Worksheet 12-2: Duality

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Worksheet 12-2: Q1

Consider the problem

\[\begin{split} \begin{aligned} \min &\quad x^2+y^2\\ \text{s.t.} &\quad x+y+1 \leq 0 \end{aligned} \end{split}\]

1. Does strong duality hold? You need to check whether the problem is convex, and whether the constraints satisfy the generalized Slater’s condition.

2. Now, determine \(q^*\) using the following steps:

   a. What is the Lagrangian, \(L(x,y,\lambda)\), and the gradient of the Lagrangian \(\nabla_{x,y}L(x,y,\lambda)\)?

   b. When is the gradient of the Lagrangian zero? Solve for \(x\) and \(y\) in terms of \(\lambda\).

   c. For a fixed \(\lambda\), what is the minimum value of \(L(x,y,\lambda)\) over \(x,y\)? Use this to write \(q(\lambda) = \inf_{x,y} L(x,y,\lambda)\).

   d. What is \(q^*=\max_{\lambda} q(\lambda)\)?

3. What do you know about \(f^*\), the optimal value of the primal problem?

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Worksheet 12-2: Q2

Consider the problem

\[\begin{split} \begin{aligned} \min &\quad 2x+y\\ \text{s.t.} &\quad -3x+3y\leq -1\\ &\quad -3x-2y \leq -4 \end{aligned} \end{split}\]

1. Does strong duality hold? You need to check whether the problem is convex, and whether the constraints satisfy the generalized Slater’s condition.

2. What are the matrices \(A\), \(\mathbf{b}\), and \(\mathbf{c}\) such that the problem can be written as

\[\begin{split} \begin{aligned} \min &\quad \mathbf{c}^\top \mathbf{x}\\ \text{s.t.} &\quad A \mathbf{x}\leq\mathbf{b} \end{aligned} \end{split}\]

3. What is the dual problem?

4. Solve the dual problem to find \(q^*\).

5. Use this desmos to visualize the primal problem. What is \(f^*\), the optimal value of the primal problem? Is it the same as your \(q^*\)?

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Worksheet 12-2: Q3

Consider the problem

\[\begin{split} \begin{aligned} \min &\quad -3x-2y-1z\\ \text{s.t.} &\quad x+y+z\leq1\\ &\quad x-z \leq -2 \end{aligned} \end{split}\]

1. This is a linear programming problem. What are the matrices \(A\), \(\mathbf{b}\), and \(\mathbf{c}\) such that the problem can be written as

\[\begin{split} \begin{aligned} \min &\quad \mathbf{c}^\top \mathbf{x}\\ \text{s.t.} &\quad A \mathbf{x}\leq\mathbf{b} \end{aligned} \end{split}\]

2. What is the dual problem?

3. Solve the dual problem to find \(q^*\).

4. What is \(f^*\), the optimal value of the primal problem? Justify your answer.