Accelerated and Improved Modified Gradient-Based Iterative Algorithms for Solving Sylvester Tensor Equations

This paper is concerned with the numerical solution of the Sylvester tensor equation, which includes the Sylvester matrix equation as special case. By applying hierarchical identification principle proposed by Ding and Chen, 2005, and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective.

Gradient-based iterative algorithms for the tensor nearness problems associated with Sylvester tensor equations

How can we identify the same or similar users from a collection of social network platforms (e.g., Facebook, Twitter, LinkedIn, etc.)? Which restaurant shall we recommend to a given user at the right time at the right location? Given a disease, which genes and drugs are most relevant? Multi-way association, which identifies strongly correlated node sets from multiple input networks, is the key to answering these questions. Despite its importance, very few multi-way association methods exist due to its high complexity. In this paper, we formulate multi-way association as a convex optimization problem, whose optimal solution can be obtained by a Sylvester tensor equation. Furthermore, we propose two fast algorithms to solve the Sylvester tensor equation, with a linear time and space complexity. We further provide theoretic analysis in terms of the sensitivity of the Sylvester tensor equation solution. Empirical evaluations demonstrate the efficacy of the proposed method.

PurposeThis paper aims to deal with the design optimization of a synchronous reluctance machine to be used in an X-ray tube, where the goal is to maximize the torque while keeping low the amount of material used, by means of gradient-based free-form shape optimization.Design/methodology/approachThe presented approach is based on the mathematical concept of shape derivatives and allows to obtain new motor designs without the need to introduce a geometric parametrization. This paper presents an extension of a standard gradient-based free-form shape optimization algorithm to the case of multiple objective functions by determining updates, which represent a descent of all involved criteria. Moreover, this paper illustrates a way to obtain an approximate Pareto front.FindingsThe presented method allows to obtain optimal designs of arbitrary, non-parametric shape with very low computational cost. This paper validates the results by comparing them to a parametric geometry optimization in JMAG by means of a stochastic optimization algorithm. While the obtained designs are of similar shape, the computational time used by the gradient-based algorithm is in the order of minutes, compared to several hours taken by the stochastic optimization algorithm.Originality/valueThis paper applies the presented gradient-based multi-objective optimization algorithm in the context of free-form shape optimization using the mathematical concept of shape derivatives. The authors obtain a set of Pareto-optimal designs, each of which is a shape that is not represented by a fixed set of parameters. To the best of the authors’ knowledge, this approach to multi-objective free-form shape optimization is novel in the context of electric machines.

Sparse principal component analysis is an important technique for simultaneous dimensionality reduction and variable selection with high-dimensional data. In this work we combine the unique geometric structure of the sparse principal component analysis problem with recent advances in convex optimization to develop novel gradient-based sparse principal component analysis algorithms. These algorithms enjoy the same global convergence guarantee as the original alternating direction method of multipliers, and can be more efficiently implemented with the rich toolbox developed for gradient methods from the deep learning literature. Most notably, these gradient-based algorithms can be combined with stochastic gradient descent methods to produce efficient online sparse principal component analysis algorithms with provable numerical and statistical performance guarantees. The practical performance and usefulness of the new algorithms are demonstrated in various simulation studies. As an application, we show how the scalability and statistical accuracy of our method enable us to find interesting functional gene groups in high-dimensional RNA sequencing data.

