\begin{align*} \max Y&=R_1X_1+R_{2}X_{2}+R_3X_3+R_4X_4+R_5X_5+R_6X_6,\\ \text{s.t}:&F_1(X_1,X_{2}, \dots, X_{6})=X_1+X_3+X_5-1\le 0,\\ &F_2(X_1,X_{2}, \dots, X_{6})=X_2+X_4+X_6-1\le 0,\\ &F_3(X_1,X_{2}, \dots, X_{6})=R_1X_1+R_{2}X_{2}-C_1\le 0,\\ &F_4(X_1,X_{2}, \dots, X_{6})=R_3X_3+R_4X_4-C_2\le 0,\\ &F_5(X_1,X_{2}, \dots, X_{6})=R_5X_5+R_{6}X_{6}-C_3\le 0,\\ X_i\in \mathbb R \forall i=1,2,\dots, 6. \end{align*} If $y^{\star}\in \mathbb R^6$ is the optimal point, and if $\mu_1,\dots, \mu_5\in \mathbb R$ are Lagrange multipliers then it is easy to see from the dual feasibility condition: \begin{align} &\nabla Y(y^{\star})-\sum\limits_{i=1}^{5}\mu_i \nabla F_i(y^{\star})=0 \text{ we get }\\ &R_1-\mu_1-\mu_3R_1=0\\ &R_2-\mu_2-\mu_3R_2=0\\ &R_3-\mu_1-\mu_4R_3=0\\ &R_4-\mu_2-\mu_4R_4=0\\ &R_5-\mu_1-\mu_5R_5=0\\ &R_6-\mu_2-\mu_5R_6=0\\ &\mu_1,\mu_2,\dots, \mu_5\ge 0. \end{align} primal feasibility: \begin{align} F_i(y^{\star})&\le 0\quad \forall i=1,2,\dots,5. \end{align} complementary slackness: \begin{align} \mu_i F_i(y^{\star})=0 \quad \forall i=1,2,\dots, 5. \end{align}
Could anyone tell me how to write Dual of the above problem and find optimal conditions in more detail from here? So far have I done the correct steps? Thank you for helping. What are dual variables here? How to find KKT?