A Comparative Study of Stochastic Model Predictive Controllers

[EN] A comparative study of two state-of-the-art stochastic model predictive controllers for linear systems with parametric and additive uncertainties is presented. On the one hand, Stochastic Model Predictive Control (SMPC) is based on analytical methods and solves an optimal control problem (OCP) similar to a classic Model Predictive Control (MPC) with constraints. SMPC defines probabilistic constraints on the states, which are transformed into equivalent deterministic ones. On the other hand, Scenario-based Model Predictive Control (SCMPC) solves an OCP for a specified number of random realizations of uncertainties, also called scenarios. In this paper, Classic MPC, SMPC and SCMPC are compared through two numerical examples. Thanks to several Monte-Carlo simulations, performances of classic MPC, SMPC and SCMPC are compared using several criteria, such as number of successful runs, number of times the constraints are violated, integral absolute error and computational cost. Moreover, a Stochastic Model Predictive Control Toolbox was developed by the authors, available on MATLAB Central, in which it is possible to simulate a SMPC or a SCMPC to control multivariable linear systems with additive disturbances. This software was used to carry out part of the simulations of the numerical examples in this article and it can be used for results reproduction.Gonzalez, E.; Sanchís Saez, J.; Garcia-Nieto, S.; Salcedo-Romero-De-Ávila, J. (2020). A Comparative Study of Stochastic Model Predictive Controllers. Electronics. 9(12):1-22. https://doi.org/10.3390/electronics9122078S12291

Tri-Criterion Model for Constructing Low-Carbon Mutual Fund Portfolios: A Preference-Based Multi-Objective Genetic Algorithm Approach

