Large-Scale Portfolio Optimization Problem Under Cardinality Constraint With Enhanced Multi-Objective Evolutionary Algorithms

Jul 16, 2026

Decision-making in financial markets becomes increasingly difficult as investors face a growing number of asset alternatives. Portfolio optimization models decide how to allocate capital across assets to balance returns and risk, but adding practical constraints often makes the problem NP-hard and limits exact solution methods. This work strengthens multi-objective evolutionary algorithms by introducing a tailored solution representation, a novel operator, and repair mechanisms under cardinality constraints that bound portfolio size. Customized mating strategies are benchmarked on market indices to improve approximation quality and convergence speed as the asset universe grows, without performance degradation.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages37

Reading Time1 h 33 min

Introduction

Portfolio selection and modern portfolio theory

Risk measures and real-world constraints

Exact and approximate solution approaches

Related exact methods for portfolio optimization

arXiv :2607 .09566v 1 [ cs .CE] 10 Jul 2026 Large-Scale Portfolio Optimization Problem Under Cardinality Constraint With Enhanced Multi-Objective Evolutionary Algorithms Danial Ramezani a , Mostafa Abouei Ardakana a Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran Abstract. Decision-making is posing an increasingly formidable challenge to investors because of the growing number of alternatives available in financial markets. A hot area of research over the past few decades has been portfolio optimization that seeks to determine how much an investor should invest in which asset. Introducing real-world conditions to the optimization model turns the problem into an NP-hard one for whose solution exact methods become inefficient; hence, researchers have turned to evolutionary algorithms to approximate solutions. In this paper, strengthening strategies are presented for multi-objective evolutionary algorithms that can provide a faster convergence rate and extensive search ability in the portfolio optimization problem under the cardinality constraint. To implement those features, a unique solution representation, a novel operator, and new repair mechanisms are introduced for solving the aforementioned problem in which lower and upper limits are set on the number of assets in the portfolio. For this purpose, new mating strategies along with the aforesaid package are implemented in well-known multi-objective evolutionary algorithms to solve the problem. The customized algorithms are subsequently tested against traditional ones using well-known market indices as benchmarks. Results indicate that the proposed strategy not only provides better approximations but also converges faster as well at no loss of performance with increasing number of assets in the market. Keywords: Cardinality-constrained portfolio optimization · Multi-objective optimization · Mixed-integer programming · Evolutionary algorithms · Asset allocation 1 Introdution: Portfolio selection is well-known to investors and fund managers and is a widely used strategy in financial markets. It, indeed, involves the selection of capital and its proper allocation to various potential assets aimed at maximizing return but minimizing risk. Investors and portfolio managers seek to identify the best investment opportunities with the highest return and lowest risk (Pedersen et al. (2021)) . Modern portfolio theory was pioneered by Harry Markowitz who put forth the concept of diversification as a solution to mitigating portfolio risk. He introduced variance as a risk measurement; his mean-variance (MV) model is based on minimizing risk for a certain level of return (Markowitz (1952)) . Although Markowitz’s revolutionary work altered quantitative finance, it suffered from certain shortcomings from a practical perspective; for instance, it ignored such real-world constraints (Ertenlice and Kalayci (2018)) as cardinality constraint (CC) that puts limits on the number of stocks in the portfolio, boundary constraint (BC) that imposes a limit on the portion of capital allowed for investment inan individual asset within a portfolio, transaction costs (Thakkar and Chaudhari (2021)), round lot constraint (Almahdi and Yang (2019)), class constraint (Almahdi and Yang (2019)), sector capitalization constraint (Golmakani and Fazel (2011)), and turnover constraint (Clarke et al. (2002)) . The portfolio optimization problem (POP) has attracted researchers’ attention and encouraged a significant amount of research work to extend the MV model or to incorporate real-world constraints. Konno and Yamazaki (1991) considered mean absolute error as a risk measure for addressing nonlinearity in the MV model. Later, value at risk (VaR) (Jorion (1997)) gained attention in modeling portfolio risks. Due to its shortcomings, namely subadditivity and convexity, Rockafellar et al. (2000) put forth the concept of minimizing conditional value at risk

FAQs

Why do cardinality and other real-world constraints make portfolio optimization harder?

Adding constraints such as cardinality bounds and other practical restrictions increases problem complexity, making the optimization NP-hard and reducing the efficiency of exact methods.

What improvements does the paper introduce to multi-objective evolutionary algorithms?

The paper proposes a unique solution representation, a novel operator, and new repair mechanisms, along with mating strategies, to handle cardinality limits on the number of assets.

How are the proposed algorithms evaluated and what do results show?

The customized algorithms are tested against traditional approaches using well-known market indices. Results show better approximations and faster convergence as the number of assets increases, without loss of performance.

Jul 16, 2026

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