Sparse Markowitz portfolio selection by using stochastic linear complementarity approach

We consider the framework of the classical Markowitz mean-variance (MV) model when multiple solutions exist, among which the sparse solutions are stable and cost-efficient. We study a two - phase stochastic linear complementarity approach. This approach stabilizes the optimization problem, finds the sparse asset allocation that saves the transaction cost, and results in the solution set of the Markowitz problem. We apply the sample average approximation (SAA) method to the two - phase optimization approach and give detailed convergence analysis. We implement this methodology on the data sets of Standard and Poor 500 index (S & P 500), real data of Hong Kong and China market stocks (HKCHN) and Fama & French 48 industry sectors (FF48). With mock investment in training data, we construct portfolios, test them in the out-of-sample data and find their Sharpe ratios outperform the \begin{document} $\ell_1$ \end{document} penalty regularized portfolios, \begin{document} $\ell_p$ \end{document} penalty regularized portfolios, cardinality constrained portfolios, and \begin{document} $1/N$ \end{document} investment strategy. Moreover, we show the advantage of our approach in the risk management by using the criteria of standard deviation (STD), Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR).

1. Introduction. Markowitz [19] ushered the era of modern portfolio management by his mean-variance model. In this model, he initiated a way for portfolio selection that balances the variance and return by mean-variance framework. Specifically, for m securities, the traditional Markowitz portfolio is to find a portfolio that has minimal variance for a given expected return ρ by solving the following problem: where w = (w 1 , w 2 , ..., w m ) T , 1 m is an m-dimensional vector with all entries being one, random variable r i : Ξ → R is the return of the ith security, µ i = E[r i (ξ)] is its expected return, the return of the securities r(ξ) = (r 1 (ξ), r 2 (ξ), ..., r m (ξ)) T , the expected return of the securities µ = (µ 1 , µ 2 , ..., µ m ) T , the covariance matrix of the returns C = E[(r(ξ)−µ)(r(ξ)−µ) T ]. Note that C is an m×m positive semi-definite matrix.
In practice, it is challenging to select portfolios computed from the Markowitz MV model. Firstly, when the assets in the investment universe are large-scale, the transaction cost and the trading complexity are considerable. Secondly, when the distribution of the stock returns is non-normal, a problem of overfitting arises in the estimation of unknown parameters, see [21]. Last but not least, the Markowitz portfolio problem is unstable, i.e., small changes in the data would result large changes in solution, as reported in [6].
Due to these facts, there is an ensemble of recently developed sparse Markowitz portfolio theories in the literature as follows.
or in [9], for p ∈ (0, 1), 2. Cardinality constrained portfolio selection (CCPS): Many researchers have investigated the CCPS by improving the branch-and-bound method, in [1], [3], developing a branch-and-bound method based on a novel geometric approach in [18], constructing a tight semidefinite program (SDP) problem approach in [33]. Several other scholars studied the CCPS by heuristic approaches in [8], [16], a positive programming approach in [27], and a nonmonotone projected gradient (NPG) method in [28]. The core of the model is to add a cardinality constraint in the Markowitz portfolio selection model to select a cardinality number of assets in the portfolio, min w T Cw s.t. w T 1 m = 1, w T µ = ρ w 0 ≤ k.
Although these several works are used to enhance the Markowitz portfolio selection, they have problems, such as the change of the objective function, imposition of additional constraints. Then there is an idea arising from those frameworks. How about finding a sparse solution from the optimal solution set of the classic Markowitz MV model? When the covariance matrix is semi-definite, i.e., when there exist multiple solutions of the Markowitz portfolio optimization problem, we study a novel method to select a sparse portfolio solution of the classic Markowitz MV model.