[EN] Sustainable finance, which integrates environmental, social and governance criteria on financial decisions rests on the fact that money should be used for good purposes. Thus, the financial sector is also expected to play a more important role to decarbonise the global economy. To align financial flows with a pathway towards a low-carbon economy, investors should be able to integrate into their financial decisions additional criteria beyond return and risk to manage climate risk. We propose a tri-criterion portfolio selection model to extend the classical Markowitz's mean-variance approach to include investor's preferences on the portfolio carbon risk exposure as an additional criterion. To approximate the 3D Pareto front we apply an efficient multi-objective genetic algorithm called ev-MOGA which is based on the concept of epsilon-dominance. Furthermore, we introduce a-posteriori approach to incorporate the investor's preferences into the solution process regarding their climate-change related preferences measured by the carbon risk exposure and their loss-adverse attitude. We test the performance of the proposed algorithm in a cross-section of European socially responsible investments open-end funds to assess the extent to which climate-related risk could be embedded in the portfolio according to the investor's preferences.Hilario Caballero, A.; Garcia-Bernabeu, A.; Salcedo-Romero-De-Ávila, J.; Vercher, M. (2020). Tri-Criterion Model for Constructing Low-Carbon Mutual Fund Portfolios: A Preference-Based Multi-Objective Genetic Algorithm Approach. International Journal of Environmental research and Public Health. 17(17):1-15. https://doi.org/10.3390/ijerph17176324S1151717Morningstar Low Carbon Designationhttps://bit.ly/2SfAFUAKrueger, P., Sautner, Z., & Starks, L. T. (2020). The Importance of Climate Risks for Institutional Investors. The Review of Financial Studies, 33(3), 1067-1111. doi:10.1093/rfs/hhz137Syam, S. S. (1998). A dual ascent method for the portfolio selection problem with multiple constraints and linked proposals. European Journal of Operational Research, 108(1), 196-207. doi:10.1016/s0377-2217(97)00048-9Li, D., Sun, X., & Wang, J. (2006). OPTIMAL LOT SOLUTION TO CARDINALITY CONSTRAINED MEAN-VARIANCE FORMULATION FOR PORTFOLIO SELECTION. Mathematical Finance, 16(1), 83-101. doi:10.1111/j.1467-9965.2006.00262.xBertsimas, D., & Shioda, R. (2007). Algorithm for cardinality-constrained quadratic optimization. Computational Optimization and Applications, 43(1), 1-22. doi:10.1007/s10589-007-9126-9Bawa, V. S. (1975). Optimal rules for ordering uncertain prospects. Journal of Financial Economics, 2(1), 95-121. doi:10.1016/0304-405x(75)90025-2Konno, H., & Yamazaki, H. (1991). Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market. Management Science, 37(5), 519-531. doi:10.1287/mnsc.37.5.519Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443-1471. doi:10.1016/s0378-4266(02)00271-6Mansini, R. (2003). LP solvable models for portfolio optimization: a classification and computational comparison. IMA Journal of Management Mathematics, 14(3), 187-220. doi:10.1093/imaman/14.3.187Hirschberger, M., Steuer, R. E., Utz, S., Wimmer, M., & Qi, Y. (2013). Computing the Nondominated Surface in Tri-Criterion Portfolio Selection. Operations Research, 61(1), 169-183. doi:10.1287/opre.1120.1140Utz, S., Wimmer, M., Hirschberger, M., & Steuer, R. E. (2014). Tri-criterion inverse portfolio optimization with application to socially responsible mutual funds. European Journal of Operational Research, 234(2), 491-498. doi:10.1016/j.ejor.2013.07.024Utz, S., Wimmer, M., & Steuer, R. E. (2015). Tri-criterion modeling for constructing more-sustainable mutual funds. European Journal of Operational Research, 246(1), 331-338. doi:10.1016/j.ejor.2015.04.035Chang, T.-J., Meade, N., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimisation. Computers & Operations Research, 27(13), 1271-1302. doi:10.1016/s0305-0548(99)00074-xMaringer, D., & Kellerer, H. (2003). Optimization of cardinality constrained portfolios with a hybrid local search algorithm. OR Spectrum, 25(4), 481-495. doi:10.1007/s00291-003-0139-1Shaw, D. X., Liu, S., & Kopman, L. (2008). Lagrangian relaxation procedure for cardinality-constrained portfolio optimization. Optimization Methods and Software, 23(3), 411-420. doi:10.1080/10556780701722542Soleimani, H., Golmakani, H. R., & Salimi, M. H. (2009). Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm. Expert Systems with Applications, 36(3), 5058-5063. doi:10.1016/j.eswa.2008.06.007Anagnostopoulos, K. P., & Mamanis, G. (2011). The mean–variance cardinality constrained portfolio optimization problem: An experimental evaluation of five multiobjective evolutionary algorithms. Expert Systems with Applications. doi:10.1016/j.eswa.2011.04.233Woodside-Oriakhi, M., Lucas, C., & Beasley, J. E. (2011). Heuristic algorithms for the cardinality constrained efficient frontier. European Journal of Operational Research, 213(3), 538-550. doi:10.1016/j.ejor.2011.03.030Meghwani, S. S., & Thakur, M. (2017). Multi-criteria algorithms for portfolio optimization under practical constraints. Swarm and Evolutionary Computation, 37, 104-125. doi:10.1016/j.swevo.2017.06.005Liagkouras, K., & Metaxiotis, K. (2016). A new efficiently encoded multiobjective algorithm for the solution of the cardinality constrained portfolio optimization problem. Annals of Operations Research, 267(1-2), 281-319. doi:10.1007/s10479-016-2377-zMetaxiotis, K., & Liagkouras, K. (2012). Multiobjective Evolutionary Algorithms for Portfolio Management: A comprehensive literature review. Expert Systems with Applications, 39(14), 11685-11698. doi:10.1016/j.eswa.2012.04.053Silva, Y. L. T. V., Herthel, A. B., & Subramanian, A. (2019). A multi-objective evolutionary algorithm for a class of mean-variance portfolio selection problems. Expert Systems with Applications, 133, 225-241. doi:10.1016/j.eswa.2019.05.018Chang, T.-J., Yang, S.-C., & Chang, K.-J. (2009). Portfolio optimization problems in different risk measures using genetic algorithm. Expert Systems with Applications, 36(7), 10529-10537. doi:10.1016/j.eswa.2009.02.062Liagkouras, K. (2019). A new three-dimensional encoding multiobjective evolutionary algorithm with application to the portfolio optimization problem. Knowledge-Based Systems, 163, 186-203. doi:10.1016/j.knosys.2018.08.025Kaucic, M., Moradi, M., & Mirzazadeh, M. (2019). Portfolio optimization by improved NSGA-II and SPEA 2 based on different risk measures. Financial Innovation, 5(1). doi:10.1186/s40854-019-0140-6Babaei, S., Sepehri, M. M., & Babaei, E. (2015). Multi-objective portfolio optimization considering the dependence structure of asset returns. European Journal of Operational Research, 244(2), 525-539. doi:10.1016/j.ejor.2015.01.025Ruiz, A. B., Saborido, R., Bermúdez, J. D., Luque, M., & Vercher, E. (2019). Preference-based evolutionary multi-objective optimization for portfolio selection: a new credibilistic model under investor preferences. Journal of Global Optimization, 76(2), 295-315. doi:10.1007/s10898-019-00782-1Anagnostopoulos, K. P., & Mamanis, G. (2010). A portfolio optimization model with three objectives and discrete variables. Computers & Operations Research, 37(7), 1285-1297. doi:10.1016/j.cor.2009.09.009Hu, Y., Chen, H., He, M., Sun, L., Liu, R., & Shen, H. (2019). Multi-Swarm Multi-Objective Optimizer Based on p-Optimality Criteria for Multi-Objective Portfolio Management. Mathematical Problems in Engineering, 2019, 1-22. doi:10.1155/2019/8418369Rangel-González, J. A., Fraire, H., Solís, J. F., Cruz-Reyes, L., Gomez-Santillan, C., Rangel-Valdez, N., & Carpio-Valadez, J. M. (2020). Fuzzy Multi-objective Particle Swarm Optimization Solving the Three-Objective Portfolio Optimization Problem. International Journal of Fuzzy Systems, 22(8), 2760-2768. doi:10.1007/s40815-020-00928-4Garcia-Bernabeu, A., Salcedo, J. V., Hilario, A., Pla-Santamaria, D., & Herrero, J. M. (2019). Computing the Mean-Variance-Sustainability Nondominated Surface by ev-MOGA. Complexity, 2019, 1-12. doi:10.1155/2019/6095712Laumanns, M., Thiele, L., Deb, K., & Zitzler, E. (2002). Combining Convergence and Diversity in Evolutionary Multiobjective Optimization. Evolutionary Computation, 10(3), 263-282. doi:10.1162/106365602760234108Matlab Central: ev-MOGA Multiobjective Evolutionary Algorithmhttps://bit.ly/3f2BYQMBlasco, X., Herrero, J. M., Sanchis, J., & Martínez, M. (2008). A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization. Information Sciences, 178(20), 3908-3924. doi:10.1016/j.ins.2008.06.01

