Let $J$ be the Functional $ J:X \rightarrow \mathbb{R}$ where $X$ is a Banach space of functions,
I would Like to minimize this functional under three constraint $F_1,F_2$ and $F_3$ such that $ F_i:X \rightarrow \mathbb{R} \;$ are linear For $i=1,2,3$
To be more precise:
$\underset{u\in X}{\min} J(u)\quad$ subject to $\;F_1(u)=0, F_2(u)\leq 0$ and $F_3(u)\leq0$
I would like to use the Lagrange multiplier but the Problem is that as all the constraints are Linear so the defferential $dF_{1}(u), dF_{2}(u)$ and $dF_{3}(u)$ are not linearly independant in $X^{'}$
What can I do in this situation ? Thank You in advance comments and references are welcome