To analyze and sketch the feasible region $S$ for the given constrained nonlinear programming problem, we start by interpreting the constraints:

Problem:

$\min f_0(x_1, x_2)$ subject to: $2x_1 + x_2 \geq 1 \quad (C_1)$ $x_1 + 3x_2 \geq 1 \quad (C_2)$ $x_1 \geq 0 \quad (C_3)$ $x_2 \geq 0 \quad (C_4)$

Step 1: Rewrite the constraints

We can rewrite the inequalities to better understand the boundary lines:

1. $2x_1 + x_2 = 1$ is the boundary line for $C_1$.
2. $x_1 + 3x_2 = 1$ is the boundary line for $C_2$.
3. $x_1 \geq 0$ restricts $x_1$ to the non-negative region (right half-plane).
4. $x_2 \geq 0$ restricts $x_2$ to the non-negative region (upper half-plane).

Step 2: Find intersection points

To sketch the region, we need to find the intersection points of the boundary lines of $C_1$ and $C_2$ with each other, as well as with the axes.

Constraint $C_1: 2x_1 + x_2 = 1$

• If $x_1 = 0$, $x_2 = 1$. So, the point is $(0, 1)$.
• If $x_2 = 0$, $x_1 = \frac{1}{2}$. So, the point is $\left( \frac{1}{2}, 0 \right)$.

Constraint $C_2: x_1 + 3x_2 = 1$

• If $x_1 = 0$, $x_2 = \frac{1}{3}$. So, the point is $\left( 0, \frac{1}{3} \right)$.
• If $x_2 = 0$, $x_1 = 1$. So, the point is $(1, 0)$.

Step 3: Sketch the region

• The boundary line $2x_1 + x_2 = 1$ passes through $\left( \frac{1}{2}, 0 \right)$ and $(0, 1)$.
• The boundary line $x_1 + 3x_2 = 1$ passes through $(1, 0)$ and $\left( 0, \frac{1}{3} \right)$.

The feasible region is the area that satisfies both inequalities, which lies above both lines and in the first quadrant (since $x_1 \geq 0$ and $x_2 \geq 0$).

The feasible region can be visualized as a polygon bounded by the lines, restricted by the axes $x_1 = 0$ and $x_2 = 0$, and where both constraints hold.

I'll now summarize the key points to plot:

1. The boundary line $2x_1 + x_2 = 1$ between $\left( \frac{1}{2}, 0 \right)$ and $(0, 1)$.
2. The boundary line $x_1 + 3x_2 = 1$ between $(1, 0)$ and $\left( 0, \frac{1}{3} \right)$.
3. The non-negative constraint regions: $x_1 \geq 0$ and $x_2 \geq 0$.

Would you like me to provide the plot for this feasible region? Let me know if you'd like more details or have any questions.

Here are some questions related to this topic:

1. What is the significance of feasible regions in optimization problems?
2. How do constraints affect the shape of the feasible region?
3. Can the feasible region ever be unbounded? Under what conditions?
4. What is the role of boundary lines in determining the optimal solution?
5. How does the choice of objective function $f_0(x_1, x_2)$ influence the solution?

Tip: When sketching feasible regions, always find the points where constraints intersect the axes and each other to identify key boundary points.