In this paper, we present a hybrid of genetic and gradient-based algorilhms, which eflciently find the global optimum without getting stuck at local minima, for iterative microwave inverse scattering(lM1S). It utilizes the global convergence of genetic algorithm(GA) and fine local tunability of gradient-based algorithm through alternation so as to $nd optimum or near-optimum in reasonable computation time. We also propose a new crossover, or 2N-parent parameter-wise crossover(N is the number of parameter) so that GA gives a faster convergence. Experimental results show that the hybrid algorithm can provide better performance eficiently. In the microwave inverse scattering, diffraction topography with the Born approximation gives an efficient reconstruction of a weak scatterer and inversion scheme with the moment method[7] in the reverse sequence may be used for a stsong scatterer[3]. Duect moment method inversion, however, suffers from the ill-posedness in the sense that a small error or noise in the measured scattered fields occurs a large error in the reconstructed profde when the total number of cells of the scatterer is larger than that of the effective propagating modes. To use the number of effective propagating modes equal to or larger than that of the cells, one may use the iterative method with the multiple incident plane waves for microwave inverse scattering of a strong scatterer[6]. An IMIS becomes a parameter optimization problem where the cost function is defined as the sum of the squared magnitudes of the difference between the measured and the calculated scattered fields from the assumed set of dielectric profile by the moment method with the multiple incidences. IMIS using the Levenberg-Marquardt Algorithm(LMA) which is one of the gradient-based Optimization algorithms gives rather limited success in the reconstruction of the large scatterer because it converges to the dielectric profile conespading to the local minimum of the cost function and has initial assumed profile dependency[6]. GA[2], a stochastic optimization algorithm having easy hybridization capability can be a candidate to overcome such local minima problem. However, GA inherently takes a long time to converge because of lack of fine local tunability although it has the advantage of not getting stuck in local minimaE83. Therefore, a hybrid search algorithm utilizing respective advantages of GA and LMA may thus be very useful when landscape on parameter space is very complex like MIS. In the hybrid algorithm, LMA can be used as the fie tuning method with GA used to escape from local minima. Additionally a new crossover, 2N-parent parameter-wise crossover(N iequals the number of parameter), is also proposed so as to increase convergence rate of GA, which eventually can reduce the total computation time. This paper describes the development of such hybrid algorithm, and examines its performance on several IMIS test problems.

Recently, the gradient-based iterative algorithms have been widely exploited for finding the (least-squares) solutions of the different kinds of (coupled) linear matrix equations. Nevertheless, so far, the convergence of the propounded gradient-based algorithms has been studied for the case where the mentioned (coupled) linear matrix equations have a unique (least-squares) solution. In the present paper, we consider the consistent general coupled linear matrix equations which incorporate many of the recently investigated (coupled) linear matrix equations as their special instances. It is demonstrated that using a gradient-based iterative algorithm for solving the mentioned coupled linear matrix equations is equivalent to extending the well-known Richardson method for solving the normal equations corresponding to the original coupled linear matrix equations. In addition, we prove the semi-convergence of the Richardson method when the coefficient matrix of the associated normal equations is singular. Finally, some numerical experiments are presented to illustrate the validity of our theoretical results.

In this paper, the simultaneous static output feedback stabilization problem for a collection of discrete‐time interval systems is considered. A sufficient condition for the existence of static output feedback simultaneously stabilizing controllers is obtained. It is shown that this problem is solvable if a matrix's spectral norm assignment problem is solvable. We proved that the admissible solution set of the matrix's spectral norm assignment problem is convex. Then, the matrix's spectral norm assignment problem is shown to be equivalent to a linear matrix inequality (LMI) feasibility problem. Finally, an example is provided to illustrate the proposed methodology.

A time-delay discrete-time fuzzy singularly perturbed modeling and fuzzy state feedback control approach are presented for a class of complex flexible nonlinear systems with time-delay. The considered flexible nonlinear system is firstly described by a time-delay standard discrete-time fuzzy singular perturbation model. A fuzzy state feedback control law is secondly design. By using a matrix spectral norm and linear matrix inequalities approach, the sufficient conditions of the controller existence are divided. The provided controller not only can stabilize the resulting closed-loop system but also overcome the effects caused by both time-delay and external disturbances. A simulation example is given to illustrate the effectiveness of the developed result.