Our approach is traced back to inspirations from [7]. Chen & Xiang [12] gave a way to find sparse solutions of the linear complementarity problem (LCP). By this method, we study a two-phase stochastic linear complementarity approach (twophase approach) to find an approximate sparse solution of MV model by solving two simple convex problems in two phases. Note that finding a sparse portfolio over the solution set of MV model is an NP-hard problem, see [23]. One of the advantages of the method is that we can approximate the NP-hard problem by solving a convex quadratic optimization problem and a linear optimization problem. Moreover, due to the discontinuity of 0 norm, it may not be easy to know how to guarantee the convergence when the SAA method is applied to the problem. Therefore, another contribution of this paper is that we establish the convergence analysis when we apply the SAA method with the two-phase approach to find a sparse portfolio.
We use the two-phase approach to compute the sparse Markowitz portfolio of a randomly generated example, and three empirical examples, which are S&P 500 portfolios, HKCHN cross market portfolios and FF48 portfolios. We test their outof-sample performance. From preliminary results, our approach has better performance compared with the reported well-performed 1 penalty regularized portfolio, p penalty regularized portfolio, cardinality constrained portfolio and 1/N investment strategy [15].
Our approach outperforms traditional Markowitz portfolios in the sense that: the sparse portfolio saves transaction fees and reduces investment complexity. Normally, the transaction cost is positive linearly related to the number of assets allocated. The sparse solutions from our approach naturally produce a cost efficient investment.
Moreover, our approach outperforms other sparse portfolios in the sense that the sparse solution of the two-phase approach is located in the solution set of the classical Markowitz portfolio optimization which facilitates risk management. The merit of variance minimization of the Markowitz portfolio is inherited by the sparse solution from the two-phase approach in the sense that the sparse solution is exactly a solution of the classic Markowitz portfolio optimization problem.
From the results of our empirical examples, our sparse portfolio could outperform the comparative portfolios from the perspective of Sharpe ratio. Therefore, by our approach, we find a sparse and stable Markowitz portfolio such that the portfolio is well-posed to achieve the goal of classical MV model and preserves the stability and sparsity properties.
Throughout this paper, we use the following notation. x T y denotes the scalar products of two vectors x and y, · 1 and · 0 denote the 1 norm and 0 norm of a vector respectively. d(x, S) := inf x ∈S x − x denotes the distance from point x to set S. For two sets S 1 and S 2 , D(S 1 , S 2 ) := sup x∈S1 d(x, S 2 ) denotes the deviation of set S 1 from set S 2 and H(S 1 , S 2 ) := max(D(S 1 , S 2 ), D(S 2 , S 1 )) denotes the Hausdorff distance between two sets S 1 and S 2 .
This paper is organized as follows. In Section 2, we introduce the two-phase approach to find a sparse Markowitz portfolio. In Section 3, we analyze the convergence of the SAA method with the two-phase approach. In Section 4, we show the numerical and empirical applications of the model. Section 5 is a conclusion.

2.
Model construction and two-phase stochastic linear complementarity approach. We consider the sparse Markowitz portfolio optimization problem as follows: min where H is the optimal solution set of the Markowitz portfolio optimization problem where η ≥ 1. Compared with problem (1), problem (5) has an additional constraint to limit the arbitrage w 1 ≤ η. Short selling, which is defined as "the sale of a security that the seller does not own or that the seller owns but does not deliver" by the SEC, is considered crucial for effective arbitrage. In fact, short sale is often constrained. In other words, arbitrage is limited, see implications in [14], [17], [26]. Moreover, when η = 1, problem (5) is equivalent to the classical Markowitz portfolio model without shortselling: and when η = ∞, problem (5) is equivalent to (1). Problem (4) is a nonconvex noncontinuous optimization problem and may not be easy to solve. Inspired by the method in [12], we provide the two-phase stochastic linear complementarity approach to approximate problem (4) in the following subsection.
2.1. Two-phase stochastic linear complementarity approach. The basic idea of this approach is that we reformulate the feasible set of problem (4) (solution set of problem (5)) as a linear system and find its least 1 norm to approximate its sparse solution. To this end, we consider the reformulation of problem (5) Firstly, we will show the equivalence between problem (5) and problem (7). Secondly, we will study how to find a sparse solution from the optimal solution set of problem (7). Proposition 1. Problem (5) and problem (7) are equivalent in the sense that (i) If w * is an optimal solution of (5), then is an optimal solution of (7), then w * = (w + ) * − (w − ) * is an optimal solution of problem (5); (iii) the least 1 norm solution and sparse solution of problem (7) must be the least 1 norm solution and sparse solution of problem (5) and vice versa.
