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Journal articles on the topic 'Portfolio optimisation'

Author: Grafiati

Published: 4 June 2021

Last updated: 26 July 2025

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1

Bulani, Vivek, Marija Bezbradica, and Martin Crane. "Improving Portfolio Management Using Clustering and Particle Swarm Optimisation." Mathematics 13, no. 10 (2025): 1623. https://doi.org/10.3390/math13101623.

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Portfolio management, a critical application of financial market analysis, involves optimising asset allocation to maximise returns while minimising risk. This paper addresses the notable research gap in analysing historical financial data for portfolio optimisation purposes. Particularly, this research examines different approaches for handling missing values and volatility, while examining their effects on optimal portfolios. For this portfolio optimisation task, this study employs a metaheuristic approach through the Swarm Intelligence algorithm, particularly Particle Swarm Optimisation and

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2

Shabe, Refiloe, Andries Engelbrecht, and Kian Anderson. "Incremental Reinforcement Learning for Portfolio Optimisation." Computers 14, no. 7 (2025): 242. https://doi.org/10.3390/computers14070242.

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Portfolio optimisation is a crucial decision-making task. Traditionally static, this problem is more realistically addressed as dynamic, reflecting frequent trading within financial markets. The dynamic nature of the portfolio optimisation problem makes it susceptible to rapid market changes or financial contagions, which may cause drifts in historical data. While reinforcement learning (RL) offers a framework that allows for the formulation of portfolio optimisation as a dynamic problem, existing RL approaches lack the ability to adapt to rapid market changes, such as pandemics, and fail to c

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3

Vasant, Jiten, Laurent Irgolic, Ryan Kruger, and Kanshukan Rajaratnam. "A Comparison Of Mean-Variance And Mean-Semivariance Optimisation On The JSE." Journal of Applied Business Research (JABR) 30, no. 6 (2014): 1587. http://dx.doi.org/10.19030/jabr.v30i6.8876.

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<p>This study investigates the effectiveness of semivariance versus mean-variance optimisation on a risk-adjusted basis on the JSE. We compare semivariance and mean-variance optimisation prior to, during and after the recent financial crisis period. Additionally, we investigate the inclusion of a fixed-income asset in the optimal portfolio. The results suggest that semivariance optimisation on the JSE in a pure equity case produces lower absolute returns, yet superior risk-adjusted returns. Further investigation suggests that semivariance metrics are effective within a certain range of p

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Yin, X. A., Z. F. Yang, and C. L. Liu. "Portfolio optimisation for hydropower producers that balances riverine ecosystem protection and producer needs." Hydrology and Earth System Sciences 18, no. 4 (2014): 1359–68. http://dx.doi.org/10.5194/hess-18-1359-2014.

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Abstract. In deregulated electricity markets, hydropower portfolio design has become an essential task for producers. The previous research on hydropower portfolio optimisation focused mainly on the maximisation of profits but did not take into account riverine ecosystem protection. Although profit maximisation is the major objective for producers in deregulated markets, protection of riverine ecosystems must be incorporated into the process of hydropower portfolio optimisation, especially against a background of increasing attention to environmental protection and stronger opposition to hydro

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5

Yin, X. A., Z. F. Yang, and C. L. Liu. "Portfolio optimisation for hydropower producers that balances riverine ecosystem protection and producer needs." Hydrology and Earth System Sciences Discussions 10, no. 12 (2013): 15841–69. http://dx.doi.org/10.5194/hessd-10-15841-2013.

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Abstract. In deregulated electricity markets, hydropower portfolio design has become an essential task for producers. The previous research on hydropower portfolio optimisation focused mainly on the maximisation of profits but did not take into account riverine ecosystem protection. Although profit maximisation is the major objective for producers in deregulated markets, protection of riverine ecosystems must be incorporated into the process of hydropower portfolio optimisation, especially against a background of increasing attention to environmental protection and stronger opposition to hydro

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6

Rutkauskas, Aleksandras Vytautas, and Grigorij Žilinskij. "Investment Portfolio Optimisation Model Based on Stocks Investment Attractiveness." Business: Theory and Practice 13, no. (3) (2012): 242–52. https://doi.org/10.3846/btp.2012.26.

