Linear Algebra - "Closest" Solution Suppose I have a combination of vectors (alpha and beta unrestricted): 
$ \gamma = \alpha\begin{pmatrix}
  3 \\
  2  \\
 5 \\
 \end{pmatrix} + \beta \begin{pmatrix}
  6 \\
  1  \\
 2 \\
 \end{pmatrix} $
and an associated value
$\lambda = 19\alpha + 13\beta$
I want to maximise $\lambda$ such that $\gamma \le \begin{pmatrix}
  33 \\
  9  \\
 19 \\
 \end{pmatrix}$
I want to minimise $\lambda$ such that $\gamma \ge \begin{pmatrix}
  33 \\
  9  \\
 19 \\
 \end{pmatrix}$
The context here is asset pricing, but felt this would be more suited as a maths question as that's what it boils down to.
Does anyone have any methods that work well?
 A: Your problems are really standard linear programs. I'm going to write $x$ for $\alpha$ and $y$ for $\beta$.
The first problem is:

maximize $19x+13y$ subject to:
\begin{eqnarray}
3x+6y&\leq 33\\
2x+ y&\leq 9\\
5x+2y&\leq 19
\end{eqnarray}

Notice the three inequalities define half-planes, and you could easily draw out the region of feasible solutions by hand, or join us in the year $2015$ and ask Wolfram to do so for you. Since your objective function $19x+13y=\lambda$ is a line with a negative slope, the optimal solution ends up occurring at that single vertex in the region of feasible solutions, which turns out to be the intersection of the lines $3x+6y=33$ and $5x+2y=19$. This intersection point is $(2,\frac{9}{2})$, and so $\lambda=\frac{193}{2}$. You can verify this also with  Wolfram Alpha
The second program is 

minimize $19x+13y$ subject to:
\begin{eqnarray}
3x+6y&\geq 33\\
2x+ y&\geq 9\\
5x+2y&\geq 19
\end{eqnarray}

Again you can just ask Wolfram to do this for you, getting an answer of $(x,y)=(\frac{7}{3},\frac{13}{3})$ where $\lambda= \frac{302}{3}.$
