Connection between Euler-Lagrange equations and KKT

While studying calculus of variations I came across the following problem: $$\min_{\substack{x\in X \\ y\in Y}} F(x,y) \text{ s.t. } L(x,y) =0$$ where $F$ is a convex functional, $L$ is a differential operator and $X$, $Y$ are Banach spaces. When, for each $x$, there is a unique $y$ such that $L(x,y)=0$ we define the operator $\phi(x) = y$ so we can study the reduced problem: $$\min_{x\in X} \tilde{F}(x) = \min_{x\in X} F(x,\phi(x))$$

The necessary and sufficient condition for optimality is $$\langle {\tilde {F}}^{\,\prime}(x) \,, x-\tilde{x}\rangle_{X^*\,X} \geq 0,\quad \forall \tilde x \in X.$$ The Euler-Lagrange equation is $\tilde F'(x) = 0$. I cannot convince myself that the Euler-Lagrange equation is the same as the KKTs for the full problem involving the adjoint: \begin{align*} \mathcal{L}_x(x,y,\lambda)&=0 \\ \mathcal{L}_y(x,y,\lambda)&=0 \\ \mathcal{L}_\lambda(x,y,\lambda)&=0 \end{align*}

where $\mathcal{L}(x,y,\lambda) = F(x,y) - \lambda L(x,y)$.

• see the Athoch's book [Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization.] ( amazon.com/Variational-Analysis-Sobolev-Spaces-Applications/dp/…) – MathOverview Aug 21 '16 at 11:42
• Which explicit manner for $\tilde{F}$? – MathOverview Aug 21 '16 at 12:03
• I have made this explicit now: $\tilde F(x) = F(x, \phi(x))$. – stephn28 Aug 21 '16 at 12:10
• The minimizatin will be $\min_{x\in X} \tilde{F}(x) = \min_{x\in X} F(x,\phi(x))?$ – MathOverview Aug 21 '16 at 12:13
• Yes, sorry for that – stephn28 Aug 21 '16 at 12:16