Investors allocate proportions of their capital among the assets they invest in. For the purpose of this study, these proportions are allocated to stocks. We denote by N the number of risky assets, R is the required return level, and \(x^0\) is the initial risky assets before rebalancing: \(x^0_k\) is the proportion of capital initially allocated to asset \(k, k=1,2,3,...,N\). Let \(x,x^b\) and \(x^s\) be N dimensional vectors of controllable variables: \(x_k\) is the portfolio invested in risky asset k after rebalancing, \(x^b_k\) is the purchases (proportion used) of risky asset k and \(x^s_k\) is the sales (proportion obtained) of risky asset k. The transaction costs incurred when buying risky assets is \(c^b\) and that of selling risky assets is \(c^s\). The financial portfolio is described by a Ndimensional vector of random returns r. The portfolio total random return is \(R_p=f(x,r)=\sum ^N_{k=1}x_k r_k\), portfolio expected return vector \(\mu _p=\mathbb {E}_x f(x,r)=\sum ^N_{k=1}\mathbb {E}_x r_k\), \(\sigma _p^2=\mathbb {E}_x(R_p\mathbb {E}_x(R_p))^2\) as variance of portfolio return and Q as the variancecovariance matrix of the portfolio return.
Roy safetyfirst principle and it’s modification
Most investors’ aim is to maximize returns and minimize risk. The Roy safetyfirst principle advocates avoiding extreme losses through the minimization of disaster probability. To optimally construct a portfolio strategy, Roy’s safetyfirst principle defines a threshold or a minimum acceptable return R, below which the portfolio wealth is considered to be a disaster. The best portfolio is one that minimizes the chances that the portfolio’s return, \(R_p\), will fall below a minimum acceptable return, R. In essence, an investor selects his portfolio by solving this optimization problem:
$$\begin{aligned} \underset{x \in \mathbb {R}^{N}}{\text {minimize}}&\quad \mathbb {P}(f(x,r) \le R)\nonumber \\ \text {subject to}&\quad e^Tx=1 \end{aligned}$$
(1)
where e is a vector with ones as entries and \(\mathbb {P}\) is a probability measure. Roy employed Bienayme–Tchebycheff's inequality as the investor is likely not to know the actual probability function and obtained an approximation
$$\begin{aligned} \mathbb {P}(R_p \le R)\le \frac{\sigma _p^2}{(\mu _pR)^2} \end{aligned}$$
Thus, the optimization problem is reformulated as
$$\begin{aligned} \underset{x \in \mathbb {R}^{N}}{\text {minimize}}&\quad \frac{\sigma _p^2}{(\mu _pR)^2} \nonumber \\ \text {subject to}&\quad e^Tx=1 \end{aligned}$$
(2)
