### Citation:

Pappas, D., Katsikis, V. N., & Stanimirović, I. P. (2017). Symbolic Computation of the Aluthge Transform. Mediterranean Journal of Mathematics, 14, 45. Springer International Publishing. Copy at http://www.tinyurl.com/y4o6vr6e

Pappas, D., Katsikis, V. N., & Stanimirović, I. P. (2017). Symbolic Computation of the Aluthge Transform. Mediterranean Journal of Mathematics, 14, 45. Springer International Publishing. Copy at http://www.tinyurl.com/y4o6vr6e

**National and Kapodistrian University of Athens**

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**Email:** vaskatsikis@econ.uoa.gr**URL:** http://scholar.uoa.gr/vaskatsikis**Google Scholar: CLICK ME!**

Stanujkić, D., Karabašević, D., Popović, G., Stanimirović, P. S., Saračević, M., Smarandache, F., Katsikis, V. N., et al. (2021). A New Grey Approach for Using SWARA and PIPRECIA Methods in a Group Decision-Making Environment. Mathematics, 9. Publisher's VersionAbstract

The environment in which the decision-making process takes place is often characterized by uncertainty and vagueness and, because of that, sometimes it is very hard to express the criteria weights with crisp numbers. Therefore, the application of the Grey System Theory, i.e., grey numbers, in this case, is very convenient when it comes to determination of the criteria weights with partially known information. Besides, the criteria weights have a significant role in the multiple criteria decision-making process. Many ordinary multiple criteria decision-making methods are adapted for using grey numbers, and this is the case in this article as well. A new grey extension of the certain multiple criteria decision-making methods for the determination of the criteria weights is proposed. Therefore, the article aims to propose a new extension of the Step-wise Weight Assessment Ratio Analysis (SWARA) and PIvot Pairwise Relative Criteria Importance Assessment (PIPRECIA) methods adapted for group decision-making. In the proposed approach, attitudes of decision-makers are transformed into grey group attitudes, which allows taking advantage of the benefit that grey numbers provide over crisp numbers. The main advantage of the proposed approach in relation to the use of crisp numbers is the ability to conduct different analyses, i.e., considering different scenarios, such as pessimistic, optimistic, and so on. By varying the value of the whitening coefficient, different weights of the criteria can be obtained, and it should be emphasized that this approach gives the same weights as in the case of crisp numbers when the whitening coefficient has a value of 0.5. In addition, in this approach, the grey number was formed based on the median value of collected responses because it better maintains the deviation from the normal distribution of the collected responses. The application of the proposed approach was considered through two numerical illustrations, based on which appropriate conclusions were drawn.

Katsikis, V. N., Mourtas, S. D., Stanimirović, P. S., & Zhang, Y. (2021). Continuous-Time Varying Complex QR Decomposition via Zeroing Neural Dynamics. Neural Processing Letters. presented at the 2021. Publisher's VersionAbstract

QR decomposition (QRD) is of fundamental importance for matrix factorization in both real and complex cases. In this paper, by using zeroing neural dynamics method, a continuous-time model is proposed for solving the time-varying problem of QRD in real-time. The proposed dynamics use time derivative information from a known real or complex matrix. Furthermore, its theoretical analysis is provided to substantiate the convergence and effectiveness of solving the time-varying QRD problem. In addition, numerical experiments in four different-dimensional time-varying matrices show that the proposed model is effective for solving the time-varying QRD problem both in the case of a real or a complex matrix as input.

Stanujkić, D., Karabašević, D., Popović, G., Stanimirović, P. S., Smarandache, F., Saračević, M., Katsikis, V. N., et al. (2021). An Innovative Grey Approach for Group Multi-Criteria Decision Analysis Based on the Median of Ratings by Using Python. Axioms, 10. Publisher's VersionAbstract

Some decision-making problems, i.e., multi-criteria decision analysis (MCDA) problems, require taking into account the attitudes of a large number of decision-makers and/or respondents. Therefore, an approach to the transformation of crisp ratings, collected from respondents, in grey interval numbers form based on the median of collected scores, i.e., ratings, is considered in this article. In this way, the simplicity of collecting respondents’ attitudes using crisp values, i.e., by applying some form of Likert scale, is combined with the advantages that can be achieved by using grey interval numbers. In this way, a grey extension of MCDA methods is obtained. The application of the proposed approach was considered in the example of evaluating the websites of tourism organizations by using several MCDA methods. Additionally, an analysis of the application of the proposed approach in the case of a large number of respondents, done in Python, is presented. The advantages of the proposed method, as well as its possible limitations, are summarized.

Katsikis, V. N., & Mourtas, S. D. (2021). Binary Beetle Antennae Search Algorithm for Tangency Portfolio Diversification. Journal of Modeling and Optimization, 13(1). Publisher's VersionAbstract

The tangency portfolio, also known as the market portfolio, is the most efficient portfolio and arises from the intercept point of the Capital Market Line (CML) and the efficient frontier. In this paper, a binary optimal tangency portfolio under cardinality constraint (BOTPCC) problem is defined and studied as a nonlinear programming (NLP) problem. Because such NLP problems are widely approached by heuristic, a binary beetle antennae search algorithm is employed to provide a solution to the BTPSCC problem. Our method proved to be a magnificent substitute to other evolutionary algorithms in real-world datasets, based on numerical applications and computer simulations.