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Firm's performance and potential return on investments in its stocks are determined by many factors. However, most of portfolio optimisation methods are oriented to decision- making based on stock price changes in the past. Recent financial crisis has showed that often the biggest downfall in the period of crisis is experienced by stocks, which had the biggest growth before crisis. So decision- making based on stock price tendencies analysis by ignoring fundamental factors can be inefficient. The variety of MCDM methods was briefly described and their application possibilities for portfolio op

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7

T, Vorkut, Bilonoh O, Petunin A, Tretynychenko Y, Kharuta V, and Chechet A. "PROJECT PORTFOLIOS OPTIMISATION OF COLLECTIVE STRATEGIES IMPLEMENTATION IN SUPPLY CHAIN NETWORKS." National Transport University Bulletin 1, no. 48 (2021): 44–62. http://dx.doi.org/10.33744/2308-6645-2021-1-48-044-062.

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The actuality of the theme arises from the need to improve and further develop the methodological support of portfolio management processes in the context of the portfolio management introduction of collective strategies implementation in supply chain networks. The purpose of the study is to develop a mathematical model for determining the optimal portfolio components as a means of single system implementation of collective strategies in supply chain networks (SCNs) to meet the target value of selected criteria of efficiency and cost-effectiveness of supply chains as integral objects, taking i

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8

T, Vorkut, Bilonoh O, Petunin A, Tretynychenko Y, Kharuta V, and Chechet A. "PROJECT PORTFOLIOS OPTIMISATION OF COLLECTIVE STRATEGIES IMPLEMENTATION IN SUPPLY CHAIN NETWORKS." National Transport University Bulletin 1, no. 48 (2021): 44–62. http://dx.doi.org/10.33744/2308-6645-2021-1-48-044-062.

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The actuality of the theme arises from the need to improve and further develop the methodological support of portfolio management processes in the context of the portfolio management introduction of collective strategies implementation in supply chain networks. The purpose of the study is to develop a mathematical model for determining the optimal portfolio components as a means of single system implementation of collective strategies in supply chain networks (SCNs) to meet the target value of selected criteria of efficiency and cost-effectiveness of supply chains as integral objects, taking i

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9

Holovatiuk, Olha. "Cryptocurrencies as an asset class in portfolio optimisation." Central European Economic Journal 7, no. 54 (2020): 33–55. http://dx.doi.org/10.2478/ceej-2020-0004.

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AbstractIn this paper, cryptocurrencies are analysed as investment instruments. The study aims to verify whether they can be classified as an asset class and what kind of benefits they may bring to the investor's portfolio. We used 6 indices as proxies for the major asset classes, including the cryptocurrency index CRIX, for all cryptographic assets.Cryptocurrencies relatively fully satisfied 7 asset class requirements, namely stable aggregation, investability, internal homogeneity, external heterogeneity, expected utility, selection skill and cost-effective access. It was found that crypto as

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10

Slate, N., E. Matwiejew, S. Marsh, and J. B. Wang. "Quantum walk-based portfolio optimisation." Quantum 5 (July 28, 2021): 513. http://dx.doi.org/10.22331/q-2021-07-28-513.

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This paper proposes a highly efficient quantum algorithm for portfolio optimisation targeted at near-term noisy intermediate-scale quantum computers. Recent work by Hodson et al. (2019) explored potential application of hybrid quantum-classical algorithms to the problem of financial portfolio rebalancing. In particular, they deal with the portfolio optimisation problem using the Quantum Approximate Optimisation Algorithm and the Quantum Alternating Operator Ansatz. In this paper, we demonstrate substantially better performance using a newly developed Quantum Walk Optimisation Algorithm in find

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11

Xu, Zhihan, Xinyue Zhang, and Zili Zhou. "Cryptocurrency Portfolio Optimisation Based on LSTM Time Series Forecasting." Applied and Computational Engineering 134, no. 1 (2025): 143–50. https://doi.org/10.54254/2755-2721/2025.22255.

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Recently, with the gradual development of machine learning technology, more and more people are trying to apply machine learning technology in various fields, and finance is one of the important fields. This work investigates the optimization of cryptocurrency portfolios by combining Long Short-Term Memory (LSTM) time series forecasting with traditional portfolio optimization methods. The focus of the paper is on using the historical price data from the past six years of Bitcoin (BTC), Ethereum (ETH), and Litecoin (LTC) to train LSTM models, which are then used to predict the prices of these c

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12

Kumar, Puneet, Amalanathan Paul, and M. Anil Kumar. "Risk Optimisation Analytics." International Journal of Social Ecology and Sustainable Development 12, no. 2 (2021): 48–62. http://dx.doi.org/10.4018/ijsesd.2021040103.