The modification to the Roy’s approach we adopt is by using a coherent downside risk measure known as Conditional ValueatRisk or Expected shortfall as it has a set of desirable properties for a risk measure (Platen and Heath 2006) leading to more accurate estimates of probability. For a detailed study on desirable properties of an ideal risk measure in portfolio theory, we refer the reader to Rachev et al. (2008).
A wellknown downside risk measure known as ValueatRisk focuses on the percentiles of loss distributions and measures the predicted maximum loss at a given probability level. Mathematically it is formulated as \(\alpha\)quantile \(VaR_\alpha (X)= min \{z\mid (F_X(z)\ge \alpha \}\), where X is a loss random variable and \(\alpha \in (0,1)\) is the given probability level. Values for \(\alpha\) often used are 90 %, 95 % and 99 %. Considering ValueatRisk (VaR) has undesirable properties such as nonsubadditive and nonsmooth etc., Rockafellar and Uryasev (2000) introduced a coherent downside risk measure termed Conditional Valueat Risk (CVaR) and for \(\alpha \in (0,1)\) represented it as
$$\begin{aligned} CVaR_\alpha (X)= \int _{\infty }^{+\infty }zdF_X^{\alpha }(z) \end{aligned}$$
where
$$\begin{aligned} \begin{array}{@{} r @{} c @{} l @{} } &F_X^{\alpha }(z)\,&\,= {\left\{ \begin{array}{ll} 0 & \quad \text {when } z<VaR_{\alpha }(X) ,\\ \frac{F_X(z)\alpha }{1\alpha }& \quad \text {when } z \ge VaR_\alpha (X). \end{array}\right. } \end{array} \end{aligned}$$
Equivalently, for \(x \in X \subseteq \mathbb {R}^N\) and random vector \(r\in \mathbb {R}^N\) which represents the actual portfolio return has a continuous density function p(r)
$$\begin{aligned} CVaR_\alpha (x)= \frac{1}{1\alpha } \int _{f(x,r) \ge VaR_\alpha (x)}f (x,r)p(r)dr \end{aligned}$$
First, we will determine the semideviation of the random return f(x, r) from the \(\alpha\)quantile \(VaR_\alpha (x)\). With respect to Bienayme–Tchebycheff's inequality, the following estimate is valid for \(VaR_{\alpha )}(x)>R\):
$$\begin{aligned} \mathbb {P}\{f(x,r) \le R \}& = \mathbb {P}\{f(x,r)\ge R\}\nonumber \\ & = \mathbb {P} \{VaR_{\alpha )}(x)f(x,r) \ge VaR_{\alpha )}(x)R\}\nonumber \\& = \mathbb {P}\{\mid f(x,r)VaR_{\alpha )}(x)\mid \ge VaR_{\alpha )}(x)R \}\nonumber \\ & \le \mathbb {P}\mid \{\min \{f(x,r)VaR_{\alpha )}(x), 0\}\mid \ge VaR_{\alpha )}(x)R \}\nonumber \\ & \le \frac{\ \mathbb {E}_x \mid \min \{f(x,r)VaR_{\alpha }(x), 0\}\mid }{VaR_{\alpha )}(x)R} \end{aligned}$$
(3)
Let us consider an \(\alpha\)quantile
$$\begin{aligned} VaR_{\alpha }(x)= \min \{z\mid \mathbb {P}(f(x,r)\le z)\ge \alpha \}\implies \mathbb {P}\{(f(x,r)\ge VaR_{\alpha }(x)\}=1\alpha \end{aligned}$$
and a measure of risk
$$\begin{aligned} CVaR_{\alpha }(x)=\frac{\mathbb {E}_x \mid \min \{f(x,r)VaR_{\alpha }(x), 0\}\mid }{\mathbb {P}\{f(x,r) \ge VaR_{\alpha }(x)\}} \end{aligned}$$