Analyzing the Nearly Optimal Solutions in a Multi-Objective Optimization Approach for the Multivariable Nonlinear Identification of a PEM Fuel Cell Cooling System

[EN] In this work, the parametric identification of a cooling system in a PEM (proton exchange membrane) fuel cell is carried out. This system is multivariable and nonlinear. In this type of system there are different objectives and the unmodeled dynamics cause conflicting objectives (prediction errors in each output). For this reason, resolution is proposed using a multi-objective optimization approach. Nearly optimal alternatives can exist in any optimization problem. Among them, the nearly optimal solutions that are significantly different (that we call nearly optimal solutions nondominated in their neighborhood) are potentially useful solutions. In identification problems, two situations arise for consideration: 1) aggregation in the design objectives (when considering the prediction error throughout the identification test). When an aggregation occurs in the design objectives, interesting non-neighboring (significantly different) multimodal and nearly optimal alternatives appear. These alternatives have different trade-offs in the aggregated objectives; 2) new objectives in decision making appear. Some models can, with similar performance in the design objectives, obtain a significant improvement in new objectives not included in the optimization phase. A typical case of additional objectives are the validation objectives. In these situations, nearly optimal solutions nondominated in their neighborhood play a key role. These alternatives allow the designer to make the final decision with more valuable information. Therefore, this work highlights, as a novelty, the relevance of considering nearly optimal models nondominated in their neighborhood in problems of parametric identification of multivariable nonlinear systems and shows an application in a complex problem.This work was supported in part by the Ministerio de Ciencia, Innovacion y Universidades, Spain, under Grant RTI2018-096904-B-I00, and in part by the Generalitat Valenciana Regional Government under Project AICO/2019/055.Pajares-Ferrando, A.; Blasco, X.; Herrero Durá, JM.; Salcedo-Romero-De-Ávila, J. (2020). Analyzing the Nearly Optimal Solutions in a Multi-Objective Optimization Approach for the Multivariable Nonlinear Identification of a PEM Fuel Cell Cooling System. IEEE Access. 8:114361-114377. https://doi.org/10.1109/ACCESS.2020.3003741S114361114377

Co-simulation platform for geometric design, trajectory control and guidance of racing drones

[EN] The design of racing drones brings quite a thrilling challenge from a flight dynamics point of view. This work aims to offer a single-based simulation platform combining its geometric design, trajectory control, and guidance of racing drones. Also, it is reckoned from a pilot¿s view in a classic FPV competition. Hence, it is an active platform for studying racing drones¿ design founded on dynamics, with fifteen different drone models. It is one of the few existing platforms that combine all aspects of racing drones in a single simulation environment. Also, it is open access via Matlab Central - File Exchange.This work was partially supported by proyect PID2020-119468RA-I00 funded by MCIN/AEI/10.13039/501100011033.Castiblanco Quintero, J.; Garcia-Nieto, S.; Simarro Fernández, R.; Salcedo-Romero-De-Ávila, J. (2022). Co-simulation platform for geometric design, trajectory control and guidance of racing drones. International Journal of Micro Air Vehicles. 14:1-20. https://doi.org/10.1177/175682932211437851201

BIBO stabilisation of continuous time takagi sugeno systems under persistent perturbations and input saturation

[EN] This paper presents a novel approach to the design of fuzzy state feedback controllers for continuous-time non-linear systems with input saturation under persistent perturbations. It is assumed that all the states of the Takagi¿Sugeno (TS) fuzzy model representing a non-linear system are measurable. Such controllers achieve bounded input bounded output (BIBO) stabilisation in closed loop based on the computation of inescapable ellipsoids. These ellipsoids are computed with linear matrix inequalities (LMIs) that guarantee stabilisation with input saturation and persistent perturbations. In particular, two kinds of inescapable ellipsoids are computed when solving a multiobjective optimization problem: the maximum volume inescapable ellipsoids contained inside the validity domain of the TS fuzzy model and the smallest inescapable ellipsoids which guarantee a minimum *-norm (upper bound of the 1-norm) of the perturbed system. For every initial point contained in the maximum volume ellipsoid, the closed loop will enter the minimum *-norm ellipsoid after a finite time, and it will remain inside afterwards. Consequently, the designed controllers have a large domain of validity and ensure a small value for the 1-norm of closed loop.The authors wish to thank the Editor-in-Chief and the anonymous reviewers for their valuable comments and suggestions. This work has been funded by Ministerio de Economia y Competitividad (Spain) through the research project DPI2015-71443-R and by Generalitat Valenciana (Valencia, Spain) through the research project GV/2017/029.Salcedo-Romero-De-Ávila, J.; Martínez Iranzo, MA.; Garcia-Nieto, S.; Hilario Caballero, A. (2018). BIBO stabilisation of continuous time takagi sugeno systems under persistent perturbations and input saturation. International Journal of Applied Mathematics and Computer Science (Online). 28(3):457-472. https://doi.org/10.2478/amcs-2018-0035S45747228