(ii) Let v 1 and v be the optimal values of problem (5) and problem (7) respectively. It is obvious that for any feasible solutionw of problem (5), (w + ,w − ) is a feasible solution of problem (7) with the same objective function value, that is be an optimal solution of problem (7), w * = (w + ) * − (w − ) * . Note that w * is a feasible solution of problem (5). Let w be an optimal solution of problem (5). Then which means (w * ) T Cw * = w T Cw and w * is an optimal solution of problem (5).
(iii) By (i) and (ii), we have that for any optimal solution w * of (5), any (w is the optimal solution of (7). Then let w * 1 be the least 1 norm solution of (5). Then we claim , which contradicts the fact that w * 1 is the least 1 norm solution of problem (5). Moreover, let (w + 1 , w − 1 ) be the least 1 norm solution of (7), it is obvious that is a feasible solution of (7), which contradicts the fact that (w + 1 , w − 1 ) is the least 1 norm solution of problem (7).
is the least 1 norm solution of (5). Otherwise, there exists an optimal solution ofw 1 of (5) such that , which contradicts the fact that (w + 1 , w − 1 ) is the 1 norm solution of problem (7). The proof for sparse solution is similar to the least 1 norm solution, we omit the details.
It is obvious that problem (7) can be rewritten as: where problem (9) can be rewritten as an LCP: We use LCP(q, M ) to denote LCP(10) with the related q and M .
Hereafter, for simplicity, we use (9) and problem (10) is a solution of (10) then w * is a solution of (9).
By Theorem 3.1.7 in [13], since M is a positive semi-definite matrix, the solution set of the LCP(q, M ), denoted by the SOL(q, M ) can be written as: By a given solution x * of the LCP(q, M ), all the solutions (w + , w − ) of the Markowitz portfolio optimization with corresponding Lagrange multipliers y are given in the SOL(q, M ). Then we try to look for a sparse solution in the solution set SOL(q, M ). We can find the sparse Markowitz portfolio of problem (4) by solving is an arbitrary solution of the LCP(q, M ).
Note that · 1 is a good approximation of · 0 solution, which has been widely used such as [6]. Then we can approximate problem (12) by the following linear programming: It will be very interesting to consider the upper bound of the 0 norm of the classical Markowitz portfolio optimization sparse solutions. We will discuss it in the following proposition.
Proposition 2. Consider problem (4), the number of nonzero entries in a solution of problem (4) is at most rank(C) + 3, where C is the covariance matrix.

2.2.
Equivalence between the least 1 norm solution and the sparse solution. Under some conditions, we can show that a solution of problem (12) can be found exactly by solving problem (13) if we replace (w + , w − ) by x in the objective functions for both two problems (In this subsection, we consider problem (12) and (13) with objective function x 0 and x 1 respectively). To this end, we need the following definition from [7]. Here |t| is the number of elements of the set t.
Definition 2.1. An m×n matrix P is said to satisfy the s-restricted isometry property (RIP) with a restricted isometry constant δ s , if for every m × |t| sub-matrix P t of P and for every vector z ∈ R |t| with |t| ≤ s, . P is said to satisfy the s, s -restricted orthogonality (RO) with a restricted orthogonality constant θ s,s for s + s ≤ n if for all sub-matrices, P t ∈ R m×|t| and P t ∈ R m×|t | of P , with |t| ≤ s and |t | ≤ s , for all the vectors z ∈ R |t| and z ∈ R |t | , |(P t z, P t z )| ≤ θ s,s z 2 z 2 holds for all the disjoint sets t and t .
The following theorem is from [12].
