Cardinality-Constrained Portfolio Selection

• Cardinality-constrained portfolio selection is an optimization framework that explicitly limits the number of active assets to improve interpretability and control risk.
• It uses mixed-integer programming, continuous reformulations, and heuristic methods to address the NP-hard challenge of selecting sparse portfolios.
• Recent advancements integrate penalty techniques, projection methods, and decomposition strategies to achieve scalable and accurate portfolio optimization.

Searching arXiv for recent and foundational papers on cardinality-constrained portfolio selection to ground the article in current research. Cardinality-constrained portfolio selection is the class of portfolio optimization problems in which the number of active positions is explicitly limited, either globally or within groups such as sectors, while the portfolio still satisfies risk, return, budget, and often trading constraints. In the classical long-only mean–variance setting, a representative formulation is

$$\min_{x \in \mathbb{R}^n}\quad x^\top A x - \tau \mu^\top x \quad \text{subject to}\quad e^\top x = 1,\; x \ge 0,\; \|x\|_0 \le k,$$

where $A \succeq 0$ is the covariance matrix, $\mu$ the expected return vector, and $k$ the maximum number of held assets (Mousavi et al., 2023). More general formulations incorporate sector-level sparsity, transaction costs, benchmark-relative risk, tracking objectives, dynamic factor structure, and even higher-order interaction terms (Zhang et al., 2018, Moeini, 2014, Meade et al., 24 Mar 2025, Gao et al., 25 Feb 2025, Serbarinov, 16 Mar 2026). The central technical feature is that the $\ell_0$-type restriction turns an otherwise convex portfolio problem into a nonconvex, combinatorial, and typically NP-hard optimization problem (Zhang et al., 2018, Mousavi et al., 2023).

1. Problem class and mathematical structure

At its core, the topic concerns sparse portfolio design under explicit support constraints. In the simplest Markowitz-style case, sparsity is imposed by $\|x\|_0 \le k$, with full investment and nonnegativity ensuring a long-only portfolio on the simplex (Mousavi et al., 2023). In more general settings, the weights are partitioned into blocks $w=[w_1,\dots,w_m]$, and the feasible set may require both sector-weight bounds and groupwise cardinality limits: $$\Omega = \Big\{p_i \le \mathbf{1}^\top w_i \le q_i,\; \|w_i\|_0 \le k_i,\; i=1,\dots,m\Big\},$$ so that sparsity is controlled simultaneously at group and total-portfolio levels (Zhang et al., 2018).

A practically important extension is single-period rebalancing with transaction costs and benchmark-relative risk. In that setting, the investor starts from a current portfolio $P$, trades through buy and sell vectors $x^b,x^s$, and obtains a new portfolio $A \succeq 0$0 satisfying

$A \succeq 0$1

while minimizing benchmark-relative variance

$A \succeq 0$2

subject to a net expected return constraint after transaction costs and a cardinality requirement $A \succeq 0$3, with linking bounds $A \succeq 0$4 (Moeini, 2014). In index tracking, the same sparsity principle appears in a different objective: the portfolio is chosen to minimize tracking error relative to an index, optionally with an additive enhancement target, while holding only $A \succeq 0$5 stocks from a much larger index universe (Meade et al., 24 Mar 2025).

This suggests that “cardinality-constrained portfolio selection” is not a single model but a family of sparse allocation problems whose common element is exact control over the support of the portfolio. The risk functional may be variance, tracking-error variance, CVaR, or a higher-order objective, but the nonconvexity arises from the same discrete support selection mechanism (Zhang et al., 2018, Serbarinov, 16 Mar 2026).

2. Mixed-integer formulations and combinatorial hardness

The standard exact modeling device is to introduce binary selection variables. In the mean–variance case, the support restriction can be written through $A \succeq 0$6 with

$A \succeq 0$7

or, more tightly, through lower and upper bound linking constraints $A \succeq 0$8 (Mousavi et al., 2023, Moeini, 2014). In benchmark-relative and rebalancing models, the resulting problem is a mixed-integer quadratic program with a quadratic objective, linear constraints, and binary variables (Moeini, 2014). In the minimum-variance setting, the classical Big-$A \succeq 0$9 formulation is

$\mu$0

which explicitly couples support selection and continuous weights (Moka et al., 15 May 2025).