Multi-Objective Optimization for Wind Estimation and Aircraft Model Identification

In this paper, a novel method for aerodynamic model identification of a micro-air vehicle is proposed. The principal contribution is a technique of wind estimation that provides information about the existing wind during flight when no air-data sensors are available. The estimation technique employs multi-objective optimization algorithms that utilize identification errors to propose the wind-speed components that best fit the dynamic behavior observed. Once the wind speed is estimated, the flight experimentation data are corrected and utilized to perform an identification of the aircraft model parameters. A multi-objective optimization algorithm is also used, but with the objective of estimating the aerodynamic stability and control derivatives. Employing data from different flights offers the possibility of obtaining sets of models that form the Pareto fronts. Deciding which model best adjusts to the experiments performed (compromise model) will be the ultimate task of the control engineer.The authors would like to thank the Spanish Ministry of Innovation and Science for providing funding through grant BES-2012-056210 and projects TIN-2011-28082 and ENE-25900. We also want to acknowledge the Generalitat Valenciana for financing this work through project PROMETEO/2012/028.Velasco Carrau, J.; García-Nieto Rodríguez, S.; Salcedo Romero De Ávila, JV.; Bishop, RH. (2015). Multi-Objective Optimization for Wind Estimation and Aircraft Model Identification. Journal of Guidance, Control, and Dynamics. 39(2):372-389. https://doi.org/10.2514/1.G001294S37238939

T-S Fuzzy Bibo Stabilisation of Non-Linear Systems Under Persistent Perturbations Using Fuzzy Lyapunov Functions and Non-PDC Control Laws

[EN] This paper develops an innovative approach for designing non-parallel distributed fuzzy controllers for continuous-time non-linear systems under persistent perturbations. Non-linear systems are represented using Takagi-Sugeno fuzzy models. These non-PDC controllers guarantee bounded input bounded output stabilisation in closed-loop throughout the computation of generalised inescapable ellipsoids. These controllers are computed with linear matrix inequalities using fuzzy Lyapunov functions and integral delayed Lyapunov functions. LMI conditions developed in this paper provide non-PDC controllers with a minimum *-norm (upper bound of the 1-norm) for the T-S fuzzy system under persistent perturbations. The results presented in this paper can be classified into two categories: local methods based on fuzzy Lyapunov functions with guaranteed bounds on the first derivatives of membership functions and global methods based on integral-delayed Lyapunov functions which are independent of the first derivatives of membership functions. The benefits of the proposed results are shown through some illustrative examples.This work has been funded by Ministerio de Economia y Competitividad, Spain (research project RTI2018-096904-B-I00) and Conselleria de Educacion, Cultura y Deporte-Generalitat Valenciana, Spain (research project AICO/2019/055).Salcedo-Romero-De-Ávila, J.; Martínez Iranzo, MA.; Garcia-Nieto, S.; Hilario Caballero, A. (2020). T-S Fuzzy Bibo Stabilisation of Non-Linear Systems Under Persistent Perturbations Using Fuzzy Lyapunov Functions and Non-PDC Control Laws. International Journal of Applied Mathematics and Computer Science (Online). 30(3):529-550. https://doi.org/10.34768/amcs-2020-0039S52955030

Experimental study on the dynamic behaviour of drones designed for racing competitions

[EN] Drones designed for racing usually feature powerful miniaturised electronics embedded in fairly light and strong geometric composite structures. The main objective of this article is to analyse the behaviour of various models of racing drones and their geometrical structures (airframes). Two approaches have been made: (i) an analysis of the information collected by a set of speed and time sensors located on an indoor race track and using a statistical technique (box and whiskers diagram) and (ii) an analysis of the know-how (flight sensations) of a group of racing pilots using a series of technical interviews on the behaviour of their drones. By contrasting these approaches, it has been possible to validate numerically the effects of varying the arm angles, as well as lengths, on a test race track and relate the geometry of these structures to racing behaviourThe author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by project RTI2018-096904-B-I00 from the Spanish Ministry of Economy, and by project AICO/2019/055 from Generalitat Valenciana.Castiblanco Quintero, JM.; Garcia-Nieto, S.; Simarro Fernández, R.; Salcedo-Romero-De-Ávila, J. (2021). Experimental study on the dynamic behaviour of drones designed for racing competitions. International Journal of Micro Air Vehicles. 13:1-22. https://doi.org/10.1177/175682932110057571221