Theorem 2.2. Letx be the optimal solution of linear programming problem (13) with x 0 ≤ s. Then 1. if H satisfies the RIP with a restricted isometry constant δ 2s < 1, thenx is the unique sparse solution of the LCP(q, M ); 2. if H satisfies the RIP and RO with δ s + θ s,s + θ s,2s < 1, thenx is the unique solution of (13) and the unique sparse solution of the LCP(q, M ). (12) and (13). Therefore, we approach an NP-hard problem by solving two simple problems (10) and (13) in two phases.

Theorem 2.2 shows the relationship between problems
Remark 1. In this subsection, we considers the x 0 where x = (w + , w − , y), but we actually want to consider w 0 . The reason we consider x 0 is that, under the conditions of Theorem 2.2, its solution can be obtained by solving problem (13). Moreover, assuming that w * is the optimal solution of problem (4) and x = ((w) + , (w) − ,ȳ) is the optimal solution of problem (12) with objective function x 0 , it is obvious that In case that the dimension of w is very large, problem (12) is a good approximation of problem (4).
But the RIP and RO conditions in Theorem 2.2 are not easy to satisfy. In the rest of paper, we consider problem (12) and (13) with objective function (w + , w − ) 0 and (w + , w − ) 1 respectively.
3. The SAA method and convergence analysis. In this section, we use the SAA method to model the randomness of the Markowitz portfolio and consider the convergence analysis between the original problem and the approximation problem.
Let {r j = (r j 1 , ..., r j m )} N j=1 be the i.i.d. samples of random variable r. Then the SAA mechanism of the LCP sparse Markowitz portfolio optimization is: where Note that the LCP(q, M N ) (15) is driven from the first order necessary condition of the following SAA form of problem (9): It follows that we consider the convergence analysis between the original problem (13) and its SAA form (14). Although problem (13) is a stochastic linear problem, some of its parameters are solutions of the LCP(q, M ) (10). So it is hard to give the convergence analysis between problem (13) and its SAA form directly. We also prove it in two steps. In the first step, we prove the convergence analysis of KKT pairs between problem (9) and its SAA form (16). In the second step, we prove the convergence between the optimal solutions and the optimal values of (13) and (14). We need the following Slater condition.
, and I is an identity matrix. Then We consider the convergence analysis between KKT pairs between problem (9) and its SAA form (16). The KKT condition of problem (9) is: and the KKT condition of problem (16) is: where y, s, y N and s N are Lagrangian multipliers corresponding to the constraints Proposition 3. Suppose Assumption 1 holds. Then there exists a sufficiently large compact set C ⊂ R 4m+5 such that (i) the intersection of C and the set of KKT pairs of the true problem (9), denoted by Y * , is nonempty; (ii) for N sufficiently large, the intersection of C and the set of KKT pairs of the SAA problem (16), denoted by Y N , is nonempty; (iii) for every > 0, there exists N ( ) > 0 such that The result is directly implied by Proposition 2.3 in [30]. Note that in Proposition 2.3 of [30], they use the condition named "no nonzero abnormal multipliers constraint qualification (NNAMCQ)" to bound the Lagrangian multipliers. This constraint qualification is well known, see [5] and [32].
Here the NNAMCQ may not hold in problem (9) since we rewrite every equality constraint in (7) as two inequality constraints. Note that problems (7) and (9) are equivalent and (y 1 − y 3 ) and (y 2 − y 4 ) are corresponding Lagrangian multipliers of equality constraints, y 5 and s are corresponding Lagrangian multipliers of inequality constraints in problem (7). Moreover, by Assumption 1, problem (7) satisfies MFCQ, which implies NNAMCQ. Then (y 1 − y 3 , y 2 − y 4 , y 5 , s) and (y 1,N − y 3,N , y 2,N − y 4,N , y 5,N , s N ) are uniformly bounded for N sufficiently large and there exists sufficiently large compact set C such that Y * and Y N are nonempty.
Note that the KKT pairs of problem (9) and problem (16) are the solutions of the LCP(q, M ) in (10) and the LCP(q, M N ) in (15), Proposition 3.1 shows the convergence analysis between the first phase problem of our two-phase method and implies ς N → ς and γ N → γ as N → ∞ almost surely.