The source of hardness is the support search. Even the simplest constraint $\mu$1 converts a convex problem on the simplex into subset selection over a nonconvex union of coordinate subspaces (Zhang et al., 2018). The combinatorial scale is immediate: choosing 10 out of 30 assets already yields $\mu$2 subsets (Zhang et al., 2018). In a CAPM-based industry universe with $\mu$3 and $\mu$4, the support count $\mu$5 is described as astronomically large, rendering exhaustive search infeasible (Gondauri, 6 Mar 2026).

This computational barrier is amplified by realistic constraints. Transaction costs, lower and upper bounds, groupwise sparsity, tracking-error restrictions, and dynamic state dependence enlarge the continuous part of the problem while leaving the discrete search intact (Moeini, 2014, Gao et al., 25 Feb 2025). The exact MIQP route therefore remains conceptually clean but can become impractical as $\mu$6, $\mu$7, or the number of auxiliary variables grows (Mousavi et al., 2023, Moka et al., 15 May 2025).

A common misconception is that sparsity can be enforced adequately by an $\mu$8 constraint. Under simplex constraints, however, $\mu$9 already holds, so an $k$0 restriction is ineffective as a cardinality surrogate in long-only portfolio models (Zhang et al., 2018).

3. Continuous reformulations and exact-penalty approaches

A major line of work replaces the discrete support decision by a continuous reformulation while preserving exact cardinality semantics. One route is difference-of-convex programming. For the rebalancing model with transaction costs and binary variables $k$1, the integrality condition is encoded by the concave penalty

$k$2

which is nonnegative on the relaxed feasible set and vanishes if and only if $k$3 for all $k$4 (Moeini, 2014). With a sufficiently large exact-penalty parameter $k$5, the original MIQP is equivalent to the continuous DC program

$k$6

where $k$7 is the convex relaxation of the original feasible set (Moeini, 2014). The associated DC Algorithm linearizes the convex part $k$8 and solves a sequence of convex quadratic programs.

Another route introduces an auxiliary sparse variable. In the relaxed framework for mean–variance and CVaR models, one keeps a continuous portfolio variable $k$9, introduces $\ell_0$0 carrying the exact $\ell_0$1 restrictions, and couples the two through a quadratic penalty: $\ell_0$2 Here the nonconvexity is confined to $\ell_0$3, while $\ell_0$4 remains in a convex simplex (Zhang et al., 2018). Projection onto $\ell_0$5 is tractable because within each sector the Euclidean projection reduces to keeping the $\ell_0$6 largest components and projecting the retained block onto a simplex or interval-simplex set (Zhang et al., 2018).

A third reformulation uses selector variables in $\ell_0$7 and a bilinear complementarity-type constraint. For a generic convex cardinality-constrained problem with nonnegative variables, the exact equivalence

$\ell_0$8

yields a continuous reformulation in $\ell_0$9 (Jiang et al., 2019). Linearizing the bilinear term $\|x\|_0 \le k$0 produces a successive convex approximation scheme in which each step solves a convex subproblem; under the paper’s assumptions, KKT points of the reformulation with $\|x\|_0 \le k$1 are local optimizers of the original problem (Jiang et al., 2019).

These constructions share a common principle: preserve the exact support logic, but move the hard combinatorics into penalties, projections, or complementarity structures that admit continuous optimization substeps.

4. Decomposition, projections, and scalable first-order methods

A different research direction seeks scalability by separating support choice from weight optimization or by working with continuous relaxations whose geometry encourages binary solutions.

Penalty decomposition is one example. For the cardinality-constrained mean–variance problem, the split formulation

$\|x\|_0 \le k$2

is penalized with an $\|x\|_0 \le k$3 coupling term

$\|x\|_0 \le k$4

For fixed $\|x\|_0 \le k$5, block coordinate descent has closed-form updates: the $\|x\|_0 \le k$6-step is an equality-constrained convex quadratic with analytic solution, and the $\|x\|_0 \le k$7-step is the hard-thresholding projection $\|x\|_0 \le k$8 (Mousavi et al., 2023). The outer penalty sequence drives $\|x\|_0 \le k$9 and $w=[w_1,\dots,w_m]$0 together, and accumulation points are proved to be local minimizers of the original mean–variance problem under the paper’s conditions (Mousavi et al., 2023).