The Wind in my Backyard (WIMBY) project is developing a web interface to aid communities in siting wind energy projects. As part of this siting tool, users will be able to find realistic wind farm layouts for any proposed site in Europe, given certain constraints. When designing this tool, there arises a need for speed: realistic layouts must be designed in computational times that are appropriate for a web interface. In this study, we compare two optimization algorithms: a gradient-based algorithm, referred to as stochastic gradient descent (SGD), and a gradient-free method, referred to as smart-start. The trade-offs between the optimal energy yield and optimization computational time are characterized via a parameter sweep, considering a site in Denmark. This analysis considered farms with 10, 25, and 50 turbines. We find that smart-start yielded the best results for very short computational times, and that SGD yielded layouts with higher energy yields when considering larger computational times.

ABSTRACTIn this paper, a new gradient-based iterative algorithm is proposed to solve the coupled Lyapunov matrix equations associated with continuous-time Markovian jump linear systems. A necessary and sufficient condition is established for the proposed gradient-based iterative algorithm to be convergent. In addition, the optimal value of the tunable parameter achieving the fastest convergence rate of the proposed algorithm is given explicitly. Finally, some numerical simulations are given to validate the obtained theoretical results.

Performance of iterative gradient-based algorithms with different intensity change models in digital image correlation

This paper presents a gradient-based iterative identification algorithm and an auxiliary-model-based multi-innovation generalized extended stochastic gradient algorithm for input nonlinear systems with autoregressive moving average (ARMA) noises, i.e., the input nonlinear Box–Jenkins (IN–BJ) systems. The estimation errors given by the gradient-based iterative algorithm are smaller than the generalized extended stochastic gradient algorithm under same data lengths. A simulation example is provided.

The Wind Farm Layout Optimization (WFLO) problem is a complex and non-convex optimization problem. Even though a lot of heuristic algorithms and mathematical programming methods have been tested and discussed, there is not a consensus about which algorithm is the most suitable one to solve the WFLO problems. Every algorithm has its own advantages and disadvantages on solving different problems, thus the multi-stage approaches have been picked up. One multi-stage approach applied in solving WFLO problems is to apply an algorithm in stage 1 to capture a coarse, initial optimized layout and import it to stage 2 as an initial condition for another algorithm for further refinement. This paper compared two types of multi-stage methods: The Heuristic-Gradient-based (H-G) model which consists of a heuristic algorithm in stage 1 and a gradient-based algorithm in stage 2; The Discrete-Continuous (D-C) model which consists of a heuristic algorithm in discrete scheme in stage 1 and an algorithm in continuous scheme in stage 2. Annual Energy Production (AEP) is used as the objective function while the computational time associated with each approach is documented. The results illustrate most of the multistage models can improve the optimization procedure both in terms of AEP and computational time. Overall, it is found that the D-C approach is better than the H-G approach. Particularly, the combination of Greedy+Random Search provides the highest AEP and the combination of Greedy and SLSQP provides the lowest computational time.

The wind farm layout optimization (WFLO) problem is a complex and nonconvex optimization problem. Even though many different heuristic algorithms and mathematical programming methods have been tested and discussed, there is no consensus about which algorithm is the most suitable approach for solving WFLO problems. Every algorithm presents its own advantages and disadvantages in solving different optimization problems; thus, multi-stage approaches may combine the advantages of multiple algorithms and offer superior performance. One multi-stage approach used for solving WFLO problems is to apply an algorithm in the first stage to produce an optimized layout which serves as the initial condition for a second-stage algorithm to perform further refinement. This paper presents a comparison between two types of multi-stage methods: the Heuristic-Gradient-based (H-G) model which consists of a heuristic algorithm in stage 1 and a gradient-based algorithm in stage 2 and the Discrete-Continuous (D-C) model which consists of a heuristic algorithm in the discrete scheme in stage 1 and an algorithm in the continuous scheme in stage 2. Annual energy production (AEP) is used as the objective function while the computational time associated with each approach is documented. Three scenarios are investigated in this paper with different complexity in the wind conditions. It was observed that the D-C models provide the optimal solutions with an average of 0.67% higher AEP and an average of 6.2% lower computational time in comparison with the H-G models. The results from this study provide a basis for selecting a proper optimization algorithm for solving WFLO problems which can lead to a significant increase in the overall annual energy production and a large reduction in computational time.