Cardinality Constrained Portfolio Optimization

• Cardinality-Constrained Mean-Variance Portfolio Optimization is a sparse portfolio optimization problem that limits the number of assets selected within the classic Markowitz framework.
• It involves nonconvex, combinatorial challenges addressed by MIQP, continuous reformulations, and various relaxations to achieve scalable and near-optimal solutions.
• Practical implementations leverage penalty decomposition, DC algorithms, and metaheuristic methods to balance risk-return trade-offs while meeting transaction cost and regulatory constraints.

Cardinality-Constrained Mean-Variance Portfolio Optimization (CCPO) is a class of sparse portfolio optimization problems that enforce an explicit constraint on the number of assets selected in the portfolio within the canonical Markowitz mean-variance framework. The objective is to balance financial risk and expected return under a cap on the number of invested assets, creating a problem that is nonconvex, combinatorial, and computationally challenging. CCPO is central to practical investment strategies, reflecting transaction cost minimization, regulatory or operational simplicity, and investor preference for portfolio tractability (Zhang et al., 2018, Iliopoulos, 23 Dec 2025).

1. Mathematical Formulation and Fundamental Properties

The CCPO model extends the classical Markowitz portfolio selection by imposing an $\ell_0$-cardinality constraint on the portfolio weights. Let $n$ denote the asset universe size, $\mu\in\mathbb R^n$ the expected return vector, $\Sigma\in\mathbb R^{n\times n}$ the covariance matrix, $x\in\mathbb R^n$ the weights, and $k\ll n$ the cardinality bound. The long-only formulation is: $$\begin{aligned} \min_{x\in\mathbb R^n}\ &x^\top\Sigma x - \gamma\,\mu^\top x\ \text{s.t.}\quad&\mathbf{1}^\top x = 1,\quad x\geq 0, \quad \|x\|_0 \leq k. \end{aligned}$$ Equivalently, introducing binary selectors $z_i \in \{0,1\}$: $$x_i \leq z_i,\quad \sum_{i=1}^n z_i \leq k, \quad z_i\in\{0,1\},\quad \sum_{i=1}^n x_i = 1,\quad x_i \geq 0.$$ This MIQP structure is NP-hard due to the combinatorial nature of the $\ell_0$-norm (Zhang et al., 2018, Wiegele et al., 2021, Branda et al., 2017).

2. Algorithmic Frameworks and Relaxations

Multiple algorithmic paradigms have been developed for CCPO, spanning exact mixed-integer optimization, continuous reformulations, nonconvex relaxations, and metaheuristic frameworks.

Exact MIQP and SDP Relaxation: Direct MIQP can solve small to moderate $n$ ($\lesssim 200$) but becomes intractable for larger problems. Tight semidefinite relaxations leverage matrix lifting and yield near-optimal solutions, achieving optimality on many literature benchmarks, with over $96\%$ of SDP relaxations producing rank-one solutions recoverable as globally optimal portfolios (Wiegele et al., 2021).

Continuous Reformulations: Auxiliary (variable splitting or complementarity) schemes facilitate the use of modern continuous optimization methods. Approaches such as the Burdakov–Kanzow–Schwartz complementarity formulation introduce auxiliary $y$ with $x_i y_i=0$, $e^\top y\geq n-k$. The Scholtes-type regularization introduces a relaxation parameter $t$ and solves a sequence of smooth NLPs, converging to S-stationary points that are local minimizers of CCPO under convexity (Branda et al., 2017). Similar structure arises in Dykstra-alternating projections and block Penalty-Decomposition (Krejić et al., 2022, Mousavi et al., 2023).

Penalty and Splitting Approaches: Penalty decomposition with BCD alternates closed-form updates for the portfolio and cardinality variables, converging to a saddle point of the penalized objective. Correct tuning of the penalty parameter ensures that the limit portfolio is feasible (i.e., $\|x\|_0 \leq k$) and locally optimal (Mousavi et al., 2023, Mousavi et al., 2024).

Difference-of-Convex (DC) Algorithms: The cardinality constraint admits a DC reformulation via $\|x\|_1 - \|x\|_{[k]}$, where $\|x\|_{[k]}$ is the sum of the $k$ largest $|x_i|$. DC Algorithms linearize the concave part at each iterate and solve a proximal convex QP, exhibiting rapid convergence and outperforming general MIP on typical problem ranges (Wang et al., 2020, Moeini, 2014, Chen et al., 2024).

Metaheuristics and Repair-Based Methods: For high-dimensional or multi-period settings, metaheuristic algorithms (GA, DE, PSO, ABC, etc.) and agentic LLM frameworks auto-generate hybrid algorithms and combine their solutions for improved efficient frontiers. Sophisticated repair operators such as CASP, based on volatility-normalized selection and Mahalanobis metric projection, enforce both the cardinality and the budget simplex, attaining significant variance reductions over standard Euclidean repairs (Paquette-Greenbaum et al., 2 Jan 2026, Iliopoulos, 23 Dec 2025).

3. Specialized Solution Schemes

Proximal Alternating Linearized Minimization (PALM): The relaxed approach minimizes

$$g(w,v) = w^\top\Sigma w - \gamma\,\mu^\top w + \frac{\nu}{2}\|w-v\|^2,$$

subject to $w$ satisfying the simplex and $v$ encoding the $\ell_0$-constraint and group bounds. Alternating minimization with strongly convex updates and top-$k$ pruning in $v$ achieves empirical global optimality for moderate $n$, and near-optimality for large-scale instances (Zhang et al., 2018).