Column-generation-type methods address cardinality indirectly by solving repeated restricted master problems over a candidate set of assets. In the benchmark-relative Markowitz extension with active-weight, sector, market-cap quintile, beta, active-share, and tracking-error considerations, the algorithm repeatedly solves a convex quadratic problem on a restricted set $w=[w_1,\dots,w_m]$1 of candidate assets, then uses dual information to compute a marginal effect

$w=[w_1,\dots,w_m]$2

for assets outside $w=[w_1,\dots,w_m]$3, where $w=[w_1,\dots,w_m]$4 is the direct contribution to the objective and $w=[w_1,\dots,w_m]$5 is the dual-cost term from the constraints (Roebers et al., 2018). Assets with the most negative $w=[w_1,\dots,w_m]$6 are added, low-weight or zero-weight names are removed, and the candidate-set size enforces the portfolio cardinality heuristically (Roebers et al., 2018).

Recent scalable first-order work reformulates sparse minimum-variance selection through a Boolean relaxation on the simplex. Defining

$w=[w_1,\dots,w_m]$7

with $w=[w_1,\dots,w_m]$8 and $w=[w_1,\dots,w_m]$9, one obtains a continuous objective that agrees exactly with the binary subset objective on $$\Omega = \Big\{p_i \le \mathbf{1}^\top w_i \le q_i,\; \|w_i\|_0 \le k_i,\; i=1,\dots,m\Big\},$$0 (Moka et al., 15 May 2025). The parameter $$\Omega = \Big\{p_i \le \mathbf{1}^\top w_i \le q_i,\; \|w_i\|_0 \le k_i,\; i=1,\dots,m\Big\},$$1 controls a convex-to-concave transition: for small $$\Omega = \Big\{p_i \le \mathbf{1}^\top w_i \le q_i,\; \|w_i\|_0 \le k_i,\; i=1,\dots,m\Big\},$$2, $$\Omega = \Big\{p_i \le \mathbf{1}^\top w_i \le q_i,\; \|w_i\|_0 \le k_i,\; i=1,\dots,m\Big\},$$3 is strictly convex on a truncated domain; for sufficiently large $$\Omega = \Big\{p_i \le \mathbf{1}^\top w_i \le q_i,\; \|w_i\|_0 \le k_i,\; i=1,\dots,m\Big\},$$4, it becomes strictly concave (Moka et al., 15 May 2025). A Frank–Wolfe continuation scheme then starts from the simplex center, follows a geometric grid in $$\Omega = \Big\{p_i \le \mathbf{1}^\top w_i \le q_i,\; \|w_i\|_0 \le k_i,\; i=1,\dots,m\Big\},$$5, and is driven toward binary extreme points corresponding to sparse portfolios.

This suggests a methodological continuum. At one end are exact MIQP formulations; at the other are purely continuous paths that use geometry, penalties, or projections to recover discrete supports without branching. The relative appeal depends on the scale of $$\Omega = \Big\{p_i \le \mathbf{1}^\top w_i \le q_i,\; \|w_i\|_0 \le k_i,\; i=1,\dots,m\Big\},$$6, the desired exactness, and the structure of the risk model.

5. Heuristics, metaheuristics, and repair-based methods

Metaheuristics remain prominent because they can directly explore the combinatorial support space without exact branching. In benchmark studies of mean–variance portfolios with exact cardinality $$\Omega = \Big\{p_i \le \mathbf{1}^\top w_i \le q_i,\; \|w_i\|_0 \le k_i,\; i=1,\dots,m\Big\},$$7 and buy-in bounds $$\Omega = \Big\{p_i \le \mathbf{1}^\top w_i \le q_i,\; \|w_i\|_0 \le k_i,\; i=1,\dots,m\Big\},$$8, Asexual Reproduction Optimization encodes a portfolio by a chromosome containing $$\Omega = \Big\{p_i \le \mathbf{1}^\top w_i \le q_i,\; \|w_i\|_0 \le k_i,\; i=1,\dots,m\Big\},$$9 asset indices and $P$0 corresponding weights (Mansouri et al., 2021). Cardinality is built into the representation, while budget and bound feasibility are restored by a repair procedure derived from Chang et al. (Mansouri et al., 2021). On five standard benchmark datasets with 31, 85, 89, 98, and 225 assets, the reported mean percentage error relative to the unconstrained efficient frontier is lower for ARO than for GA, SA, TS, and PSO on four of the five datasets, and the average error is reduced by approximately 20 percent relative to the minimum average error of those alternatives (Mansouri et al., 2021).