which is termed expected shortfall from the \(\alpha\)quantile \(VaR_\alpha (x)\) value. The estimate (3) can be written as
$$\begin{aligned} \begin{aligned} \mathbb {P}\{f(x,r) \le R \}&\le \frac{ \mathbb {P}\{f(x,r) \ge VaR_{\alpha }(x)\}}{VaR_{\alpha }(x)R}\frac{\mathbb {E}_x \mid \min \{f(x,r)VaR_{\alpha }(x), 0\}\mid }{\mathbb {P}\{f(x,r) \ge VaR_{\alpha }(x)\}}\\&\le \frac{(1\alpha )CVaR_{\alpha }(x)}{VaR_{\alpha }(x)R} \end{aligned} \end{aligned}$$
(4)
Therefore (1) can be reformulated by considering the approximation of the right hand side of (4) and obtain the following
$$\begin{aligned} \underset{x \in \mathbb {R}^{N}}{\text {minimize}}&\quad \frac{(1\alpha )CVaR_{\alpha }(x)}{VaR_{\alpha }(x)R} \nonumber \\ \text {subject to}&\quad VaR_{\alpha }(x)> R\nonumber \\&\quad e^Tx=1 \end{aligned}$$
(5)
Telser (1955) considered a portfolio strategy by maximizing portfolio returns under the the constraint of Roy safetyfirst principle. He solved the optimization problem:
$$\begin{aligned} \underset{x \in \mathbb {R}^{N}}{\text {maximize}}&\quad \mu ^Tx \nonumber \\ \text {subject to}&\quad \mathbb {P}(f(x,r) \le R) \ge {1\epsilon } \end{aligned}$$
(6)
Inspired by Telser’s approach, we constrain (5) with the minimum mean return vector \(\mu ^Tx \ge L\) from below where L is the lower bound of \(u^Tx\), where \(L>R\). Thus we obtain the optimization problem :
$$\begin{aligned} P_0:\quad \underset{x \in \mathbb {R}^{N}}{\text {minimize}}&\quad \frac{(1\alpha )CVaR_{\alpha }(x)}{VaR_{\alpha }(x)R}\nonumber \\ \text {subject to}&\quad \mu ^Tx\ge L\nonumber \\&\quad VaR_{\alpha }(x)> R\nonumber \\&\quad e^Tx=1 \end{aligned}$$
(7)
In another approach, we consider variance and the modified Roy’s safety firstprinciple as a consolidated risk measure in a meanrisk framework. To this end, we propose the optimization problem:
$$\begin{aligned} P_1:\quad \underset{x \in \mathbb {R}^{N}}{\text {minimize}}&\quad x^TQx+\frac{(1\alpha )CVaR_{\alpha }(x)}{VaR_{\alpha }(x)R}\nonumber \\ \text {subject to}&\quad \mu ^Tx\ge L\nonumber \\&\quad VaR_{\alpha }(x)> R\nonumber \\&\quad e^Tx=1 \end{aligned}$$
(8)
The rest of the modifications is geared towards investigating realistic constraints such as transaction costs, sparsity, and stability to \(P_0\) and \(P_1\) on the financial market.
Portfolio revision
We consider an extension of problems (7) and (8) in which transaction costs are incurred to rebalance or revise the initial portfolio \(x^0\), into an efficient portfolio x. A portfolio of investments may require rebalancing on periodical basis because of updated risk, and return information is generated over time. We make the following assumptions on the transaction cost function c.
Assumption 1
The transaction cost function satisfies the following:

(i)
c(x) is a convex function of x

(ii)
c(0) = 0

(iii)
c(x)
\(\ge 0, \quad \forall x\)
To achieve portfolio \(x_k\) from the previous or initial portfolio \(x_k^0\), we make a payment of transaction costs \(c(xx^0)\). We incorporate proportional transaction costs (Kellerer et al. 2000; Muthuraman and Kumar 2006; Mitchell and Braun 2013) which are induced by liquidity costs, tax, brokerage fees (Dumas and Luciano 1991; Kellerer et al. 2000; Lobo et al. 2007) into our model. Therefore, proportional transaction cost follows this structure:
$$\begin{aligned} c(xx^0)=\sum _{k=1}^N c_k \left( x_kx_k^0\right) \end{aligned}$$
where
$$\begin{aligned} \begin{array}{@{} l @{} l @{} l @{} } &c_k (x_kx^0_k)\;&\;= {\left\{ \begin{array}{ll} c_k^b (x_kx^0_k) & \quad \text {if } \quad x_k \ge x^0_k ,\\ c_k^s (x_k^0x_k) & \quad \text {if } \quad \mathrm{otherwise}. \end{array}\right. } \end{array} \end{aligned}$$
where cost of buying is \(c_k^b \>0\) and cost of selling is \(c_k^s>0\).
The \(P_1\) model with proportional transaction costs \((P_{1t})\) is the optimization problem
$$\begin{aligned} P_{1t}:\quad \underset{x,x^s,x^b \in \mathbb {R}^{N}}{\text {minimize}}&\quad x^TQx+ \frac{(1\alpha )CVaR_{\alpha }(x)}{VaR_{\alpha }(x)R} \end{aligned}$$
(9)
$$\begin{aligned} \text {subject to}&\quad \mu ^Tx\sum _{k=1}^N \left( c^bx_k^b+c^s x_k^s\right) \ge L \end{aligned}$$
(10)
$$\begin{aligned}&\quad \sum _{k=1}^N x_k +c^b \sum _{k=1}^N x_k^b+c^s \sum _{k=1}^N x_k^s=1 \end{aligned}$$
(11)
$$\begin{aligned}&\quad x_k=x_k^0+x_k^bx^s_k, \quad \forall k=1,\ldots ,N \end{aligned}$$
(12)
$$\begin{aligned}&\quad x_k^b \cdot x_k^s=0, \quad \forall k=1,\ldots ,N \end{aligned}$$
(13)
$$\begin{aligned}&\quad VaR_{\alpha }(x)> R \end{aligned}$$
(14)
$$\begin{aligned}&\quad x_k^b\ge 0 ,\quad x_k^s \ge 0, \quad \forall k=1,\ldots ,N \end{aligned}$$
(15)
The model \(P_{1t}\) minimizes the upper bound estimate (5) w.r.t x and superimposes the lower constraint \(L \le \mu 'x\sum _{k=1}^N (c^bx_k^b+c^s x_k^s)\) on the average return after the deduction of transaction costs.
Explaining the constraints with respect to transaction costs, the above optimization problem is subjected to a set of linear constraints. Constraint (10) requires the net return of the portfolio after the deduction of transaction costs to be greater or equal to a threshold level L. Constraint (11) is the budget constraint: the capital available to cover transaction costs and shares of stocks. Constraint (12) shows that \(x_k\) represents the portfolio position to be chosen explicitly through sold shares \(x^s_k\) and purchased shares \(x^b_k\) that are rebalanced adjustments to the initial position \(x^0_k\) of stock k. Constraint (13) and constraint (15) are the complementary constraint and nonnegative constraint respectively. They prevent any possibility of concurrent purchases and sales (Dybvig 2005). Note that \(P_0\) model with proportional cost \((P_{0t})\) is defined similarly but without the variance term in the objective function.
Stable and sparse portfolio
In a portfolio selection strategy where the dimensionality of the set of candidate assets is high, sparsity is desired. When the number of assets is large, a nonregularized numerical approach will intensify the effects of estimation risk, leading to an unstable and unreliable estimate of the vector x. Typically, portfolio managers want to set up portfolios with suitable balance between risk and return by investing in a small number of assets, thereby limiting their transaction, management, and monitoring costs.