Then we move to the convergence analysis of the second phase problem (13) and its SAA problem (14). We use the Chapter 6, Proposition 6 and Remark 8 in [4] to show the result. Let v * and v N denote the optimal value of the true problem (13) and the SAA problem (14) respectively. Proposition 4. Suppose Assumption 1 holds. Then there exists a sufficiently large compact set K ∈ R 2m+5 such that (i) the intersection of K and the optimal solution set of problem (13), denoted by S, is nonempty; (ii) for N sufficiently large the intersection of K and optimal solution set of problem (14), denoted by S N , is nonempty; (iii) we have v N → v * and D(S N , S) → 0 w.p.1 as N → ∞ w.p.1.
Proof. Obviously, the optimal solution set of problem (13) is nonempty and v * is finite. Note that the feasible sets of problem (9) and problem (16) are closed and belong to the compact set {w + , w − ∈ R m : 1 T m (w + + w − ) ≤ η, w + , w − ≥ 0}. Moreover, we have uniform boundness of (y 1 −y 3 , y 2 −y 4 , y 5 ), and (y 1,N −y 3,N , y 2,N − y 4,N , y 5,N ) for N sufficiently large by Proposition 2.2 in [30] and the discussion below Proposition 3, we have S and S N are nonempty for N sufficiently large.
Moreover, by Proposition 3, we have that ς N → ς and γ N → γ as N → ∞ w.p.1. Then by the uniform law of large numbers [4, Chapter 6, Proposition 7], the constraint functions of problem (14) uniformly converge to the constraint functions of problem (13). Note that problem (13) and problem (14)

4.
Applications. In this section, we demonstrate the two-phase approach by a randomly generated example and three empirical examples. We use Matlab R2014a, in a computer with Intel Core 2 Due CPU E8500 3.16GHz for the randomly generated example portfolio construction, Hong Kong and China cross market portfolio construction and FF48 portfolio construction; Matlab R2015a in the service machine with the Intel Xeon E7-4890v2 processor, 2.8GHz, 37.5M Cache, 15 Cores per CPU, 4 CPU, 60 Core in total for S&P500 portfolio construction.
In the examples, we compute the first phase optimization problem by "quadprog" and the second phase optimization problem by "linprog". For the comparative portfolios, we compute the 1 norm penalty regularized portfolio by "CVX" 1 . For portfolios with cardinality constraint, we compute with a Matlab tool box Yalmip "optimize" 2 . For the p penalty regularized portfolio, we approximate the nonsmooth objective function with a smoothing function which has been extensively investigated in [2], [10] and compute it by "fmincon". The distance of the comparative portfolios to the Markowitz portfolio feasible solution set is also calculated. Together, we also compute and compare the VaR and CVaR of these portfolios. 4.1. Randomly generated example. We randomly generate sample returns of nine asset and apply the two-phase method to construct sparse portfolios. Example 1. For nine assets in the investment universe, we generate the multivariate normal random variables with mean µ and covariance Σ as the sample of the asset returns. We set every three assets in a group with the same expected return, the same STD, and fully correlated.
Explicitly, our assets have been arranged in three groups. Asset 1, asset 2 and asset 3 in Group a; asset 4, asset 5, and asset 6 in Group b; asset 7, asset 8 and asset 9 in Group c. The mean returns of the nine assets are µ = [10 10 10 1 1 1 3 3 3], and the variance of the nine assets are σ 2 = [5 5 5 1 1 1 2 2 2]. The correlations for assets in the same group is 1 and correlations of assets inter-groups are ρ ab = 0.5, ρ ac = 0.3 and ρ bc = 0.8.
We calculate and analyze the true optimal value and the approximate optimal value. The true optimal value and the SAA optimal value (STD) for various sample sizes, are displayed respectively in Table 1.
