided distributed optimization architecture designed for tackling large-scale QP problems. First, we combine the state-of-the-art Operator Splitting QP (OSQP) method with a consensus approach to derive DistributedQP, a new method tailored for network-structured problems, with convergence guarantees to optimality. Subsequently, we unfold this optimizer into a deep learning framework, leading to DeepDistributedQP, which leverages learned policies to accelerate reaching to desired accuracy within a restricted amount of iterations. Our approach is also theoretically grounded through Probably Approximately Correct (PAC)-Bayes theory, providing generalization bounds on the expected optimality gap for unseen problems. The proposed framework, as well as its centralized version DeepQP, significantly outperform their standard optimization counterparts on a variety of tasks such as randomly generated problems, optimal control, linear regression, transportation networks and others. Notably, DeepDistributedQP demonstrates strong generalization by training on small problems and scaling to solve much larger ones (up to 50K variables and 150K constraints) using the same policy. Moreover, it achieves orders-of-magnitude improvements in wall-clock time compared to OSQP. The certifiable performance guarantees of our approach are also demonstrated, ensuring higher-quality solutions over traditional optimizers.