To obtain meaningful and sparse (zero components) results, a regularization procedure is usually adopted. A standard approach is to augment the objective function of interest with a \(\textit{l}_0\)norm penalty or adding a cardinality constraint \(\Vert x \Vert _0 \le N'\) to optimization problems \(P_{0t}\) and \(P_{1t}\), where \(\Vert x \Vert _0\) is the number of the nonzero entries of x and \(N'\) is the upper bound limitation of assets to be managed in the portfolio. However, with the inclusion of cardinality constraint, the portfolio selection strategy becomes NPhard (MoralEscudero et al. 2006). We therefore impose its equivalent \(\textit{l}_1\)norm penalty as employed by Brodie et al. (2009) among others. The \(\textit{l}_1\)norm is a convex function of x, and such convex relaxation makes portfolio selection strategy more tractable.
To this end, we suggest to evaluate the portfolio weights by
$$\begin{aligned} S_{1t}:\quad \underset{x,x^s,x^b \in \mathbb {R}^{N}}{\text {minimize}}&x^TQx+\frac{(1\alpha )CVaR_{\alpha }(x)}{VaR_{\alpha }(x)R} +\tau _1 \Vert x\Vert _1\nonumber \\ \text {subject to}&\quad \mu 'x\sum _{k=1}^N \left( c^bx_k^b+c^s x_k^s\right) \ge L\nonumber \\&\quad \sum _{k=1}^N x_k +c^b \sum _{k=1}^N x_k^b+c^s \sum _{k=1}^N x_k^s=1\nonumber \\&\quad x_k=x_k^0+x_k^bx^s_k, \quad \forall k=1,\ldots ,N\nonumber \\&\quad x_k^b \cdot x_k^s=0, \quad \forall k=1,\ldots ,N\nonumber \\&\quad VaR_{\alpha }(x)> R\nonumber \\&\quad x^b_k\ge 0 ,\quad x^s_k \ge 0,\quad \forall k=1,\ldots ,N \end{aligned}$$
(16)
where the \(\textit{l}_1\)norm of a vector \(x \in \mathbb {R}^N\) is defined by \(\Vert x\Vert _1{:}{=}\sum _{k=1}^N \mid x_k \mid\) and \(\tau _1\) is an adjustable parameter that controls the sparsity of the portfolios. Similarly, \(S_{0t}\) can be formulated without the variance term in the objective function.
Optimization models \(S_{0t}\) and \(S_{1t}\) have estimation risk or overfitting problem. The \(l_1\)norm penalty expedites sparsity of x and leads to a subset of assets receiving zero weights. Such sparsity may result in underdiversification and extreme weights of the portfolio. On the contrary, convex norm ball does not produce sparsity but it can efficiently regularize size of portfolio weight vector. Thus, the norm ball constraint used by DeMiguel et al. (2009) can function as a solution to alleviate the problems of underdiversification and extreme weights of the portfolio aside estimation risk. Following DeMiguel et al.’s (2009) work and specifying the general squared \(l_2\)norm under no short sale constraint, we propose to formulate the portfolio weights by
$$\begin{aligned} RSMt:\quad \underset{x,x^b,x^s \in \mathbb {R}^{N}}{\text {minimize}}&\quad \frac{(1\alpha )CVaR_{\alpha }(x)}{VaR_{\alpha }(x)R}+\tau _1 \Vert x\Vert _1+\tau _2 \Vert x \Vert _2^2 \nonumber \\ \text {subject to}&\quad \mu ^Tx\sum _{k=1}^N \left( c^bx_k^b+c^s x_k^s\right) \ge L\nonumber \\&\quad \sum _{k=1}^N x_k +c^b \sum _{k=1}^N x_k^b+c^s \sum _{k=1}^N x_k^s=1\nonumber \\&\quad x_k=x_k^0+x_k^bx^s_k, \quad \forall k=1,\ldots ,N\nonumber \\&\quad x_k^b \cdot x_k^s=0, \quad \forall k=1,\ldots ,N\nonumber \\&\quad VaR_{\alpha }(x)> R\nonumber \\&\quad x^b_k\ge 0 ,\quad x^s_k \ge 0,\quad \forall k=1,\ldots ,N \end{aligned}$$
(17)
and
