# Solve $\max_{x_1,x_2,x_3} \{ \alpha \min \{a x_1,b x_2,c x_3\}\}$ s.t. $p_1 x_1 + p_2 x_2 + p_3 x_3 = w$

Consider the objective function \begin{equation*} f(x_1,x_2,x_3)=\alpha \min \{a x_1,b x_2,c x_3\} \end{equation*} where $\alpha, a, b, c \in \mathbb{R}$ are arbitrary constants.

We wish to maximize $f$, i.e. solve \begin{equation*} \max_{\{(x_1,x_2,x_3) \in \mathbb{R}^3\}} \{\alpha \min \{a x_1,b x_2,c x_3\} \} \end{equation*} subject to the constraint \begin{equation*} p_1 x_1 +p_2 x_2 + p_3 x_3 = w \end{equation*} where $p_1,p_2,p_3,w \in \mathbb{R}$ are arbitrary constants.

(To visualize this problem, we wish to find the vector $\mathbf{x} \in \mathbb{R}^3$ such that the plane $p_1 x_1 + p_2 x_2 + p_3 x_3 = w$ intersects the 3-dimensional "box" $f(x_1,x_2,x_3)=\alpha \min \{a x_1,b x_2, c x_3\}$ at its lower "corner".)

Remark

To solve an constrained optimization problem, you would normally write a Lagrangian to turn the constrained maximization problem into an unconstrained maximization problem: \begin{align*} \mathcal{L}(x_1,x_2,x_3,\lambda) &=f(x_1,x_2,x_3)+\lambda(w - p_1 x_1 - p_2 x_2 - p_3 x_3) \\ & \\ &=\alpha \min \{a x_1,b x_2,c x_3\} - \lambda (p_1 x_1 + p_2 x_2 + p_3 x_3 - w) \end{align*} whose solutions are given by the first order conditions \begin{equation*} \frac{\partial f(x_1,x_2,x_3)}{\partial x_i}=\lambda p_i \end{equation*} The problem is, in this particular example, our objective function isn't everywhere differentiable (since $\min$ has a "kink"). As this answer points out, the above partial derivatives of our objective function with respect to $x_i$ will only exist on the "sides" of this 3-dimensional box. So how, mathematically, can one take the first order conditions to solve this problem?

The first order conditions require differentiability so you can't use those. What you really need is the generalized KKT conditions, that deal with sub-differentiability. Basically, rather than the gradient of the Lagrangian to be zero at $x^*$, you need the zero vector to be a subgradient at $x^*$. This is sufficient since your problem is still concave, despite it being nondifferentiable.

Note: this assumes $\alpha \geq 0$. But in that case, you can just drop $\alpha$.

Note2: problems like this are normally solved using linear programming, not through optimality condition algebra.

Note3: To convert your program into a linear program, the following problem will model the $\min$ in your objective function:

$\max z$ s.t. $z \leq ax_1, z \leq bx_2, z \leq cx_3, p^tx = w$

To clarify: You want to maximize $\min\{a,b,c\}$. You could see it as wanting to find THREE functions of which the smallest one is as large as possible. Let's denote $z := \min\{a,b,c\}$.

What do we know of $z$? It can't be larger than any of $a,b,c$. It is easy to see that this is a necessary condition to model your problem - what might be less obvious is that it is also a sufficient condition.

This is due to us trying to MAXIMIZE $z$ - if we can choose ANY feasible $z$ that is not greater than any of $a,b,c$, then the largest we ever can make $z$ is bounded by the smallest of $a,b$ and $c$. So

$$\text{maximize}\ z = \min\{a,b,c\} \\ @ \ p^tx = w$$

is equivalent to

$$\text{maximize}\ z = \min\{a,b,c\} \\ @ \ p^tx = w \\ z \leq a, z \leq b, z\leq c$$

But as I explained, defining $z$ as the $\min$ of a bunch of terms is redundant. We can just use $z$ as a decision variable instead:

$$\text{maximize}\ z \\ @ \ p^tx = w \\ z \leq a, z \leq b, z\leq c.$$

You are correct that there will be three additional Lagrange multipliers (but for inequality constraints) and one additional decision variable. Letting $z := x_4$ is completely fine here - there's nothing special about $z$.

• Hi Benjamin, thanks for your reply. Since I don't have any prior experience with Linear Programming, could you show me what you mean, i.e. how to apply linear programming to solve this problem, for the purpose of this example? – Mathemanic Nov 4 '15 at 19:58
• Meanwhile, I'll try applying the Lagrangian method to the new constrained maximization problem you wrote. I have three questions, however: (1) Where did this new CMP problem come from? I don't quite see why it is equivalent to the original. (2) For this new problem, there will be 4 Lagrange multipliers, correct? (3) What does $p^t$ denote in your equation? – Mathemanic Nov 4 '15 at 19:58
• $p^tx = p_1x_1 p_2 x_2 \dots$ – Benjamin Lindqvist Nov 5 '15 at 7:09
• Linear programming is kind of annoying to do by hand, and a little bit off-topic anyway - Patrickjmt has some nice videos on the Simplex method on Youtube if you really want to, however. Computers can solve LP:s in polynomial time, so it's basically considered trivial. – Benjamin Lindqvist Nov 5 '15 at 7:13
• Will edit answer to clarify equivalent problem. – Benjamin Lindqvist Nov 5 '15 at 7:15