In this work, a convergence lemma for function $f$ being finite compositions of analytic mappings and the maximum operator is proved. The lemma shows that the set of $\delta$-stationary points near an isolated local minimum point $x^*$ is shrinking to $x^*$ as $\delta\to 0$. It is a natural extension of the version for strongly convex $C^1$ functions. However, the correctness of the lemma is subtle. Analytic mappings are necessary for the lemma in the sense that replacing it with differentiable or $C^\infty$ mappings makes the lemma false. The proof is based on stratification theorems of semi-analytic sets by {\L}ojasiewicz. An extension of this proof presents a geometric characterization of the set of stationary points of $f$. Finally, a notion of stability on stationary points, called convergence stability, is proposed. It asks, under small numerical errors, whether a reasonable convergent optimization method started near a stationary point should eventually converge to the same stationary point. The concept of convergence stability becomes nontrivial qualitatively only when the objective function is both nonsmooth and nonconvex. Via the convergence lemma, an intuitive equivalent condition for convergence stability of $f$ is proved. These results together provide a new geometric perspective to study the problem of "where-to-converge" in nonsmooth nonconvex optimization.

Mixed-precision Deep Neural Networks achieve the energy efficiency and throughput needed for hardware deployment, particularly when the resources are limited, without sacrificing accuracy. However, the optimal per-layer bit precision that preserves accuracy is not easily found, especially with the abundance of models, datasets, and quantization techniques that creates an enormous search space. In order to tackle this difficulty, a body of literature has emerged recently, and several frameworks that achieved promising accuracy results have been proposed. In this paper, we start by summarizing the quantization techniques used generally in literature. Then, we present a thorough survey of the mixed-precision frameworks, categorized according to their optimization techniques such as reinforcement learning and quantization techniques like deterministic rounding. Furthermore, the advantages and shortcomings of each framework are discussed, where we present a juxtaposition. We finally give guidelines for future mixed-precision frameworks.

Recently, blockchain-based federated learning (BFL) has attracted intensive research attention due to that the training process is auditable and the architecture is serverless avoiding the single point failure of the parameter server in vanilla federated learning (VFL). Nevertheless, BFL tremendously escalates the communication traffic volume because all local model updates (i.e., changes of model parameters) obtained by BFL clients will be transmitted to all miners for verification and to all clients for aggregation. In contrast, the parameter server and clients in VFL only retain aggregated model updates. Consequently, the huge communication traffic in BFL will inevitably impair the training efficiency and hinder the deployment of BFL in reality. To improve the practicality of BFL, we are among the first to propose a fast blockchain-based communication-efficient federated learning framework by compressing communications in BFL, called BCFL. Meanwhile, we derive the convergence rate of BCFL with non-convex loss. To maximize the final model accuracy, we further formulate the problem to minimize the training loss of the convergence rate subject to a limited training time with respect to the compression rate and the block generation rate, which is a bi-convex optimization problem and can be efficiently solved. To the end, to demonstrate the efficiency of BCFL, we carry out extensive experiments with standard CIFAR-10 and FEMNIST datasets. Our experimental results not only verify the correctness of our analysis, but also manifest that BCFL can remarkably reduce the communication traffic by 95-98% or shorten the training time by 90-95% compared with BFL.

Conic optimization has recently emerged as a powerful tool for designing tractable and guaranteed algorithms for non-convex polynomial optimization problems. On the one hand, tractability is crucial for efficiently solving large-scale problems and, on the other hand, strong bounds are needed to ensure high quality solutions. In this research, we investigate the strengthening of RLT relaxations of polynomial optimization problems through the addition of nine different types of constraints that are based on linear, second-order cone, and semidefinite programming to solve to optimality the instances of well established test sets of polynomial optimization problems. We describe how to design these conic constraints and their performance with respect to each other and with respect to the standard RLT relaxations. Our first finding is that the different variants of nonlinear constraints (second-order cone and semidefinite) are the best performing ones in around $50\%$ of the instances. Additionally, we present a machine learning approach to decide on the most suitable constraints to add for a given instance. The computational results show that the machine learning approach significantly outperforms each and every one of the nine individual approaches.

This is one of the most applied courses on Genetic Algorithms (GA), which presents an integrated framework to solve real-world optimization problems in the most simple way. For the first time, we have presented a practical course in the domain of metaheuristics algorithms required for students, researchers and practitioners. Firstly, we will introduce the basic theory of GA, then implement the simplest version of GA, namely Binary GA, into Matlab, and then present the continuous version, real GA, of it. Therefore, the main focus will be on the Genetic Algorithm as the most well-regarded optimization algorithm in the literature. In the following sections, we will introduce some well-known operation research problems, including transportation problems, hub location problems (HLP), quadratic assignment problems and travelling salesman problems (TSP) and try to solve them via GA.

