Solution type	$x_1^{*}$	$x_2^{*}$
Lower boundary, x1	$\text{max}(x_1^{\mathit{\text{min}}},\frac{{\beta }_1}{{\alpha }_1})$	$\text{min}(x_2^{\mathit{\text{max}}},x_2^{\mathit{\text{bound}}})$
Lower boundary, x2	$\text{min}(x_1^{\mathit{\text{max}}},x_1^{\mathit{\text{bound}}})$	$\text{max}(x_2^{\mathit{\text{min}}},\frac{{\beta }_2}{{\alpha }_2})$
Interior	Numerically solve for $(x_1^{*},x_2^{*})$: $-2k_Q{\alpha }_1{(x_1^{*})}^2+(2a_2{\alpha }_1+k_Q({\beta }_1-C_1))x_1^{*}+a_2(C_1-{\beta }_1)$ $=-2k_Q{\alpha }_2{(x_2^{*})}^2+(2a_1{\alpha }_2+k_Q({\beta }_2-C_2))x_2^{*}+a_1(C_2-{\beta }_2)$ or $-2k_{IA}{\alpha }_1{(x_1^{*})}^2+(2b_2{\alpha }_1+k_{IA}({\beta }_1-C_1))x_1^{*}+b_2(C_1-{\beta }_1)$ $=-2k_{IA}{\alpha }_2{(x_2^{*})}^2+(2b_1{\alpha }_2+k_{IA}({\beta }_2-C_2))x_2^{*}+b_1(C_2-{\beta }_2)$ $x_2^{*}=\frac{{\beta }_2-C_2\pm \sqrt{{(C_2-{\beta }_2)}^2-4{\alpha }_2[{\alpha }_1{(x_1^{*})}^2+(C_1-{\beta }_1)x_1^{*}-B]}}{2{\alpha }_2}$