Katsikis, V. N., & Mourtas, S. D. (2021). Portfolio Insurance and Intelligent Algorithms. In S. Patnaik, Tajeddini, K., & Jain, V. (Eds.), Computational Management: Applications of Computational Intelligence in Business Management (pp. 305 - 323). presented at the 2021, Cham: Springer International Publishing. Publisher's VersionAbstract

Minimizing portfolio insurance (PI) costs is an investment strategy of great importance. In this chapter, by converting the classical minimum-cost PI (MCPI) problem to a multi-period MCPI (MPMCPI) problem, we define and investigate the MPMCPI under transaction costs (MPMCPITC) problem as a nonlinear programming (NLP) problem. The problem of MCPI gets more genuine in this way. Given the fact that such NLP problems are widely handled by intelligent algorithms, we are introducing a well-tuned approach that can solve the challenging MPMCPITC problem. In our portfolios’ applications, we use real-world data and, along with some of the best memetic meta-heuristic and commercial methods, we provide a solution to the MPMCPITC problem, and we compare their solutions to each other.

Stanimirović, P. S., Katsikis, V. N., Jin, L., & Mosić, D. (2021). Properties and computation of continuous-time solutions to linear systems. Applied Mathematics and Computation, 405. presented at the 2021. Publisher's VersionAbstract

We investigate solutions to the system of linear equations (SoLE) in both the time-varying and time-invariant cases, using both gradient neural network (GNN) and Zhang neural network (ZNN) designs. Two major limitations should be overcome. The first limitation is the inapplicability of GNN models in time-varying environment, while the second constraint is the possibility of using the ZNN design only under the presence of invertible coefficient matrix. In this paper, by overcoming the possible limitations, we suggest, in all possible cases, a suitable solution for a consistent or inconsistent linear system. Convergence properties are investigated as well as exact solutions.

Katsikis, V. N., Mourtas, S. D., Stanimirović, P. S., Li, S., & Cao, X. (2021). Time-Varying Mean-Variance Portfolio Selection under Transaction Costs and Cardinality Constraint Problem via Beetle Antennae Search Algorithm (BAS). Operations Research Forum, 2(2), 18. presented at the 2021. Publisher's VersionAbstract

The Markowitz mean-variance portfolio selection is widely acclaimed as a very important investment strategy. A popular option to solve the static mean-variance portfolio selection (MVPS) problem is based on the use of quadratic programming (QP) methods. On the other hand, the static portfolio selection under transaction costs (PSTC) problem is usually approached with nonlinear programming (NLP) methods. In this article, we define and study the time-varying mean-variance portfolio selection under transaction costs and cardinality constraint (TV-MVPSTC-CC) problem as a time-varying nonlinear programming (TVNLP) problem. The TV-MVPSTC-CC also comprises the properties of a moving average. These properties make the TV-MVPSTC-CC an even greater analysis tool suitable to evaluate investments and identify trading opportunities across a continuous-time period. Using the Beetle Antennae Search (BAS) algorithm, we also provide an online solution to the static NLP problem. To the best of our knowledge, this is an innovative approach that incorporates modern meta-heuristic optimization techniques to provide an online, thus more realistic, solution to the TV-MVPSTC-CC problem. In this way, we present an online solution to a time-varying financial problem while eliminating the restrictions of static methods. Our approach is also verified by numerical experiments and computer simulations as an excellent alternative to traditional approaches.

Li, Z., Zhang, Y., Ming, L., Guo, J., & Katsikis, V. N. (2021). Real-Domain QR Decomposition Models Employing Zeroing Neural Network and Time-Discretization Formulas for Time-Varying Matrices. Neurocomputing. presented at the 2021. Publisher's VersionAbstract

This study investigated the problem of QR decomposition for time-varying matrices. We transform the original QR decomposition problem into an equation system using its constraints. Then, we propose a continuous-time QR decomposition (CTQRD) model by applying zeroing neural network method, equivalent transformations, Kronecker product, and vectorization techniques. Subsequently, a high-precision ten-instant Zhang et al discretization (ZeaD) formula is proposed. A ten-instant discrete-time QR decomposition model is also proposed by using the ten-instant ZeaD formula to discretize the CTQRD model. Moreover, three discrete-time QR decomposition models are proposed by applying three other ZeaD formulas, and three examples of QR decomposition are presented. The experimental results confirm the effectiveness and correctness of the proposed models for the QR decomposition of time-varying matrices.

Katsikis, V. N., Mourtas, S. D., Stanimirović, P. S., & Zhang, Y. (2021). Solving Complex-Valued Time-Varying Linear Matrix Equations via QR Decomposition With Applications to Robotic Motion Tracking and on Angle-of-Arrival Localization. IEEE Transactions on Neural Networks and Learning Systems, 1 - 10.

Mosić, D., Stanimirović, P. S., Sahoo, J. K., Behera, R., & Katsikis, V. N. (2020). One-sided weighted outer inverses of tensors. Journal of Computational and Applied Mathematics. presented at the 2020. Publisher's VersionAbstract

In this paper, for the first time in literature, we introduce one-sided weighted inverses and extend the notions of one-sided inverses, outer inverses and inverses along given elements. Although our results are new and in the matrix case, we decided to present them in tensor space with reshape operator. For this purpose, a left and right (M,N)-weighted (B,C)-inverse and the (M,N)-weighted (B,C)-inverse of a tensor are defined. Additionally, necessary and sufficient conditions for the existence of these new inverses are presented. We describe the sets of all left (or right) (M,N)-weighted (B,C)-inverses of a given tensor. As consequences of these results, we consider the one-sided (B,C)-inverse, (B,C)-inverse, one-sided inverse along a tensor and inverse along a tensor. Further, we introduce a (M,N)-weighted (B,C)-outer inverse and a W-weighted (B,C)-outer inverse of tensors with a few characterizations. Then, corresponding algorithms for computing various types of outer inverses of tensors are proposed, implemented and tested. The prowess of the proposed inverses are demonstrated for finding the solution of Poisson problem and the restoration of 3D color images.