Approximating the Koopman operator from data is numerically challenging when many lifting functions are considered. Even low-dimensional systems can yield unstable or ill-conditioned results in a high-dimensional lifted space. In this paper, Extended Dynamic Mode Decomposition (DMD) and DMD with control, two methods for approximating the Koopman operator, are reformulated as convex optimization problems with linear matrix inequality constraints. Asymptotic stability constraints and system norm regularizers are then incorporated as methods to improve the numerical conditioning of the Koopman operator. Specifically, the H-infinity norm is used to penalize the input-output gain of the Koopman system. Weighting functions are then applied to penalize the system gain at specific frequencies. These constraints and regularizers introduce bilinear matrix inequality constraints to the regression problem, which are handled by solving a sequence of convex optimization problems. Experimental results using data from an aircraft fatigue structural test rig and a soft robot arm highlight the advantages of the proposed regression methods.

The discovery of drug-target interactions (DTIs) is a very promising area of research with great potential. The accurate identification of reliable interactions among drugs and proteins via computational methods, which typically leverage heterogeneous information retrieved from diverse data sources, can boost the development of effective pharmaceuticals. Although random walk and matrix factorization techniques are widely used in DTI prediction, they have several limitations. Random walk-based embedding generation is usually conducted in an unsupervised manner, while the linear similarity combination in matrix factorization distorts individual insights offered by different views. To tackle these issues, we take a multi-layered network approach to handle diverse drug and target similarities, and propose a novel optimization framework, called Multiple similarity DeepWalk-based Matrix Factorization (MDMF), for DTI prediction. The framework unifies embedding generation and interaction prediction, learning vector representations of drugs and targets that not only retain higher-order proximity across all hyper-layers and layer-specific local invariance, but also approximate the interactions with their inner product. Furthermore, we develop an ensemble method (MDMF2A) that integrates two instantiations of the MDMF model, optimizing the area under the precision-recall curve (AUPR) and the area under the receiver operating characteristic curve (AUC) respectively. The empirical study on real-world DTI datasets shows that our method achieves statistically significant improvement over current state-of-the-art approaches in four different settings. Moreover, the validation of highly ranked non-interacting pairs also demonstrates the potential of MDMF2A to discover novel DTIs.

This paper develops the method of Continuous Pontryagin Differentiable Programming (Continuous PDP), which enables a robot to learn an objective function from a few sparsely demonstrated keyframes. The keyframes, labeled with some time stamps, are the desired task-space outputs, which a robot is expected to follow sequentially. The time stamps of the keyframes can be different from the time of the robot's actual execution. The method jointly finds an objective function and a time-warping function such that the robot's resulting trajectory sequentially follows the keyframes with minimal discrepancy loss. The Continuous PDP minimizes the discrepancy loss using projected gradient descent, by efficiently solving the gradient of the robot trajectory with respect to the unknown parameters. The method is first evaluated on a simulated robot arm and then applied to a 6-DoF quadrotor to learn an objective function for motion planning in unmodeled environments. The results show the efficiency of the method, its ability to handle time misalignment between keyframes and robot execution, and the generalization of objective learning into unseen motion conditions.

In this chapter, an integer linear programming formulation for the problem of obtaining task-relevant, multi-resolution, environment abstractions for resource-constrained autonomous agents is presented. The formulation leverages concepts from information-theoretic signal compression, specifically, the information bottleneck (IB) method, to pose an abstraction problem as an optimal encoder search over the space of multi-resolution trees. The abstractions emerge in a task-relevant manner as a function of agent information-processing constraints. We detail our formulation, and show how hierarchical tree structures, signal encoders, and information-theoretic methods for signal compression can be unified under a common theme. A discussion delineating the benefits and drawbacks of our formulation is presented, as well as a detailed explanation how our approach can be interpreted within the context of generating abstractions for resource-constrained autonomous systems. It is shown that the resulting information-theoretic abstraction problem over the space of multi-resolution trees can be formulated as a integer linear programming (ILP) problem. We demonstrate the approach on a number of examples, and provide a discussion detailing the differences of the proposed framework compared to existing methods. Lastly, we consider a linear program relaxation of the ILP problem, thereby demonstrating that multi-resolution information-theoretic tree abstractions can be obtained by solving a convex program.

The online assignment problem plays an important role in operational research and computer science which is why immense attention has been given to improving its solution quality. Due to the incomplete information about the input, it is difficult for online algorithms to produce the optimal solution. The quality of the solution of an online algorithm is measured using a competitive ratio. No online deterministic algorithm can achieve a competitive ratio better than (2n-1). It has been shown that advice in online computation improves the lower bound of the competitive ratio of online problems. Advice in online computation can be interpreted as additional information for the online algorithm to compensate for the lack of information about the whole input sequence. In this study, we investigate how introducing machine-learned advice could improve the competitive ratio for this problem. We provide an online algorithm for the online assignment problem by simulating a machine learning algorithm that predicts the whole input in advance. We utilize an optimal offline algorithm to provide a matching solution from the predicted input. Furthermore, we investigate how the prediction error of machine learning affects the competitive ratio of the online algorithm. We utilize a benchmark data set to perform our empirical analysis. We show that as the Machine Learning prediction error increases, the solution quality decreases. Moreover, the magnitude of error is directly proportional to the size of the input. This result is analogous to the competitive ratio of the best deterministic algorithm for the online assignment problem which is dependent also on the parameter n.