Successive Convex Approximation (SCA): SCA methods for continuous reformulations or DC surrogates linearize the nonconvex part and solve a penalized convex subproblem at each iteration. SCA with the Jiang–Wu–Hu reformulation $x_i-x_iy_i\leq 0$, $\sum y_i\leq k$, yields exact cardinality feasibility and avoids integer variables, converging rapidly in practice (Jiang et al., 2019).

Column Generation: By restricting the active asset set to cardinality $k$ and iteratively updating the set based on reduced cost computed through dual multipliers, one solves a sequence of small convex QPs. This enables scalable implementations in institutional settings with tight real-time constraints (Roebers et al., 2018).

Factor Model and Index Tracking: For factor-structured covariances, CCPO can be further reduced using piecewise linear approximations, clique-based combinatorial models, and heuristics exploiting model structure, yielding nearly optimal solutions with significant computational savings (Monge, 2017).

4. Extensions and Empirical Performance

Risk Measures Beyond Variance: CVaR, mean-variance-CVaR, and high-order moment (skewness, kurtosis) extensions generalize CCPO for tail-risk and distributional characteristics. Penalty and BCD algorithms generalize straightforwardly, with empirical performance comparable or superior to state-of-the-art PADM and MIQP solvers. Closed-form block updates and aggressive variable screening enhance scalability (Mousavi et al., 2024, Wang et al., 2020).

Robust and Sparse Portfolios: Robustification under ellipsoidal uncertainty and fixed transaction cost (cardinality penalization)—RSMV—admits a DC-type algorithmic treatment. There is a provable one-to-one correspondence between risk aversion and robustness parameter, and cardinality is governed by a tradeoff between uncertainty set size and transaction penalty. The local convergence rate is linear, enabling efficient grid-search over hyperparameters (Chen et al., 2024).

High-Dimensional and Out-of-Sample Efficiency: In ultrahigh dimensions, CCPO strategies with SAA and convex proxies (e.g., $\ell_1$-regularization plus screening) yield sparsity, outperformance relative to equally-weighted portfolios, and substantial reduction in estimation and optimization error, with theoretical guarantees for out-of-sample generalization. Safe screening and pathwise updates keep computational cost affordable for $n\gg 1000$ (Du et al., 2022).

Dynamic and Multiperiod Settings: Multi-period CCPO can be solved semi-analytically in factor-driven markets via dynamic programming and Bellman recursion, yielding piecewise-linear feedback policies and time-consistency in efficiency characterized by martingale properties of the variance-optimal signed supermartingale measure (Gao et al., 25 Feb 2025).

5. Empirical Evaluation and Benchmarking

Benchmark studies consistently show that continuous relaxation, penalty decomposition, DC-programming, and SDP-relaxation based algorithms for CCPO:

• Match or exceed in-sample and out-of-sample Sharpe ratios and risk levels of commercial MIQP solvers and PADM methods, often at $5$–$100\times$ lower wall-clock time (Mousavi et al., 2023, Zhang et al., 2018, Wiegele et al., 2021).
• Scale effectively to hundreds or thousands of assets, maintaining low optimality gaps and supporting parametric sweeps needed for efficient frontier construction.
• Produce portfolios whose realized cardinality, return, and variance characteristics closely track the prescribed constraints, with negligible empirical compromise for tractability.
• Outperform generic metaheuristics, especially when repair operators exploit the covariance structure or when LLM/agentic frameworks pool algorithmic diversity for frontier coverage (Iliopoulos, 23 Dec 2025, Paquette-Greenbaum et al., 2 Jan 2026).

6. Practical Implementation Considerations

Key methodological choices and technical guidelines include:

• Penalty parameter warm-start and continuation strategies to avoid poor local minima, and adaptive updates to ensure constraint satisfaction.
• Exploiting problem structure (sparse or low-rank $\Sigma$, group constraints) for computational efficiency.
• Aggressive feature screening and convex proxy usage in high-dimensional settings.
• Modularization via separation of asset selection and weight optimization, supporting integration with existing metaheuristic or factor-based optimizers.
• Grid search over risk aversion $\gamma$, cardinality $k$, robustification parameter $\rho$, and transaction penalty $\phi$ to align the method with investor objectives and regulatory requirements (Chen et al., 2024, Du et al., 2022, Moeini, 2014).
• For robust, multi-period or conditional risk extensions, solution reuse and state-space discretization facilitates tractable embedding in dynamic programs (Gao et al., 25 Feb 2025).

7. Research Directions and Extensions

Current research avenues in CCPO include:

• Extending CASP-style covariance-aware repair to multi-period and robust optimization regimes (Iliopoulos, 23 Dec 2025).
• Integrating LLM-driven agentic frameworks for automated metaheuristics discovery and hyper-heuristics coordination (Paquette-Greenbaum et al., 2 Jan 2026).
• SDP-relaxation theory for objective gap quantification and scalable extraction of rank-one solutions from relaxed moment matrices (Wiegele et al., 2021).
• Dynamic programming for time-consistency in efficiency under nonconvex and factor-driven cones (Gao et al., 25 Feb 2025).
• Portfolio-efficient frontiers in the presence of transaction costs, factor exposures, sector bounds, and regulatory cardinality—an area progressively addressed through penalty and SCA methods (Mousavi et al., 2024, Mousavi et al., 2023).

Cardinality-constrained mean-variance optimization continues to evolve as a technically rich class of portfolio selection problems, integrating nonconvex combinatorics with convex optimization, providing scalable algorithms that directly address practical investment constraints (Zhang et al., 2018, Iliopoulos, 23 Dec 2025, Mousavi et al., 2023).