Global Condition Number Research Articles

An accelerated decentralized stochastic optimization algorithm with inexact model

This paper considers the decentralized stochastic optimization problems where each node of network has only access to the local large data samples and local functions, which are distributed to the computational nodes. We extend the centralized fast adaptive gradient method with inexact model to deal with the large scale problem in the decentralized manner. Moreover, we propose an accelerated decentralized stochastic optimization algorithm with reconstructing parameter equations and defining new approximate local functions. Further, we provide the convergence analysis of the proposed algorithm and illustrate that our algorithm can achieve both the optimal stochastic oracle complexity and communication complexity that depend on the global condition number. Finally, the numerical experiments validate the convergence results of the proposed algorithm.

An accelerated distributed method with inexact model of relative smoothness and strong convexity

AbstractDistributed optimisation methods are widely applied in many systems where agents cooperate with each other to minimise a sum‐type problem over a connected network. An accelerated distributed method based on the inexact model of relative smoothness and strong convexity is introduced by the authors. The authors demonstrate that the proposed method can converge to the optimal solution at the linear rate and achieve the optimal gradient computation complexity and the near optimal communication complexity, where κg denotes the global condition number. Finally, the numerical experiments are provided to validate the theoretical results and further show the efficacy of the proposed method.

Performance based design optimization of an intrinsically compliant 6-dof parallel robot

Parallel robots are preferred over serial robots owing to their enhanced accuracy and rigidity which comes from their higher stiffness. However, there are applications both in industry and in healthcare where higher accuracy is required alongside high compliance (reduced stiffness). Accuracy and compliance being conflicting to each other are difficult to achieve simultaneously. To address this issue, an intrinsically compliant 6-dof parallel robot is proposed in this work. Kinematic and analytical modeling is performed for its conceptual design to obtain the Jacobian matrix and thereby map the joint and Cartesian spaces. Robot’s structure design is analyzed, and the wrench analysis is also carried out to estimate the link forces and stiffness. It is shown that by small changes in the proposed robot design; its compliance can be altered making it suitable for a range of applications. It is also shown mathematically that the robot design can be optimized to maintain higher accuracy together with higher compliance. To carry out design optimization, three important performance criteria, namely; global condition number (for higher accuracies), norm of link forces (to reduce actuator power requirement) and robot compliance (for response to an external wrench) are mathematically formulated. Later, a multi-criteria optimization is performed using an evolutionary algorithm to simultaneously optimize these performance criteria. From the final robot design selected, it is evident that a higher robot compliance with optimal condition number and link forces can be achieved.

Low-Rank Factorizations in Data Sparse Hierarchical Algorithms for Preconditioning Symmetric Positive Definite Matrices

We consider the problem of choosing low-rank factorizations in data sparse matrix approximations for preconditioning large-scale symmetric positive definite (SPD) matrices. These approximations are memory-efficient schemes that rely on hierarchical matrix partitioning and compression of certain sub-blocks of the matrix. Typically, these matrix approximations can be constructed very fast, and their matrix product can be applied rapidly as well. The common practice is to express the compressed sub-blocks by low-rank factorizations, and the main contribution of this work is the numerical and spectral analysis of SPD preconditioning schemes represented by $2\times2$ block matrices, whose off-diagonal sub-blocks are low-rank approximations of the original matrix off-diagonal sub-blocks. We propose an optimal choice of low-rank approximations which minimizes the condition number of the preconditioned system, and demonstrate that the analysis can be applied to the class of hierarchically off-diagonal low-rank matrix approximations. Spectral estimates that take into account the error propagation through levels of the hierarchy which quantify the impact of the choice of low-rank compression on the global condition number are provided. The numerical results indicate that the properties of the preconditioning scheme using proper low-rank compression are superior to employing standard choices for low-rank compression. A major goal of this work is to provide an insight into how proper reweighted prior to low-rank compression influences the condition number for a simple case, which would lead to an extended analysis for more general and more efficient hierarchical matrix approximation techniques.

Optimal design of dynamic and control performance for planar manipulator

k]A design and optimization approach of dynamic and control performance for a two-DOF planar manipulator was proposed. After the kinematic and dynamic analysis, several advantages of the mechanism were illustrated, which made it possible to obtain good dynamic and control performances just through mechanism optimization. Based on the idea of design for control (DFC), a novel kind of multi-objective optimization model was proposed. There were three optimization objectives: the index of inertia, the index describing the dynamic coupling effects and the global condition number. Other indexes to characterize the designing requirements such as the velocity of end-effector, the workspace size, and the first mode natural frequency were regarded as the constraints. The cross-section area and length of the linkages were chosen as the design variables. NSGA-II algorithm was introduced to solve this complex multi-objective optimization problem. Additional criteria from engineering experience were incorporated into the selecting of final parameters among the obtained Pareto solution sets. Finally, experiments were performed to validate the linear dynamic structure and control performances of the optimized mechanisms. A new expression for measuring the dynamic coupling degree with clear physical meaning was proposed. The results show that the optimized mechanism has an approximate decoupled dynamics structure, and each active joint can be regarded as a linear SISO system. The control performances of the linear and nonlinear controllers were also compared. It can be concluded that the optimized mechanism can achieve good control performance only using a linear controller.