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This is question 4.35 from Introduction to Linear Optimization by Bertsimas and Tsitsiklis.

The problem

Consider two non-empty polyhedra $P=\{\mathbf{x}\in\mathbb{R}^n\ |\ \mathbf{Ax}\leq\mathbf{b}\}$ and $Q=\{\mathbf{x}\in\mathbb{R}^n\ |\ \mathbf{Dx}\leq\mathbf{d}\}$.

(a) Devise a linear programming problem such that: if $P\cap Q\neq\varnothing$, it returns a point in $P\cap Q$; if $P\cap Q=\varnothing$, the linear programming problem is infeasible.

(b) Suppose that $P\cap Q$ is empty. Use the dual of the problem you have constructed in part (a) to show that there exists a vector $\mathbf{c}$ such that $\mathbf{c}^\text{T}\mathbf{x}<\mathbf{c}^\text{T}\mathbf{y}$ for all $\mathbf{x}\in P$ and $\mathbf{y}\in Q$.

Attempted solution

I think I have an answer for part (a): $$ \begin{array}{rl} \text{max}\quad&\mathbf{0}^\text{T}\mathbf{x} \\ \text{s.t.}\quad&\mathbf{Ax}\leq\mathbf{b} \\ & \mathbf{Dx}\leq\mathbf{d} \end{array} $$

This seems to accomplish the requirements of part (a). I am stuck on part (b). The dual of the above linear program is $$ \begin{array}{rl}\text{min}\quad&\mathbf{p}_1^\text{T}\mathbf{b}+\mathbf{p}_2^\text{T}\mathbf{d}\\\text{s.t.}\quad&\mathbf{p}_1^\text{T}\mathbf{A}+\mathbf{p}_2^\text{T}\mathbf{D}=\mathbf{0}^\text{T}\\ & \mathbf{p}_1,\mathbf{p}_2\geq\mathbf{0} \end{array} $$ If we assume the primal is infeasible (i.e. $P\cap Q=\varnothing$), then the dual is either infeasible or unbounded. Since $\mathbf{p}_1=\mathbf{0}$, $\mathbf{p}_2=\mathbf{0}$ is feasible for the dual, it must be unbounded. I think this might be a useful fact, but I can't see exactly how.

Geometrically, this question is very straightforward. But I'm struggling to come up with the algebra to show the existence of this $\mathbf{c}$.

Similar question

The question found here is similar, but I'm not interested in computing the separating hyperplane, just proving its existence.

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    Before I proceed to help you answering the question, you should have a look at your "dual" LP. I don't think that it is correct. – 2017-02-16
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    Ah, I seem to have flipped my inequalities on the bounds for $\mathbf{p}_1$ and $\mathbf{p}_2$. Is that what you were referring to? Thanks for tip! – 2017-02-18
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    There is still more. The transpose of $Ax \leq b, Dx \leq d$ is also incorrect. Try to fix that and you will see, I'm quite sure. – 2017-02-18
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    Hint: The correct form of the dual LP is: minimize $p_1^Tb+p_2^Td$ s.t. $A^Tp_1 + D^Tp_2 = 0, p_1 \geq 0, p_2\geq 0$. – 2017-02-19
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    Sorry for slow responses. I see why that dual is correct--each variable in the primal is a constraint in the dual, so we just want one constraint, corresponding to the primal variable $\mathbf{x}$. I feel like I now apply either Farkas' lemma or the separating hyperplane theorem, since that's the most obvious thing I can think of to give the existence of some $\mathbf{c}$. Am I headed in the right direction? – 2017-02-21
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    @DavidM. You don't need them. Just some linear algebraic manipulations will suffice. See my answer for details. – 2017-02-21

1 Answers 1

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(a) The linear program

\begin{array}{rl} \max &\mathbf{0}^\text{T}\mathbf{x} \\ \text{s.t.}&\mathbf{Ax}\leq\mathbf{b} \tag{P}\label{P}\\ & \mathbf{Dx}\leq\mathbf{d} \end{array}

suggested by the OP responds to the problem.

(b) Since $P \cap Q = \varnothing$, each of these two points satisfies exactly one of the two constraints in \eqref{P} (from the definition of $P$ and $Q$).

The OP's deduction is correct:

The dual for \eqref{P} is \begin{array}{rl} \min & \mathbf{p}_1^\text{T}\mathbf{b}+\mathbf{p}_2^\text{T}\mathbf{d} \\ \text{s.t.} & \mathbf{p}_1^\text{T}\mathbf{A}+\mathbf{p}_2^\text{T}\mathbf{D}=\mathbf{0}^\text{T} \tag{D} \label{D} \\ & \mathbf{p}_1,\mathbf{p}_2\geq\mathbf{0} \end{array} The infeasibility of \eqref{P} and the feasibility of \eqref{D} (with $\mathbf{p}_1 = {\bf 0}, \mathbf{p}_2 = {\bf 0}$) implies that \eqref{D} is unbounded.

From the unboundedness of \eqref{D}, we deduce that for each $z \in \Bbb R$, there exists $\mathbf{p}_1,\mathbf{p}_2\geq\mathbf{0}$ such that $\mathbf{p}_1^\text{T}\mathbf{b}+\mathbf{p}_2^\text{T}\mathbf{d} \le z$. In particular, we fix $\epsilon > 0$ and take $z = -\epsilon$, so that $$\mathbf{p}_1^\text{T}\mathbf{b}+\mathbf{p}_2^\text{T}\mathbf{d} < 0 \tag{*} \label{*}.$$

For all ${\bf x} \in P$ and ${\bf y} \in Q$, we have (from the definition of $P$ and $Q$)

\begin{align} \mathbf{Ax} &\leq \mathbf{b} \tag{1}\label{1} \\ \mathbf{Dy} &\leq \mathbf{d} \tag{2}\label{2} \end{align}

Multiply $\mathbf{p}_1^\text{T}$ and $\mathbf{p}_2^\text{T}$ on both sides of \eqref{1} and \eqref{2} respectively. (We can do so since $\mathbf{p}_1 \ge {\bf 0}, \mathbf{p}_2 \ge {\bf 0}$.)

\begin{align} \mathbf{p}_1^\text{T}\mathbf{Ax} &\leq \mathbf{p}_1^\text{T}\mathbf{b} \tag{3}\label{3} \\ \mathbf{p}_2^\text{T}\mathbf{Dy} &\leq \mathbf{p}_2^\text{T}\mathbf{d} \tag{4}\label{4} \end{align}

Substitute the constraint in \eqref{D} into \eqref{4}.

$$(-\mathbf{p}_1^\text{T}\mathbf{A}){\bf y} \leq \mathbf{p}_2^\text{T}\mathbf{d} \tag{4'}\label{4'}$$

Add \eqref{3} and \eqref{4'}.

$$\mathbf{p}_1^\text{T}\mathbf{A} ({\bf x} - {\bf y}) \le \mathbf{p}_1^\text{T}\mathbf{b}+\mathbf{p}_2^\text{T}\mathbf{d} \stackrel{\eqref{*}}< 0 \tag{$\star$} \label{fin}$$

It's evident from \eqref{fin} that ${\bf c} = (\mathbf{p}_1^\text{T}\mathbf{A})^\text{T} = \mathbf{A}^\text{T} \mathbf{p}_1$ satisfies $$\mathbf{c}^\text{T}\mathbf{x}<\mathbf{c}^\text{T}\mathbf{y} \quad \forall \mathbf{x} \in P, \forall \mathbf{y} \in Q.$$