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\begin{aligned} min_{β, g, h, {γ_{k}}_{k = 1}^{4}, {z_{k}}_{k = 1}^{4} \in R^{m}} & - \frac{1}{2} y^{T} X β - \frac{1}{2} n α λ 1^{T} g - \frac{1}{2} λ 1^{T} γ_{4} (2) \\ subject to & X^{T} X β - X^{T} y + γ_{1} - γ_{2} = 0 \\ n g - γ_{1} - γ_{2} = 0 \\ n (α - 1) g + n h - γ_{3} + γ_{4} - n α λ 1 = 0 \\ 0 \leq γ_{1} \leq M z_{1}, 0 \leq h - β \leq M (1 - z_{1}) \\ 0 \leq γ_{2} \leq M z_{2}, 0 \leq h + β \leq M (1 - z_{2}) \\ 0 \leq γ_{3} \leq M z_{3}, 0 \leq g \leq M (1 - z_{3}) \\ 0 \leq γ_{4} \leq M z_{4}, 0 \leq λ 1 - g \leq M (1 - z_{4}) \\ z_{1}, z_{2}, z_{3}, z_{4} \in {0, 1}^{m} \end{aligned}

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\begin{align} & \min \limits _{\bbeta , \mb{g}, \mb{h}, \{ {\bgamma }_ k\} _{k=1}^4, \{ \mb{z}_ k\} _{k=1}^4 \in R^ m} & \qquad & - \frac{1}{2} \mb{y}^ T \bX \bbeta - \frac{1}{2} n \alpha \lambda \mb{1}^ T \mb{g} - \frac{1}{2} \lambda \mb{1}^ T {\bgamma }_4 \hspace{1in}\mbox{(2)}\\ & \text {subject to} & & \bX ^ T \bX \bbeta - \bX ^ T \mb{y} + {\bgamma }_1 - {\bgamma }_2 =\mb{0} \nonumber \\ & & & n \mb{g} - {\bgamma }_1 - {\bgamma }_2 =\mb{0} \nonumber \\ & & & n(\alpha -1) \mb{g} + n \mb{h} - {\bgamma }_3 + {\bgamma }_4 - n \alpha \lambda \mb{1} =\mb{0} \nonumber \\ & & & \mb{0}\leq {\bgamma }_1 \leq \mathcal{M} \mb{z}_1, \quad \mb{0}\leq \mb{h} - \bbeta \leq \mathcal{M} (\mb{1} - \mb{z}_1) \nonumber \\ & & & \mb{0}\leq {\bgamma }_2 \leq \mathcal{M} \mb{z}_2, \quad \mb{0}\leq \mb{h} + \bbeta \leq \mathcal{M} (\mb{1} - \mb{z}_2) \nonumber \\ & & & \mb{0}\leq {\bgamma }_3 \leq \mathcal{M} \mb{z}_3, \quad \mb{0}\leq \mb{g} \leq \mathcal{M} (\mb{1} - \mb{z}_3) \nonumber \\ & & & \mb{0}\leq {\bgamma }_4 \leq \mathcal{M} \mb{z}_4, \quad \mb{0}\leq \lambda \mb{1} - \mb{g} \leq \mathcal{M} (\mb{1} - \mb{z}_4) \nonumber \\ & & & \mb{z}_1, \mb{z}_2, \mb{z}_3, \mb{z}_4 \in \{ 0, 1\} ^ m \nonumber \end{align}