Predictive LPV control of a liquid-gas separation process

[EN] The problem of controlling a liquid-gas separation process is approached by using LPV control techniques. An LPV model is derived from a nonlinear model of the process using differential inclusion techniques. Once an LPV model is available, an LPV controller can be synthesized. The authors present a predictive LPV controller based on the GPC controller [Clarke D, Mohtadi C, Tuffs P. Generalized predictive control - Part I. Automatica 1987;23(2):137-48; Clarke D, Mohtadi C, Tuffs P. Generalized predictive control - Part II. Extensions and interpretations. Automatica 1987;23(2):149-60]. The resulting controller is denoted as GPC-LPV. This one shows the same structure as a general LPV controller [El Gahoui L, Scorletti G. Control of rational systems using linear-fractional representations and linear matrix inequalities. Automatica 1996;32(9):1273-84; Scorletti G, El Ghaoui L. Improved LMI conditions for gain scheduling and related control problems. International Journal of Robust Nonlinear Control 1998;8:845-77; Apkarian P, Tuan HD. Parametrized LMIs in control theory. In: Proceedings of the 37th IEEE conference on decision and control; 1998. p. 152-7; Scherer CW. LPV control and full block multipliers. Automatica 2001;37:361-75], which presents a linear fractional dependence on the process signal measurements. Therefore, this controller has the ability of modifying its dynamics depending on measurements leading to a possibly nonlinear controller. That controller is designed in two steps. First, for a given steady state point is obtained a linear GPC using a linear local model of the nonlinear system around that operating point. And second, using bilinear and linear matrix inequalities (BMIs/LMIs) the remaining matrices of GPC-LPV are selected in order to achieve some closed loop properties: stability in some operation zone, norm bounding of some input/output channels, maximum settling time, maximum overshoot, etc., given some LPV model for the nonlinear system. As an application, a GPC-LPV is designed for the derived LPV model of the liquid-gas separation process. This methodology can be applied to any nonlinear system which can be embedded in an LPV system using differential inclusion techniques. (C) 2006 Elsevier Ltd. All rights reserved.Partially supported by projects: CICYT DPI2004-08383-C03-02 and DPI2005-07835.Salcedo-Romero-De-Ávila, J.; Martínez Iranzo, MA.; Ramos Fernández, C.; Herrero Durá, JM. (2007). Predictive LPV control of a liquid-gas separation process. Advances in Engineering Software. 38(7):466-474. https://doi.org/10.1016/j.advengsoft.2006.10.003S46647438