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Risk optimization using business analytics is gaining momentum in India over the last decade. In order to tap this huge opportunity, most of the startups are getting into the analytics field engaging in financial market survey to provide their customers with valid data. The objective of this study is to help “Sai Builders” in solving their portfolio investment problem as well as sinking funds problem using linear programming and to obtain the total optimum returns by satisfying all constraints. The authors solve the problem of minimizing portfolio risk measures. In addition, the expected retur

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13

Silvester, Colin. "Portfolio Optimisation in Uniper." Impact 2021, no. 1 (2021): 46–50. http://dx.doi.org/10.1080/2058802x.2020.1846318.

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14

King, David. "Portfolio optimisation and diversification." Journal of Asset Management 8, no. 5 (2007): 296–307. http://dx.doi.org/10.1057/palgrave.jam.2250082.

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15

Lee, Stephen, and Simon Stevenson. "Time weighted portfolio optimisation." Journal of Property Investment & Finance 21, no. 3 (2003): 233–49. http://dx.doi.org/10.1108/14635780310481667.

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16

Srivatsa, Rahul, Andrew Smith, and Jon Lekander. "Portfolio optimisation and bootstrapping." Journal of Property Investment & Finance 28, no. 1 (2010): 24–33. http://dx.doi.org/10.1108/14635781011020029.

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17

Scherer, Bernd. "Can robust portfolio optimisation help to build better portfolios?" Journal of Asset Management 7, no. 6 (2007): 374–87. http://dx.doi.org/10.1057/palgrave.jam.2250049.

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18

Zhang, Yanfeng. "Application of Financial Mathematical Models Combined with Root Algorithms in Finance." Scalable Computing: Practice and Experience 25, no. 4 (2024): 2146–58. http://dx.doi.org/10.12694/scpe.v25i4.2447.

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Investors in the financial markets must deal with various hazards, for which they must create prudent investment portfolios and risk management plans. A multi-objective optimisation approach is proposed using the root algorithm to create a multi-objective root system growth model based on many clusters. An investment risk management optimisation model based on root system growth is built into the study using distributed decision-making. To create a multi-objective root algorithm-based portfolio optimisation model, the Markowitz mean-variance model and a multi-objective root algorithm are emplo

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19

Di Persio, Luca, and Nicola Fraccarolo. "Exploring Optimisation Strategies Under Jump-Diffusion Dynamics." Mathematics 13, no. 3 (2025): 535. https://doi.org/10.3390/math13030535.

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This paper addresses the portfolio optimisation problem within the jump-diffusion stochastic differential equations (SDEs) framework. We begin by recalling a fundamental theoretical result concerning the existence of solutions to the Black–Scholes–Merton partial differential equation (PDE), which serves as the cornerstone for subsequent analysis. Then, we explore a range of financial applications, spanning scenarios characterised by the absence of jumps, the presence of jumps following a log-normal distribution, and jumps following a distribution of greater generality. Additionally, we delve i

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20

Mats, Vladyslav. "Hedge performance of different asset classes in varying economic conditions." Radioelectronic and Computer Systems 2024, no. 1 (2024): 217–34. http://dx.doi.org/10.32620/reks.2024.1.17.

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In the realm of long-term investment, strategic portfolio allocation is an essential tool, especially in relation to risk management and return optimisation. There are many ways to pursue optimal portfolio composition, and their effectiveness depends on many factors, including the investor’s goals, risk appetite, and investment horizon. One of the primary means of portfolio optimisation is diversification. The core idea of diversification is to maintain a diverse portfolio with weakly correlated assets that can vastly reduce portfolio exposure to different market stress factors. Diversificatio

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21

Narsoo, Jason. "Performance Analysis of Portfolio Optimisation Strategies: Evidence from the Exchange Market." International Journal of Economics and Finance 9, no. 6 (2017): 124. http://dx.doi.org/10.5539/ijef.v9n6p124.

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Portfolio allocation is embedded in many decisional tasks for ensuring best returns under the constraint of minimising risk. In this paper, we implement several strategies in order to generate a holistic assessment of portfolio evaluation. The study analyses the performance of an extended framework of the classical tangency and targeted portfolio strategies. The extension is essentially the use of the skewed student-t distribution for the individual assets’ log-return. Our investigation is based on 15 currencies with US dollar as the base currency for the period spanning from 1999 to 2015. A c

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22

Chau Li, Wan, Yue Wu, and Udechukwu Ojiako. "Using portfolio optimisation models to enhance decision making and prediction." Journal of Modelling in Management 9, no. 1 (2014): 36–57. http://dx.doi.org/10.1108/jm2-11-2011-0057.

