3.1.2. Minimizing the Mean and Variance

Referring to the above subsections, we define the formula of MDSVC as follows

$$\begin{array}{ll}\underset{R,\mathfrak{a}}{\min} & R^2 + \lambda\_1 \bar{\gamma} + \lambda\_2 \stackrel{\scriptstyle \chi}{\dot{\gamma}} + \mathbb{C} \sum\_{i=1}^m \xi\_i \\ \text{s.t. } & ||\phi(\mathbf{x}\_i) - \mathbf{a}||^2 \le R^2 + \xi\_{i\prime}^{\times} \\ & \xi\_i^{\times} \ge 0 \end{array} \tag{9}$$

Consider that the center *a* of the sphere is closer to the denser part in the feature space as minimizing the mean, and then we minimize the value of *λ*<sup>2</sup> to make more points closer to *a*, resulting in fewer support vector points. Next, we simplify Equation (9).

Based on Theorem 1, Equation (9) leads to

$$\begin{cases} \min\_{\mathbf{R},\alpha} \mathbf{R}^2 + \mathfrak{a}^T (\lambda\_1 \mathbf{Q} + H + P) \mathfrak{a} - \frac{2\lambda\_1}{\mathfrak{m}} \mathbf{e}^T \mathbf{Q} \mathfrak{a} + \mathbb{C} \sum\_{i=1}^{\mathfrak{m}} \mathcal{J}\_i^{\mathbb{Z}} \\ \text{s.t. } \left\| \boldsymbol{\phi}(\mathbf{x}\_i) - \mathbf{X} \mathbf{a} \right\|^2 \le R^2 + \xi\_{i\boldsymbol{\nu}} \\ \xi\_{i\boldsymbol{i}}^{\mathbb{Z}} \ge 0 \end{cases} \tag{10}$$

By introducing Lagrange multipliers *βi*, *μi*, the Lagrange function of Equation (12) is given as follows

$$\begin{split} L(\mathbb{R}, \mathfrak{a}, \mathfrak{x}, \mathfrak{f}, \mathfrak{f}, \mathfrak{p}) &= \mathfrak{a}^{T} ( (\lambda + 1)\_{1} \mathbb{Q} + H + P ) \mathfrak{a} \\ &- (\frac{2\lambda\_{1}}{\mathfrak{m}} \mathfrak{e}^{T} \mathbb{Q} + 2\mathfrak{z}^{T} \mathbb{Q}) \mathfrak{a} + R^{2} (1 - \sum\_{i=1}^{m} \beta\_{i}) + \sum\_{i=1}^{m} (\mathbb{C} - \mu\_{i} - \beta\_{i}) \mathfrak{z}\_{i} \end{split} \tag{11}$$

By setting the partial derivatives {*R*, *α*, *ξ*} to zero for satisfying the KKT conditions, we have the following equations of derivatives

$$\frac{\partial L}{\partial R} = 2R - 2R\sum\_{i=1}^{m} \beta\_i = 0\tag{12}$$

$$\frac{\partial L}{\partial \mathbf{a}} = 2\mathbf{a}^T((\lambda\_1 + 1)\mathbf{Q} + H + P) - \left(\frac{2\lambda\_1}{\mathbf{m}}\mathbf{e}^T\mathbf{Q} + 2\mathbf{g}^T\mathbf{Q}\right) = \mathbf{0} \tag{13}$$

$$\frac{\partial L}{\partial \xi\_i^{\alpha}} = \mathbb{C} - \mu\_i - \beta\_i = 0 \tag{14}$$

Thus, we adopt *G* = ((*λ*<sup>1</sup> + 1)*Q* + *H* + *P*) −1 *Q*, where ((*λ*<sup>1</sup> + 1)*Q* + *H* + *P*) <sup>−</sup><sup>1</sup> refers to the inverse matrix of ((*λ*<sup>1</sup> + 1)*Q* + *H* + *P*). On the basis of these equations, we obtain vector *A* as follows

$$A = \frac{\lambda\_1}{\text{m}}((\lambda\_1 + 1)Q + H + P)^{-1}Qe = \frac{\lambda\_1}{\text{m}}Ge \tag{15}$$

Substituting Equation (15) into Equation (13), we thus have

$$
\mathfrak{a} = \mathfrak{A} + \mathfrak{G}\mathfrak{B} \tag{16}
$$

By substituting Equations (12)–(14) into Equation (11), Equation (11) is re-written as follows

$$\begin{array}{l} L(\mathfrak{F}) = (A + G\mathfrak{F})^T ( (\lambda\_1 + 1)Q + H + P )(A + G\mathfrak{F}) - (\frac{2\lambda\_1}{\mathfrak{m}} \mathfrak{e}^T Q + 2\mathfrak{f}^T Q)(A + G\mathfrak{f}) \\ = \min\_{\mathfrak{F}} \frac{1}{2} \mathfrak{f}^T D \mathfrak{f} + F \mathfrak{f} \end{array} \tag{17}$$

We notice that *G* = ((*λ*<sup>1</sup> + 1)*Q* + *H* + *P*) −1 *Q*, so *D* and *F* have the following form

$$\begin{array}{l}D = 4QG - 2G^TQ = 2G^TQ = 2QG\\ F = \frac{2\lambda\_1}{m}e^TQG \end{array} \tag{18}$$

Referring to the above equations, thus, we derive our formula of MDSVC as follows

$$\begin{aligned} \min\_{\boldsymbol{\beta}} & \frac{1}{2} \boldsymbol{\beta}^T \boldsymbol{D} \boldsymbol{\beta} + \boldsymbol{F} \boldsymbol{\beta} \\ \text{s.t. } & 0 \le \boldsymbol{\beta}\_i \le \boldsymbol{C} \end{aligned} \tag{19}$$

Based on Theorem 2, *D* is symmetric and consists of positive elements. We can then make a conclusion that Equation (19) is a convex quadratic problem resulting from the convex objective function and convex domain *β* ∈ [0, *C*]. Thus, we can solve the objective function with convex quadratic programming.