Data missing case: Consider a case where the price data of some trading days is missing for some assets, which is very often encountered in the real practice. Then the covariance matrix of the underlying assets could not be easily constructed. In [22], a quadratically convergent Newton method is introduced to compute the nearest covariance matrix. Using the above sample, we randomly delete some data, then apply the method in [22] to construct the nearest covariance matrix of the portfolio optimization problem. We then use the two-phase approach to solve the reconstructed problem.
The optimal value (variance) at phase 2 in the case of data missing is 6.23 (the equal STD 2.4960) in Table 1 Column 11. The two-phase stochastic approach could be accustomed to the case of asset prices data missing.
Convergence analysis: We also conduct tests for different sample size N = 100, 500, 1000, · · · , 10000. For each sample size, we conduct 100 independent exercises. The true optimal value and approximate optimal values with N increasing are shown in Table 1. The convergence of the optimal values with N increasing is shown in Figure 1. We use a rolling window procedure for out-of-sample comparison. T is the total period of returns in the data set; τ is the length of the rolling window; os is the length of the out-of-sample testing window, which is the holding period. The data frequency is daily or monthly, depending on the data set.
Our testing scheme is that we use τ returns as the training data, then use the subsequent os returns as the forecasting data to compute the out-of-sample return, STD, Sharpe ratio and sparsity. We do the exercise from the beginning of our sampling data and roll ahead. For example, our first portfolio selection takes place at the end of the first τ trading days or months. We use the τ historical returns to estimate covariance matrix C and mean µ by SAA method. We then solve the portfolio optimization problem by using the estimated parameters, targeting the required return and compute the weights of optimal solutions. Once a portfolio is thus determined, it is held for the subsequent os trading days or months from (τ +1) to (τ + os), and its returns are recorded. We repeat the same process, using returns from 2 to (τ + 1) to construct the portfolio in test 2. The portfolio is observed for the subsequent os trading days or months from (τ + 2) to (τ + os + 1) and their returns are recorded.
To reduce the effects of chance factors, we repeat the same exercise always with a rolling window as described above for 20 rolling times (out-of-sample performance tests). For a given period (whether it is the full period, or the subperiods), all the daily or monthly returns corresponding to this period are used to compute the average return and its STD.
The criteria that we pay attention to are portfolio STD, Sharpe ratio. Sharpe ratio, which measures the risk-adjusted return, is a ratio of return and STD in [25], where r p is portfolio return; σ p is portfolio STD. We compute the with r p = 1 20 It measures the trade-off between returns and volatilities of the portfolios. We also compute each test Sharpe ratio, where r t p is portfolio return; σ t p is portfolio STD at test t. We compute the with r t p = 1 os We compare the value of risk measure between different portfolios by using VaR and CVaR. VaR is a widely used risk measure to evaluate the market risk.
The definition of VaR [20] is: which says that VaR at confidence level α is given by the smallest number l so that the probability that the loss L exceeds l is at most (1 − α) and the definition of CVaR [24] is: We use the historical simulation method to calculate VaR and CVaR. For the computation of VaR and CVaR, the portfolios are thus held for the subsequent os days or months after the training period and their daily or monthly out-of-sample returns are observed. The confidence level of the VaR and CVaR is α = 99%.
We also investigate the distance of the 1 penalty regularized portfolio, cardinality constrained portfolio solution and p penalty regularized portfolio to the optimal Markowitz portfolio solution set, which is defined as the shortest distance between the comparative portfolio solution to the optimal solution set: where H is the solution set of the Markowitz portfolio optimization problem and w is a solution of the comparative portfolio. The optimal value of the optimization problem (23) is the distance of the comparative portfolio solution to the optimal solution set.
Example 2. S&P 500. The data set contains the returns of S&P 500 index component stocks of big companies by market capitalization listed on the NYSE or NASDAQ.
Our sampling data is from Compustat. Sampling data are daily returns of S&P 500 from January 2001 to August 2001. The required return is set as the average of the training sample returns. Our empirical analysis relies on a "rolling-sample" procedure. In this example, τ = 100, os = 5 for STD, Sharpe ratio computation, os = 20, α = 99% for VaR and CVaR computation.
