# finding extreme points for Lagrangian with multiple inequality constraints

I am trying to find maximum of

$$$$f(x, y) = x^2 - xy + y - 4x$$$$

$$$$\label{constraints} \text{s.t. } 0 \leq x \leq 2 \text{ and } 0 \leq y \leq 1$$$$

I got an advice to first check for the interior solution (it has none there is one saddle point at [1,-2] and afterwards check each constraint separately, but I am not sure if what I did is correct and how to proceed with finding extreme points at the boundaries

\begin{aligned} 0=x; 0=y \\ 0=x; 1=y \\ 2=x; 0=y \\ 2=x; 1=y \\ \end{aligned}

Now we can set up 4 Lagrangians to check for each boundary:

\subsubsection*{First Lagrangian:}

$$$$\mathcal{L} = x^2 - xy + y - 4x - \lambda_1(-x) - \lambda_2 (-y)$$$$

with FOCs

\begin{aligned} \frac{\partial \mathcal{L} }{\partial x} & = 2x - y -4 + \lambda_1 = 0\\ \frac{\partial \mathcal{L}}{ \partial y} & = -x + 1 + \lambda_2 = 0 \\ x&=0\\ y&=0 \end{aligned}

hence here $$x=y=0$$ and $$\lambda_1 = 4$$ and $$\lambda_2=-1$$

\subsubsection*{Second Lagrangian:}

$$$$\mathcal{L} = x^2 - xy + y - 4x - \lambda_1(-x) - \lambda_2 (-y)$$$$

with FOCs

\begin{aligned} \frac{\partial \mathcal{L} }{\partial x} & = 2x - y -4 + \lambda_1 = 0\\ \frac{\partial \mathcal{L}}{ \partial y} & = -x + 1 + \lambda_2 = 0 \\ x&=0\\ y&=1 \end{aligned}

hence here $$x=y=0$$ and $$\lambda_1 = 3$$ and $$\lambda_2=-1$$

\subsubsection*{Third Lagrangian:}

$$$$\mathcal{L} = x^2 - xy + y - 4x - \lambda_1(-x) - \lambda_2 (-y)$$$$

with FOCs

\begin{aligned} \frac{\partial \mathcal{L} }{\partial x} & = 2x - y -4 + \lambda_1 = 0\\ \frac{\partial \mathcal{L}}{ \partial y} & = -x + 1 + \lambda_2 = 0 \\ x&=0\\ y&=1 \end{aligned}

hence here $$x=y=0$$ and $$\lambda_1 = 3$$ and $$\lambda_2=-1$$

\subsubsection{Fourth Lagrangian}

$$$$\mathcal{L} = x^2 - xy + y - 4x - \lambda_1(-x) - \lambda_2 (-y)$$$$

with FOCs

\begin{aligned} \frac{\partial \mathcal{L} }{\partial x} & = 2x - y -4 + \lambda_1 = 0\\ \frac{\partial \mathcal{L}}{ \partial y} & = -x + 1 + \lambda_2 = 0 \\ x&=2\\ y&=1 \end{aligned}

• You seem to check the corner points, not the boundaries; why are all your Lagrangians the same? Why not write one Lagrangian with all four constraints? Oct 5 '18 at 22:56
• Given the optimal value of $y$, the function is convex in $x$ and its maximum over $x \in [0,2]$ must lie on the boundary, i.e., either $x=0$ or $x=2$. So the global max $(x^*,y^*)$ is quite easy to find without Lagrange multipliers, as we know it suffices to assume $x^*\in \{0,2\}$. Oct 5 '18 at 22:57
• @LinAlg well I just was told this would be simpler but I am not opposed to it I am just not sure how to solve it with so many unknowns Oct 5 '18 at 22:59
• @Michael by the way how would I prove that function is convex in this case? I know without lagrangian I would look at the eigenvalues of hessian but I am not sure how the matrix should look with inequality constraints Oct 5 '18 at 23:01
• Since the feasible set is compact, and the cost convex, the $\max$ must occur at an extreme point of the feasible set. Hence there are only 4 points to check. Forget the Lagrangian. Oct 5 '18 at 23:21

Since $$f$$ is a quadratic function having a saddle point it is not convex.

In order to find the maximum of $$f$$ on the boundary $$\partial R$$ of the given rectangular domain $$R$$ just pull back $$f$$ to the four edges of $$R$$, i.e., consider the four auxiliary functions \eqalign{&\phi_0(y):=f(0,y)=y,\quad\phi_2(y)=f(2,y)=-y-4\qquad\qquad(0\leq y\leq 1),\cr &\psi_0(x):=f(x,0)=(x-2)^2-4,\quad \psi_1(x):=f(x,1)=(x-2.5)^2-5.25\qquad(0\leq x\leq2)\ .\cr} Since all four of these functions are monotone on the relevant intervals we can conclude that $$\max_{(x,y)\in R}f(x,y)=\max\bigl\{f(0,0),f(2,0),f(0,1),f(2,1)\bigr\}=1\ .$$