$$\begin{aligned} RSMVt:\quad \underset{x,x^b,x^s \in \mathbb {R}^{N}}{\text {minimize}}&\quad x^TQx+\frac{(1\alpha )CVaR_{\alpha }(x)}{VaR_{\alpha }(x)R}+\tau _1 \Vert x\Vert _1+\tau _2 \Vert x \Vert _2^2 \nonumber \\ \text {subject to}&\quad \mu ^Tx\sum _{k=1}^N \left( c^bx_k^b+c^s x_k^s\right) \ge L\nonumber \\&\quad \sum _{k=1}^N x_k +c^b \sum _{k=1}^N x_k^b+c^s \sum _{k=1}^N x_k^s=1\nonumber \\&\quad x_k=x_k^0+x_k^bx^s_k, \quad \forall k=1,\ldots ,N\nonumber \\&\quad x_k^b \cdot x_k^s=0, \quad \forall k=1,\ldots ,N\nonumber \\&\quad VaR_{\alpha }(x)> R\nonumber \\&\quad x^b_k\ge 0 ,\quad x^s_k \ge 0,\quad \forall k=1,\ldots ,N \end{aligned}$$
(18)
where \(\tau _1\) and \(\tau _2\) are tuning parameters controlling sparsity and stability respectively, with \(\Vert x \Vert _2^2 =x'x\) as the squared \(l_2\)norm of a vector. We estimate tuning parameters \(\tau _1\) and \(\tau _2\) by a method of crossvalidation. We perform crossvalidation for various possible values of the parameters and select the parameter value that produces the minimum crossvalidation average error. A combination of the \(l_2\)norm penalty and \(l_1\)norm penalty is referred to as elastic net (Zou and Hastie 2005).
Necessary and sufficient conditions for optimal problems
In this section we would like to identify the necessary and sufficient conditions for optimality of problems (17) and (18). We investigate the KarushKuhnTucker (KKT) conditions for these problems under the assumption of normality and study whether the constrained problems have optimal solutions.
KKT conditions for optimal problem
The KKT conditions provide necessary conditions for a point to be optimal point for a constrained nonlinear optimal problem. The system
$$\begin{aligned}&x_k=x_k^0+x_k^bx^s_k, \quad \forall k=1,\ldots ,N\nonumber \\&x_k^b \cdot x_k^s=0, \quad \forall k=1,\ldots ,N\nonumber \\&x_k^b\ge 0 ,\quad x_k^s \ge 0 \end{aligned}$$
(19)
has a unique solution
$$\begin{aligned} x_k^b &= \left\{ \begin{array}{@{}l@{\quad }l@{}} x_kx_k^0, & \quad \text {if} \quad x_kx_k^0 \ge 0\\ 0, & \quad \text {else}\\ \end{array}\right. \\ x_k^s & = \left\{ \begin{array}{@{}l@{\quad }l@{}} x_k^0x_k, & \quad \text {if} \quad x_kx_k^0 < 0\\ 0, & \quad \text {else}\\ \end{array}\right. \end{aligned}$$
i.e.
$$\begin{aligned} \begin{aligned} x_k^b=max\{x_kx_k^0,0\}\\ x_k^s=max\{x_k^0x_k,0\} \end{aligned} \end{aligned}$$
From the budget constraint \(\sum _{k=1}^N x_k +c^b \sum _{k=1}^N x_k^b+c^s \sum _{k=1}^N x_k^s=1\), we get that \(c^b \sum _{k=1}^N x_k^b+c^s \sum _{k=1}^N x_k^s=1\sum _{k=1}^N x_k\). Implementing the first constraint,\(\mu ^Tx\sum _{k=1}^N (c^bx_k^b+c^s x_k^s)\ge L\) we get that \(\mu ^Tx(1\sum _{k=1}^N x_k)\ge L\). Thus, RSMt can be represented as
$$\begin{aligned} \underset{x \in \mathbb {R}^{N}}{\text {minimize}}&\quad f_1(x)=\frac{(1\alpha )CVaR_{\alpha }(x)}{VaR_{\alpha }(x)R}+\tau _1 \Vert x\Vert _1+\tau _2 \Vert x \Vert _2^2 \nonumber \\ \text {subject to}&\quad g_1(x)= \mu ^Tx\left( 1\sum _{k=1}^N x_k\right) \ge L\nonumber \\&\quad h_1(x)=c^b \sum _{k=1}^N max\left\{ x_kx_k^0,0\right\} +c^s \sum _{k=1}^N max\left\{ x_k^0x_k,0\right\} =1\sum _{k=1}^N x_k\nonumber \\&\quad g_2(x)= VaR_{\alpha }(x)> R \end{aligned}$$
(20)
Similarly, RSMVt can be represented with addition of \(x^TQx\) to the objective function.
Let x be a regular point for the problem RSMt. Then the point x is a local minimum of f subject to constraints (20) if there exists Lagrange multipliers \(\lambda _1, \lambda _2\) and \(\lambda _3\) for the Lagrangian function \(L=f_1(x)+\lambda _1 g_1(x)+ \lambda _2 h_1(x)+\lambda _3 g_2(x)\) such that the following are true.