We likewise carry out computation for our comparative portfolios. The empirical results of the forecasting return, STD, Sharpe ratio, sparsity, Var and CVaR are in Table 2. The sparsity of the MV model is contained in the bracket of the LCP sparse portfolio column. Abbreviations are LCP sparse portfolio (LCPSP), 1 penalty regularized portfolio with tuning parameter λ = 0.1 ( 1 0.1), cardinality constrained portfolio with cardinality number 100 (CCPS100) and 1/N investment strategy (1/N). The one-by-one test results of Sharpe ratio and sparsity are displayed in Figure 2.  We use the same data rolling scheme. The required return is set as the average of the total returns. The training data window length is τ = 25. The holding period data length is os = 10. For VaR and CVaR computation, we use os = 20 for observation and set confidence level α = 99%.
The empirical results of the forecasting return, STD, Sharpe ratio, sparsity, Var and CVaR are in Table 3. The sparsity of the MV model is contained in the bracket. The one-by-one rolling results of the tests are displayed in Figures 3(a)   Our sampling data is from Fama and French. Sampling data are monthly returns of 48 industry sector value-weighted portfolios in percentage from September 2005 to January 2011.
We use the same data rolling scheme. The required return is set as the average of the total returns on each test. The size of the rolling window is τ = 25. The out-of-sample performance tests used os = 10. For VaR and CVaR computation, we use os = 20 for observation and set confidence level α = 99%. Our sample data is rolling ahead monthly. We roll the window and repeat the procedure for 20 tests to generate the average aggregate results.
The empirical results of the forecasting return, STD, Sharpe ratio, sparsity, Var and CVaR are in Table 4. The sparsity of the MV model is contained in the bracket. The one-by-one test results of Sharpe ratio and sparsity are displayed in Figures 4(a) and 4(b). Our portfolio outperforms 1 portfolio (with tuning parameter 0.1), cardinality constrained portfolio (with the cardinality number 18 and 24), p portfolio (with p=0.5, tuning parameter 0.015) and 1/N investment strategy, denoted by 1 0.  Table 4. FF48 Portfolio return, STD, Sharpe Ratio, VaR, CVaR, sparsity and distance. 4.3. Discussion of the empirical results. In this subsection, we discuss the out-of-sample performance of our approach and our comparing approaches. Tables 2-4 report the out-of-sample performance of the portfolios constructed from the S&P 500, Hong Kong and mainland China cross-market stocks and FF48 data sets. We illustrate each test's Sharpe ratio performance and sparsity by bar graph in Figures 2-4. In the case when the MV model has multiple solutions, our method combines the advantage of the MV model and its sparsity modification methods ( 1 regularization, p regularization, cardinality constrained portfolio selection methods and so on).
On the one hand, we can observe from the tables and figures in Section 4.2 that the number of assets selected in the our approach are sparser than that in the classical Markowitz portfolios. That means we could construct sparse Markowitz mean-variance portfolios with reduced transaction cost in this way. Note that the simple structure of sparse portfolios enables the portfolio construction to depend less on the human management. Moreover, the sparsity of the sparse Markowitz portfolio solution could provide a reference for the setting of k in the cardinality constrained portfolio selection model.
On the other hand, the empirical results of the examples show that the twophase optimization scheme could find sparse solutions in the optimal solution set of the Markowitz MV model. The out-of-sample results indicate that the LCP sparse portfolio from our approach can keep the property of the classical Markowitz portfolio very well. That is in all the tests, the STDs of LCP sparse portfolio are reasonably small and in most of the tests, its STDs are the smallest. Moreover, not only the performance of the Sharpe radio, VaR and CVaR are reasonably good, but also the Sharpe radios in all the periods are stabler than the other portfolios in Figures 2(a), 3(a) and 4(a).

Conclusions.
We study a two-phase stochastic linear complementarity approach to seek for a sparse asset allocation of the Markowitz portfolio. In contrast to