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Purpose – The purpose of this paper is to analyse and compare the performances of portfolio optimisation models including Markowitz's mean-variance model (MV model), Konno and Yamazaki's mean-absolute deviation portfolio optimisation model (MAD model), Young's minimax portfolio model and the VaR model. Design/methodology/approach – Historical data on 43 constituent shares listed on the Hong Kong Hang Seng Index (HSI) covering a four-year period are obtained. The paper then tests the performance of each model under different scenarios and against different sets of historical data. Findings – Th

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23

Pienaar, Ruwan, and Gary Van Vuuren. "Enhancing Portfolio Asset Allocation Efficiency using Blended Covariance Matrices." International Journal of Economics and Financial Issues 15, no. 4 (2025): 343–55. https://doi.org/10.32479/ijefi.18883.

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The Mahalanobis distance (MD) is employed as a threshold for outlier identification in the construction of a blended covariance matrix, aimed at enhancing portfolio allocation and risk management during periods of elevated market volatility. During such times, sample covariance matrices often become unstable, resulting in unrealistic and unconstrained efficient frontiers. This study addresses the inadequacy of the sample covariance matrix for portfolio optimisation under turbulent conditions, where estimation errors can significantly undermine investment performance. By comparing the performan

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24

Fontana, Claudio, and Martin Schweizer. "Simplified mean-variance portfolio optimisation." Mathematics and Financial Economics 6, no. 2 (2012): 125–52. http://dx.doi.org/10.1007/s11579-012-0067-4.

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25

Zagst, Rudi, and Michaela Pöschik. "Inverse portfolio optimisation under constraints." Journal of Asset Management 9, no. 3 (2008): 239–53. http://dx.doi.org/10.1057/jam.2008.20.

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26

Wahyuputro, Bernardus, Steve Begg, and Graeme Bethune. "Characterisation of petroleum assets for portfolio management." APPEA Journal 50, no. 2 (2010): 721. http://dx.doi.org/10.1071/aj09085.

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There is growing use of modern portfolio management methods that integrate risks, strategic goals and optimisation techniques to aid investment decision-making in the exploration and production industry. This modern approach consists of stages of analysis that include asset analysis, strategic goals definition and portfolio selection to maximise the probability of meeting the strategic goals. To date, most work in this area has focussed on the portfolio management requirements of oil and gas operators. However, the approach has the potential to help decision-making surrounding the management o

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27

Kumar, Nand, Archana Singh, Ranganath M. S., and Amandeep Kaur. "Portfolio Optimization: Indifference Curve Approach." International Journal of Advance Research and Innovation 2, no. 1 (2014): 9–15. http://dx.doi.org/10.51976/ijari.211402.

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The study examines the monthly stock prices of 45 SENSEX companies for the period ranging from February 2002 to January 2012. Also the study includes the Indian G-SEC long term bonds with maturities ranging from 15 to 25 years. The set of all efficient portfolios is called the efficient frontier. All risk-averse investors who act to maximize expected utility have an optimal portfolio on this frontier. Based on the risk-aversion factor and the investment time horizon of each individual investor, an attempt is being made to select the optimal portfolio for that particular investor. Given a utili

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28

de Jong, Marielle. "Portfolio optimisation in an uncertain world." Journal of Asset Management 19, no. 4 (2017): 216–21. http://dx.doi.org/10.1057/s41260-017-0066-3.

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29

MacDonald, Erin, Alexis Lubensky, Bryon Sohns, and Panos Y. Papalambros. "Product semantics and wine portfolio optimisation." International Journal of Product Development 7, no. 1/2 (2009): 73. http://dx.doi.org/10.1504/ijpd.2009.022277.

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30

Loukeris, N., I. Eleftheriadis, and E. Livanis. "The Portfolio Heuristic Optimisation System (PHOS)." Computational Economics 48, no. 4 (2016): 627–48. http://dx.doi.org/10.1007/s10614-015-9552-1.

