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Problem 1

# In Exercises $$I-8,$$ write down (without solving) the dual LP problem. $$\begin{array}{ll} \text { Maximize } & p=2 x+y \\ \text { subject to } & x+2 y \leq 6 \\ & -x+y \leq 2 \\ & x \geq 0, y \geq 0 . \end{array}$$

Expert verified
The dual linear programming problem corresponding to the given primal problem is: $$\min$$ w = 6u + 2v, subject to: (1) u - v ≥ 2, (2) 2u + v ≥ 1, (3) u, v ≥ 0.
See the step by step solution

## Step 1: Identify the dual problem type

Since the primal problem is a maximization problem, the dual problem will be a minimization problem.

## Step 2: Write the objective function

In the dual problem, we have one variable per constraint (excluding x, y ≥ 0) in the primal problem. Let's denote the variables in the dual problem as u and v, respectively, for constraints (1) and (2). Therefore, the objective function in the dual problem would be: $$\min$$ w = 6u + 2v.

## Step 3: Write the constraints

Now, let's write the constraints for the dual problem. For each variable in the primal problem, we have a constraint in the dual problem. The primal problem has two variables (x and y), so the dual problem will have two constraints: (1) u - v ≥ 2, (2) 2u + v ≥ 1.

## Step 4: Write the bounds for the dual variables

Since the primal problem has “≤” constraints (excluding x, y ≥ 0), the dual variables (u, v) will have non-negative constraints: u, v ≥ 0. Now, we can write the entire dual problem as follows: $$\min$$ w = 6u + 2v, subject to: (1) u - v ≥ 2, (2) 2u + v ≥ 1, (3) u, v ≥ 0. This is the dual linear programming problem corresponding to the given primal problem.

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