The motivation for these problems is to enable efficient computation of heuristic estimators to formally evaluate and reason about different quantities of deep neural networks in the interest of AI alignment [3]. Our solutions involve reframing the matrix completion problems as a semidefinite program (SDP) and using recent advances in spectral bundle methods for fast, efficient, and scalable SDP solving. Proving that this task is at least as hard as dense matrix multiplication or positive semidefinite testing would count as a resolution. Question 2 (fast "approximate squaring"): Given A R The core idea is to formulate both questions as semidefinite programs (SDP) and use a spectral bundle method [1, 5, 9-11] to implicitly solve the SDP or obtain a certificate of infeasibility. In the case where the SDP is infeasible, our method computes an upper bound quantifying the degree to which the SDP is infeasible.

Many real world optimization problems are formulated as mixed-variable optimization problems (MVOPs) which involve both continuous and discrete variables. MVOPs including dimensional variables are characterized by a variable-size search space. Depending on the values of dimensional variables, the number and type of the variables of the problem can vary dynamically. MVOPs and variable-size MVOPs (VMVOPs) are difficult to solve and raise a number of scientific challenges in the design of metaheuristics. Standard metaheuristics have been first designed to address continuous or discrete optimization problems, and are not able to tackle (V)MVOPs in an efficient way. The development of metaheuristics for solving such problems has attracted the attention of many researchers and is increasingly popular. However, to our knowledge there is no well established taxonomy and comprehensive survey for handling this important family of optimization problems. This paper presents a unified taxonomy for metaheuristic solutions for solving (V)MVOPs in an attempt to provide a common terminology and classification mechanisms. It provides a general mathematical formulation and concepts of (V)MVOPs, and identifies the various solving methodologies than can be applied in metaheuristics. The advantages, the weaknesses and the limitations of the presented methodologies are discussed. The proposed taxonomy also allows to identify some open research issues which needs further in-depth investigations.

Optimizing service schedules is pivotal to the reliable, efficient, and inclusive on-demand mobility. This pressing challenge is further exacerbated by the increasing needs of an aging population, the over-subscription of existing services, and the lack of effective solution methods. This study addresses the intricacies of service scheduling, by jointly optimizing rider trip planning and crew scheduling for a complex dynamic mobility service. The resulting optimization problems are extremely challenging computationally for state-of-the-art methods. To address this fundamental gap, this paper introduces the Joint Rider Trip Planning and Crew Shift Scheduling Problem (JRTPCSSP) and a novel solution method, called AGGNNI-CG (Attention and Gated GNN- Informed Column Generation), that hybridizes column generation and machine learning to obtain near-optimal solutions to the JRTPCSSP with the real-time constraints of the application. The key idea of the machine-learning component is to dramatically reduce the number of paths to explore in the pricing component, accelerating the most time-consuming component of the column generation. The machine learning component is a graph neural network with an attention mechanism and a gated architecture, that is particularly suited to cater for the different input sizes coming from daily operations. AGGNNI-CG has been applied to a challenging, real-world dataset from the Paratransit system of Chatham County in Georgia. It produces dramatic improvements compared to the baseline column generation approach, which typically cannot produce feasible solutions in reasonable time on both medium-sized and large-scale complex instances. AGGNNI-CG also produces significant improvements in service compared to the existing system.

In this work we present deep learning implementations of two popular theoretical constrained optimization algorithms in infinite dimensional Hilbert spaces, namely, the penalty and the augmented Lagrangian methods. We test these algorithms on some toy problems originating in either calculus of variations or physics. We demonstrate that both methods are able to produce decent approximations for the test problems and are comparable in terms of different errors produced. Leveraging the common occurrence of the Lagrange multiplier update rule being computationally less expensive than solving subproblems in the penalty method, we achieve significant speedups in cases when the output of the constraint function is itself a function.

In recent years, multi-objective optimization (MOO) emerges as a foundational problem underpinning many multi-agent multi-task learning applications. However, existing algorithms in MOO literature remain limited to centralized learning settings, which do not satisfy the distributed nature and data privacy needs of such multi-agent multi-task learning applications. This motivates us to propose a new federated multi-objective learning (FMOL) framework with multiple clients distributively and collaboratively solving an MOO problem while keeping their training data private. Notably, our FMOL framework allows a different set of objective functions across different clients to support a wide range of applications, which advances and generalizes the MOO formulation to the federated learning paradigm for the first time. For this FMOL framework, we propose two new federated multi-objective optimization (FMOO) algorithms called federated multi-gradient descent averaging (FMGDA) and federated stochastic multi-gradient descent averaging (FSMGDA). Both algorithms allow local updates to significantly reduce communication costs, while achieving the {\em same} convergence rates as those of their algorithmic counterparts in the single-objective federated learning. Our extensive experiments also corroborate the efficacy of our proposed FMOO algorithms.