In index tracking, another heuristic strategy avoids the NP-hard joint problem by separating asset pre-selection from weight estimation. Eight pre-selection procedures are studied, combining forward selection or backward elimination with OLS or LAD regression, with or without an intercept (Meade et al., 24 Mar 2025). The preferred variant is BE-OLS(n), and for S&P 500 tracking the paper reports that out-of-sample tracking errors are roughly proportional to $P$1 (Meade et al., 24 Mar 2025). For enhanced index tracking, by contrast, cardinalities of the order 10 to 20 are reported as most effective (Meade et al., 24 Mar 2025).

Metaheuristics also require repair operators. A covariance-aware repair framework for long-only portfolios with $P$2, simplex, and box constraints introduces the distance

$P$3

whose square equals the tracking-error variance between portfolios $P$4 and $P$5 (Iliopoulos, 23 Dec 2025). The proposed Covariance-Aware Simplex Projection first selects the active set using volatility-normalized scores $P$6, then projects onto the selected simplex by minimizing $P$7 rather than Euclidean distance (Iliopoulos, 23 Dec 2025). On S&P 500 data from 2020–2024, CASP-Basic is reported to deliver materially lower portfolio variance than standard Euclidean repair, with most of the variance reduction driven by volatility-normalized selection and an additional consistent improvement from the covariance-aware projection (Iliopoulos, 23 Dec 2025).

A separate controversy concerns higher-order portfolio objectives. In native cubic cardinality-constrained portfolio optimization, where the objective contains quadratic Markowitz terms plus cubic three-way sector co-movement penalties, SA and tabu search are applied only after Rosenberg quadratization expands the variable count from $P$8 to $P$9 and adds penalty terms for auxiliary consistency and cardinality (Serbarinov, 16 Mar 2026). Under matched 60-second CPU budgets, Hyper-Adaptive Momentum Dynamics operates directly on the native cubic objective with exact cardinality-preserving projection and iterated local search, and the paper reports substantially lower decoded native cubic objective values than SA and tabu search at $x^b,x^s$0 (Serbarinov, 16 Mar 2026). The paper’s decoded-feasibility analysis shows that SA can satisfy all exact cardinality and Rosenberg auxiliary constraints yet still decode to a native objective 80–88% worse than HAMD at $x^b,x^s$1, which the authors interpret as a surrogate-distortion effect rather than simple infeasibility (Serbarinov, 16 Mar 2026).

6. Extensions: CVaR, tracking, tangent portfolios, dynamic factors, and global optimization

Cardinality constraints are now used well beyond static variance minimization. In CVaR optimization, sparsity can be handled by introducing both a sparse auxiliary portfolio $x^b,x^s$2 and an auxiliary hinge variable $x^b,x^s$3, leading to a relaxed problem in $x^b,x^s$4 solved by PALM with FISTA-style acceleration (Zhang et al., 2018). For small instances, the paper reports that the method finds a very good local minimum for the cardinality-constrained CVaR model, while for larger dimensions it yields feasible portfolios nearly as efficient as their unconstrained counterparts (Zhang et al., 2018).

In tangent portfolio optimization, sparsity is imposed directly on the Sharpe-ratio maximizer: $x^b,x^s$5 A Cholesky-based heuristic exploits the characterization that Sharpe ratios are ordered by angles in the transformed space $x^b,x^s$6 (Bae et al., 17 Feb 2025). The surrogate problem is solved by selecting the $x^b,x^s$7 largest absolute components of $x^b,x^s$8, where $x^b,x^s$9 is the unconstrained tangency portfolio, and then re-optimizing on the reduced universe (Bae et al., 17 Feb 2025). The paper further introduces a covariance-based diagonal-dominance score and reports a strong positive correlation, about 0.8265, between this score and heuristic performance relative to CPLEX (Bae et al., 17 Feb 2025).