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31

Chang, T. J., N. Meade, J. E. Beasley, and Y. M. Sharaiha. "Heuristics for cardinality constrained portfolio optimisation." Computers & Operations Research 27, no. 13 (2000): 1271–302. http://dx.doi.org/10.1016/s0305-0548(99)00074-x.

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32

Fahrenwaldt, Matthias A., and Chaofan Sun. "Expected utility approximation and portfolio optimisation." Insurance: Mathematics and Economics 93 (July 2020): 301–14. http://dx.doi.org/10.1016/j.insmatheco.2020.05.010.

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33

Beraldi, Patrizia, Lucio Grandinetti, Italo Eipoco, Antonio Violi, and Maria Elena Bruni. "An advanced system for portfolio optimisation." International Journal of Grid and Utility Computing 5, no. 1 (2014): 21. http://dx.doi.org/10.1504/ijguc.2014.058253.

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34

Lutgens, Frank, and Peter C. Schotman. "Robust Portfolio Optimisation with Multiple Experts*." Review of Finance 14, no. 2 (2008): 343–83. http://dx.doi.org/10.1093/rof/rfn028.

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35

Tiwary, Mrityunjay Kumar, and Vaibhav Aggarwal. "Portfolio optimisation in global equity markets." International Journal of Business Innovation and Research 37, no. 1 (2025): 1–18. https://doi.org/10.1504/ijbir.2025.146036.

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36

Martín García, Rodrigo, Enrique Ventura Pérez, and Raquel Arguedas Sanz. "Temporal optimisation of signals emitted automatically by securities exchange indicators." Cuadernos de Gestión 20, no. 3 (2020): 61–71. http://dx.doi.org/10.5295/cdg.170851rm.

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Stock exchange indicators deliver buy/sell signals that enable analysts to improve the results of a strategy based strictly on fundamental analysis. Nonetheless, since the automatic implementation of signals as they appear may not yield optimal returns, the present paper analysed the suitability of using a series of technical indicators as guidance for portfolio results. A second aim pursued was to study how delaying the implementation of indicator signals may enhance profitability. A simulation was performed for the years 2005-2016 using the most representative index for the Spanish stock exc

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37

Agrawal, Preeti Bai, and Dr Anuradha Samal. "Mapping portfolio optimisation: a systematic and bibliometric review." Global Journal of Mathematical Analysis 11, no. 1 (2024): 1–10. http://dx.doi.org/10.14419/nfmybb32.

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Purpose: This study provides a comprehensive analysis of the evolution of portfolio optimization over the last three decades, employing systematic review and advanced bibliometric techniques to map key trends, influential works, and significant contributors in the field. Design/Methodology/Approach: Adhering to PRISMA guidelines, we conducted a systematic review and bibliometric analysis of 1,000 articles sourced from the Web of Science database, spanning from 1989 to 2023. Advanced bibliometric tools, including citation analysis, co-occurrence analysis, and network visualization, were utilize

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38

Alotaibi, Tahani S., Luciana Dalla Valle, and Matthew J. Craven. "The Worst Case GARCH-Copula CVaR Approach for Portfolio Optimisation: Evidence from Financial Markets." Journal of Risk and Financial Management 15, no. 10 (2022): 482. http://dx.doi.org/10.3390/jrfm15100482.

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Portfolio optimisation aims to efficiently find optimal proportions of portfolio assets, given certain constraints, and has been well-studied. While portfolio optimisation ascertains asset combinations most suited to investor requirements, numerous real-world problems impact its simplicity, e.g., investor preferences. Trading restrictions are also commonly faced and must be met. However, in adding constraints to Markowitz’s basic mean-variance model, problem complexity increases, causing difficulties for exact optimisation approaches to find large problem solutions inside reasonable timeframes

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39

Iqbal, Javed, Moeed Ahmad Sandhu, Shaheera Amin, and Aliya Manzoor. "Portfolio Selection and Optimization through Neural Networks and Markowitz Model: A Case of Pakistan Stock Exchange Listed Companies." Review of Economics and Development Studies 5, no. 1 (2019): 183–96. http://dx.doi.org/10.26710/reads.v5i1.354.