Computing the Mean-Variance-Sustainability Nondominated Surface by ev-MOGA

[EN] Despite the widespread use of the classical bicriteria Markowitz mean-variance framework, a broad consensus is emerging on the need to include more criteria for complex portfolio selection problems. Sustainable investing, also called socially responsible investment, is becoming a mainstream investment practice. In recent years, some scholars have attempted to include sustainability as a third criterion to better reflect the individual preferences of those ethical or green investors who are willing to combine strong financial performance with social benefits. For this purpose, new computational methods for optimizing this complex multiobjective problem are needed. Multiobjective evolutionary algorithms (MOEAs) have been recently used for portfolio selection, thus extending the mean-variance methodology to obtain a mean-variance-sustainability nondominated surface. In this paper, we apply a recent multiobjective genetic algorithm based on the concept of epsilon-dominance called ev-MOGA. This algorithm tries to ensure convergence towards the Pareto set in a smart distributed manner with limited memory resources. It also adjusts the limits of the Pareto front dynamically and prevents solutions belonging to the ends of the front from being lost. Moreover, the individual preferences of socially responsible investors could be visualised using a novel tool, known as level diagrams, which helps investors better understand the range of values attainable and the tradeoff between return, risk, and sustainability.This work was funded by "Ministerio de Economia y Competitividad" (Spain), research project RTI2018-096904B-I00, and "Conselleria de Educacion, Cultura y DeporteGeneralitat Valenciana" (Spain), research project AICO/2019/055Garcia-Bernabeu, A.; Salcedo-Romero-De-Ávila, J.; Hilario Caballero, A.; Pla Santamaría, D.; Herrero Durá, JM. (2019). Computing the Mean-Variance-Sustainability Nondominated Surface by ev-MOGA. Complexity. 2019:1-12. https://doi.org/10.1155/2019/6095712S1122019Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77. doi:10.2307/2975974Hirschberger, M., Steuer, R. E., Utz, S., Wimmer, M., & Qi, Y. (2013). Computing the Nondominated Surface in Tri-Criterion Portfolio Selection. Operations Research, 61(1), 169-183. doi:10.1287/opre.1120.1140Utz, S., Wimmer, M., Hirschberger, M., & Steuer, R. E. (2014). Tri-criterion inverse portfolio optimization with application to socially responsible mutual funds. European Journal of Operational Research, 234(2), 491-498. doi:10.1016/j.ejor.2013.07.024Utz, S., Wimmer, M., & Steuer, R. E. (2015). Tri-criterion modeling for constructing more-sustainable mutual funds. European Journal of Operational Research, 246(1), 331-338. doi:10.1016/j.ejor.2015.04.035Qi, Y., Steuer, R. E., & Wimmer, M. (2015). An analytical derivation of the efficient surface in portfolio selection with three criteria. Annals of Operations Research, 251(1-2), 161-177. doi:10.1007/s10479-015-1900-yGasser, S. M., Rammerstorfer, M., & Weinmayer, K. (2017). Markowitz revisited: Social portfolio engineering. European Journal of Operational Research, 258(3), 1181-1190. doi:10.1016/j.ejor.2016.10.043Qi, Y. (2018). On outperforming social-screening-indexing by multiple-objective portfolio selection. Annals of Operations Research, 267(1-2), 493-513. doi:10.1007/s10479-018-2921-0Nathaphan, S., & Chunhachinda, P. (2010). Estimation Risk Modeling in Optimal Portfolio Selection: An Empirical Study from Emerging Markets. Economics Research International, 2010, 1-10. doi:10.1155/2010/340181DeMiguel, V., Garlappi, L., & Uppal, R. (2007). Optimal Versus Naive Diversification: How Inefficient is the 1/NPortfolio Strategy? Review of Financial Studies, 22(5), 1915-1953. doi:10.1093/rfs/hhm075Metaxiotis, K., & Liagkouras, K. (2012). Multiobjective Evolutionary Algorithms for Portfolio Management: A comprehensive literature review. Expert Systems with Applications, 39(14), 11685-11698. doi:10.1016/j.eswa.2012.04.053Bertsimas, D., & Shioda, R. (2007). Algorithm for cardinality-constrained quadratic optimization. Computational Optimization and Applications, 43(1), 1-22. doi:10.1007/s10589-007-9126-9Chang, T.-J., Yang, S.-C., & Chang, K.-J. (2009). Portfolio optimization problems in different risk measures using genetic algorithm. Expert Systems with Applications, 36(7), 10529-10537. doi:10.1016/j.eswa.2009.02.062Woodside-Oriakhi, M., Lucas, C., & Beasley, J. E. (2011). Heuristic algorithms for the cardinality constrained efficient frontier. European Journal of Operational Research, 213(3), 538-550. doi:10.1016/j.ejor.2011.03.030Chen, B., Lin, Y., Zeng, W., Xu, H., & Zhang, D. (2017). 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