Heterogeneity across devices in federated learning (FL) typically refers to statistical (e.g., non-i.i.d. data distributions) and resource (e.g., communication bandwidth) dimensions. In this paper, we focus on another important dimension that has received less attention: varying quantities/distributions of labeled and unlabeled data across devices. In order to leverage all data, we develop a decentralized federated domain adaptation methodology which considers the transfer of ML models from devices with high quality labeled data (called sources) to devices with low quality or unlabeled data (called targets). Our methodology, Source-Target Determination and Link Formation (ST-LF), optimizes both (i) classification of devices into sources and targets and (ii) source-target link formation, in a manner that considers the trade-off between ML model accuracy and communication energy efficiency. To obtain a concrete objective function, we derive a measurable generalization error bound that accounts for estimates of source-target hypothesis deviations and divergences between data distributions. The resulting optimization problem is a mixed-integer signomial program, a class of NP-hard problems, for which we develop an algorithm based on successive convex approximations to solve it tractably. Subsequent numerical evaluations of ST-LF demonstrate that it improves classification accuracy and energy efficiency over state-of-the-art baselines.

Dynamic graph embeddings, inductive and incremental learning facilitate predictive tasks such as node classification and link prediction. However, predicting the structure of a graph at a future time step from a time series of graphs, allowing for new nodes has not gained much attention. In this paper, we present such an approach. We use time series methods to predict the node degree at future time points and combine it with flux balance analysis -- a linear programming method used in biochemistry -- to obtain the structure of future graphs. Furthermore, we explore the predictive graph distribution for different parameter values. We evaluate this method using synthetic and real datasets and demonstrate its utility and applicability.

Principal component analysis (PCA), along with its extensions to manifolds and outlier contaminated data, have been indispensable in computer vision and machine learning. In this work, we present a unifying formalism for PCA and its variants, and introduce a framework based on the flags of linear subspaces, \ie a hierarchy of nested linear subspaces of increasing dimension, which not only allows for a common implementation but also yields novel variants, not explored previously. We begin by generalizing traditional PCA methods that either maximize variance or minimize reconstruction error. We expand these interpretations to develop a wide array of new dimensionality reduction algorithms by accounting for outliers and the data manifold. To devise a common computational approach, we recast robust and dual forms of PCA as optimization problems on flag manifolds. We then integrate tangent space approximations of principal geodesic analysis (tangent-PCA) into this flag-based framework, creating novel robust and dual geodesic PCA variations. The remarkable flexibility offered by the 'flagification' introduced here enables even more algorithmic variants identified by specific flag types. Last but not least, we propose an effective convergent solver for these flag-formulations employing the Stiefel manifold. Our empirical results on both real-world and synthetic scenarios, demonstrate the superiority of our novel algorithms, especially in terms of robustness to outliers on manifolds.

Bayesian optimization has become a powerful tool for safe online optimization of systems, due to its high sample efficiency and noise robustness. For further speed-up reduced physical models of the system can be incorporated into the optimization to accelerate the process, since the models are able to offer an approximation of the actual system, and sampling from them is significantly cheaper. The similarity between model and reality is represented by additional hyperparameters and learned within the optimization process. Safety is an important criteria for online optimization methods like Bayesian optimization, which has been addressed by recent literature, which provide safety guarantees under the assumption of known hyperparameters. However, in practice this is not applicable. Therefore, we extend the robust Gaussian process uniform error bounds to meet the multi-task setting, which involves the calculation of a confidence region from the hyperparameter posterior distribution utilizing Markov chain Monte Carlo methods. Then, using the robust safety bounds, Bayesian optimization is applied to safely optimize the system while incorporating measurements of the models. Simulations show that the optimization can be significantly accelerated compared to other state-of-the-art safe Bayesian optimization methods depending on the fidelity of the models.

This paper proposes a framework that optimizes the linkage mechanism of the quasi-serial manipulator for target tasks. This process is explained through a case study of 2-degree-of-freedom linkage mechanisms, which significantly affect the workspace of the quasi-serial manipulator. First, a vast quasi-serial mechanism is generated with a workspace satisfying a target task and it converts it into a 3D CAD model. Then, the workspace and required torque performance of each mechanism are evaluated through kinematic and dynamic analysis. A deep learning-based surrogate model is leveraged to efficiently predict mechanisms and performance during the optimization process. After model training, a multi-objective optimization problem is formulated under the mechanical and dynamic conditions of the manipulator. The design goal of the manipulator is to recommend quasi-serial mechanisms with optimized kinematic (workspace) and dynamic (joint torque) performance that satisfies the target task. To investigate the underlying physics from the obtained Pareto solutions, various data mining techniques are performed to extract design rules that can provide practical design guidance. Finally, the manipulator was designed in detail for realization with 3D printed parts, including topology optimization. Also, the task-based optimized manipulator is verified through a payload test. Based on these results, the proposed framework has the potential for other real applications as realized cases and provides a reasonable design plan through the design rule extraction.