1.
\(\frac{\partial L}{\partial x}={(1\alpha )CVaR'_{\alpha }(x)}{VaR_{\alpha }(x)R}\frac{(1\alpha )CVaR_{\alpha }(x) VaR'_{\alpha }(x)}{(VaR_{\alpha }(x)R)^2}+\tau _1 c^1 +2\tau _2x\lambda _1(\mu +I^{N \times 1})+ \lambda _2 c^2\lambda _3 VaR'_{\alpha }(x)=0\)

2.
\(\lambda _1(L\mu ^Tx(1\sum _{k=1}^N x_k)=0\)

3.
\(\lambda _3(RVaR_{\alpha }(x))=0\)

4.
\(\lambda _1,\lambda _3 \ge 0\)

5.
\(\mu ^Tx(1\sum _{k=1}^N x_k)\ge L\)

6.
\(c^b \sum _{k=1}^N max\{x_kx_k^0,0\} +c^s \sum _{k=1}^N max\{x_k^0x_k,0\}=1\sum _{k=1}^N x_k\)

7.
\(VaR_{\alpha }(x)> R\)
where
$$\begin{aligned} c_k^1 &= \left\{ \begin{array}{@{}l@{\quad }l@{}} 1, & \quad \text {if } \quad x_k>0\\ 1, & \quad \text {if } \quad x_k<0\\ \in [1,1], & \quad \text {if } \quad x_k=0\\ \end{array}\right. \\ c_k^2 & = \left\{ \begin{array}{@{}l@{\quad }l@{}} c^b, & \quad \text {if }\quad x_k>x_k^0\\ c^s, & \quad \text {if } \quad x_k<x_k^0\\ \in [c^s,c^b], & \quad \text {if }\quad x_k=x_k^0\\ \end{array}\right. \end{aligned}$$
Remark 1
Since the function \(h_1\) in (20) is linear and the functions \(g_1\) and \(g_2\) are convex, then the feasible region \(\Omega = \{x : h_1,g_1,\) and \(g_2\}\) is a convex set. On the other hand, \(f_1\) is a convex function subject to the variable x. We see that any local minimum for problem (20) is a global minimum too and KKT conditions are also sufficient.
Similarly, the above KKT conditions holds for RSMVt for when \(f_1(x)\) in (20) is \(f_2(x)=x^TQx+\frac{(1\alpha )CVaR_{\alpha }(x)}{VaR_{\alpha }(x)R}+\tau _1 \Vert x\Vert _1+\tau _2 \Vert x \Vert _2^2\) and when the following are true

1.
\(\frac{\partial L}{\partial x}=2Qx+{(1\alpha )CVaR'_{\alpha }(x)}{VaR_{\alpha }(x)R}\frac{(1\alpha )CVaR_{\alpha }(x) VaR'_{\alpha }(x)}{(VaR_{\alpha }(x)R)^2}+\tau _1 c^1 +2\tau _2x\lambda _1(\mu +I^{N \times 1})+ \lambda _2 c^2\lambda _3 VaR'_{\alpha }(x)=0\)

2.
\(\lambda _1(L\mu ^Tx(1\sum _{k=1}^N x_k)=0\)

3.
\(\lambda _3(RVaR_{\alpha }(x))=0\)

4.
\(\lambda _1,\lambda _3 \ge 0\)

5.
\(\mu ^Tx(1\sum _{k=1}^N x_k)\ge L\)

6.
\(c^b \sum _{k=1}^N max\{x_kx_k^0,0\} +c^s \sum _{k=1}^N max\{x_k^0x_k,0\}=1\sum _{k=1}^N x_k\)

7.
\(VaR_{\alpha }(x)> R\)
where
$$\begin{aligned} c_k^1 & = \left\{ \begin{array}{@{}l@{\quad }l@{}} 1, & \quad \text {if } \quad x_k>0\\ 1, & \quad \text {if } \quad x_k<0\\ \in [1,1], & \quad \text {if } \quad x_k=0\\ \end{array}\right. \\ c_k^2 & = \left\{ \begin{array}{@{}l@{\quad }l@{}} c^b, & \quad \text {if }\quad x_k>x_k^0\\ c^s, & \quad \text {if }\quad x_k<x_k^0\\ \in [c^s,c^b], & \quad \text {if } \quad x_k=x_k^0\\ \end{array}\right. \end{aligned}$$
Remark 2
Since the feasible region \(\Omega = \{x : h_1,g_1,\) and \(g_2\}\) in (20) and the objective function \(f_2(x)\) are convex, then the feasible region is a convex set. We can see that the KKT conditions are also sufficient and any local minimum for problem (20) with objective function \(f_2(x)\) is a global minimum as well.