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This paper used artificial neural networks (ANNs) time series predictor for approximating returns of Pakistan Stock Exchange (PSX) listed 100 companies. These projected returns are then substituted into expected returns in the Markowitz’s Mean Variance (MV) portfolio Model. For comparison empirical data used is closing prices of PSX listed stocks, Karachi Inter Bank Offer Rates (KIBOR) as risk free rate and KSE-all share index as benchmark. The Portfolio returns are compared for two datasets by employing various constraints like budget, transaction costs, and turnover constraints. The value of

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Velasco, José Carlos Javier, Sergio Gerardo De los Cobos Silva, Eric Alfredo Rincón García, et al. "PSO-3P for the portfolio optimisation problem." International Journal of Business Continuity and Risk Management 8, no. 3 (2018): 219. http://dx.doi.org/10.1504/ijbcrm.2018.094175.

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Lara Velázquez, Pedro, Antonin Ponsich, Roman Anselmo Mora Gutiérrez, et al. "PSO-3P for the portfolio optimisation problem." International Journal of Business Continuity and Risk Management 8, no. 3 (2018): 219. http://dx.doi.org/10.1504/ijbcrm.2018.10015208.

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42

Yin, Jiangyong, and Xinyi Xu. "Portfolio optimisation using constrained hierarchical bayes models." Statistical Theory and Related Fields 1, no. 1 (2017): 112–20. http://dx.doi.org/10.1080/24754269.2017.1347310.

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Sivaramakrishnan, Kartik, Vishv Jeet, and Dieter Vandenbussche. "Multi-period portfolio optimisation with alpha decay." International Journal of Financial Engineering and Risk Management 2, no. 4 (2018): 283. http://dx.doi.org/10.1504/ijferm.2018.094030.

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Mulvey, John M., Han Hao, and Nongchao Li. "Machine learning, economic regimes and portfolio optimisation." International Journal of Financial Engineering and Risk Management 2, no. 4 (2018): 260. http://dx.doi.org/10.1504/ijferm.2018.094043.

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Sivaramakrishnan, Kartik, Vishv Jeet, and Dieter Vandenbussche. "Multi-period portfolio optimisation with alpha decay." International Journal of Financial Engineering and Risk Management 2, no. 4 (2018): 283. http://dx.doi.org/10.1504/ijferm.2018.10015275.

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46

Hao, Han, Nongchao Li, and John M. Mulvey. "Machine learning, economic regimes and portfolio optimisation." International Journal of Financial Engineering and Risk Management 2, no. 4 (2018): 260. http://dx.doi.org/10.1504/ijferm.2018.10015289.

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47

Ortí, Francesc J., and José B. Sáez. "Portfolio optimisation: A fuzzy multi-objective approach." Journal of Asset Management 9, no. 2 (2008): 138–48. http://dx.doi.org/10.1057/jam.2008.16.

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48

Bernardi, Mauro, and Leopoldo Catania. "Portfolio optimisation under flexible dynamic dependence modelling." Journal of Empirical Finance 48 (September 2018): 1–18. http://dx.doi.org/10.1016/j.jempfin.2018.05.002.

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49

Meister, Bernhard K., and Henry C. W. Price. "A Quantum Double-or-Nothing Game: An Application of the Kelly Criterion to Spins." Entropy 26, no. 1 (2024): 66. http://dx.doi.org/10.3390/e26010066.

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A quantum game is constructed from a sequence of independent and identically polarised spin-1/2 particles. Information about their possible polarisation is provided to a bettor, who can wager in successive double-or-nothing games on measurement outcomes. The choice at each stage is how much to bet and in which direction to measure the individual particles. The portfolio’s growth rate rises as the measurements are progressively adjusted in response to the accumulated information. Wealth is amassed through astute betting. The optimal classical strategy is called the Kelly criterion and plays a f

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50

Kevin, Choon Liang Yew, Kuan Wong Wai, Sim Hong Seng, Goh Yong Kheng, Pan Wei Yeing, and Sim Shin Zhu. "l0- Norm Sparse Portfolio Optimisation using Proximal Spectral Gradient Method on Malaysian Stock Market." Sains Malaysiana 54, no. 2 (2025): 601–9. https://doi.org/10.17576/jsm-2025-5402-24.

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In this paper, we introduce a modified norm-constraint mean-variance portfolio selection method. First, we use the Augmented Lagrangian method (ALM) to convert the objective function to an unconstrained objective function. Then we apply the proximal spectral gradient method (PSG) onto the unconstrained objective function to find an optimal sparse portfolio. This novel sparse portfolio optimization procedure encourages sparsity in the entire portfolio using norm. The PSG utilizes a multiple damping gradient (MDG) method to solve the smooth terms of the function. The step size is computed using

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