Dynamic settings generalize the support constraint across time. In a multi-period mean–variance problem under dynamic factor models, cardinality appears as a cone-type constraint

$A \succeq 0$00

which limits the number of active risky positions each period (Gao et al., 25 Feb 2025). Despite the non-convexity, the paper derives a semi-analytical optimal policy as a piecewise linear feedback rule in wealth, with factor dependence embedded in slope vectors $A \succeq 0$01 obtained from backward stochastic optimization recursions (Gao et al., 25 Feb 2025). In a Markov regime-switching illustration with no-short-selling and cardinality at most 2 out of 4 assets, the active asset set changes with regime and sign regime, showing that sparse dynamic portfolios can exhibit discrete support switching over time (Gao et al., 25 Feb 2025).

At the other extreme, globally optimal sparse quadratic optimization can be pursued by interval branch-and-bound. For the generic convex quadratic problem

$A \succeq 0$02

with $A \succeq 0$03 and box constraints $A \succeq 0$04, an interval branch-and-bound method branches only at 0 and encodes each coordinate’s status by flags indicating fixed zero, undecided, or definitely nonzero (Singh et al., 5 Apr 2025). The paper states that the proposed algorithm can also accommodate linear inequality constraints while maintaining global convergence, so a budget constraint can be incorporated by linearization into inequalities (Singh et al., 5 Apr 2025). In regression benchmarks, the method is competitive with GUROBI’s quadratic mixed-integer solver and scales to dimensions up to $A \succeq 0$05 for small $A \succeq 0$06, which suggests applicability to sparse portfolios when exact support control is required (Singh et al., 5 Apr 2025).

A complementary theoretical line studies relaxations of quadratically constrained cardinality minimization. For the problem

$A \succeq 0$07

where $A \succeq 0$08 counts nonzero components, continuous relaxations derived from mixed-integer formulations are shown to have arbitrarily poor approximation ratios, whereas optimized diagonal relaxations often give much stronger lower bounds and greatly reduce branch-and-bound complexity (Wei, 2012). In portfolio terms, this means that risk ellipsoids close to diagonal, diagonally dominant, or nearly coordinate-aligned structures are especially favorable for diagonal relaxation quality (Wei, 2012).

7. Computational trade-offs, misconceptions, and open directions

Several broad conclusions emerge. First, exact cardinality control matters. Regularization and surrogate penalties may induce sparsity, but they generally do not let the investor prescribe the support size directly (Mousavi et al., 2023). This is why many recent methods favor exact $A \succeq 0$09 constraints handled through penalties, projections, split variables, or discrete neighborhoods rather than through tuning-only surrogates (Zhang et al., 2018, Moka et al., 15 May 2025).

Second, the main algorithmic trade-off is between global optimality and scalability. MIQP and global branch-and-bound methods can produce exact or provably optimal solutions, but their cost rises rapidly with problem size and support complexity (Singh et al., 5 Apr 2025, Mousavi et al., 2023). Continuous methods such as DCA, PALM, SCA, Boolean relaxation, and penalty decomposition typically converge only to stationary points or local minimizers, yet they can solve medium- and large-scale instances in fractions of a second to a few seconds and often match exact solvers closely on benchmark sets (Moeini, 2014, Zhang et al., 2018, Jiang et al., 2019, Moka et al., 15 May 2025).

Third, sparsity interacts strongly with model specification. For pure index tracking, larger cardinality improves out-of-sample tracking error, transaction volume, and return–risk ratios, approaching the index as $A \succeq 0$10 rises (Meade et al., 24 Mar 2025). For enhancement, small cardinalities preserve alpha better (Meade et al., 24 Mar 2025). For tangent portfolios, low correlation and diagonal dominance improve the quality of Cholesky-based support selection (Bae et al., 17 Feb 2025). For cubic models, surrogate quadratization can distort the search landscape severely (Serbarinov, 16 Mar 2026). For dynamic factor models, cardinality constraints produce regime-dependent support switching, but the optimal policy retains a tractable piecewise linear feedback form (Gao et al., 25 Feb 2025).

A plausible implication is that future progress will depend less on a single dominant algorithmic paradigm than on matching the support-handling mechanism to the surrounding portfolio model. Static mean–variance, CVaR, tracking, dynamic factor allocation, and higher-order co-movement models all impose cardinality in different ways, and the most effective methods exploit that structure directly rather than treating sparsity as a generic afterthought.