To solve the problem of maximizing $f(x, y) = y - x^2$ with constraints $x^2 + y^2 \leq 4$, $x + y \leq 2$, and $x + y \geq -2$, we need to set up a Lagrangian with multiple inequality constraints and verify that the Necessary Directional Constraint Qualification (NDCQ) holds.

Step 1: Define the Lagrangian with Multiple Inequality Constraints

Given:

• Objective function: $f(x, y) = y - x^2$
• Constraints:
  1. $g_1(x, y) = x^2 + y^2 - 4 \leq 0$
  2. $g_2(x, y) = x + y - 2 \leq 0$
  3. $g_3(x, y) = -x - y - 2 \leq 0$

The Lagrangian for this problem, incorporating the inequality constraints, is: $\mathcal{L}(x, y, \lambda_1, \lambda_2, \lambda_3) = y - x^2 + \lambda_1 (4 - x^2 - y^2) + \lambda_2 (2 - x - y) + \lambda_3 (2 + x + y)$ where $\lambda_1, \lambda_2,$ and $\lambda_3$ are the Lagrange multipliers associated with each constraint.

Step 2: Show That the Necessary Directional Constraint Qualification (NDCQ) Holds

The NDCQ condition is met if, at any point in the constraint set, the gradients of the active constraints are not only linearly independent but also have feasible directions in the constraint set.

1. Compute the Gradients of the Constraints:

   • $\nabla g_1(x, y) = (2x, 2y)$
   • $\nabla g_2(x, y) = (1, 1)$
   • $\nabla g_3(x, y) = (-1, -1)$

2. Check Linear Independence of Active Constraints: Since each gradient vector points in a different direction, no two of these vectors are scalar multiples of each other. Thus, at any feasible point in the constraint set, the gradients of the active constraints will be linearly independent, satisfying the NDCQ.

Therefore, we have established the Lagrangian and verified that the NDCQ holds for all points within the constraint set. This allows us to proceed with using the KKT conditions or related optimization techniques to solve the problem analytically.

Would you like further details on how to apply the KKT conditions or any specific part of the solution?

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Related Questions:

1. How does the NDCQ differ from other constraint qualifications, such as the LICQ?
2. What are the KKT conditions, and how do they apply to this problem?
3. How does adding inequality constraints affect the maximization process?
4. Can this problem be solved graphically, and if so, what would the feasible region look like?
5. How would the solution change if the objective function were modified, e.g., to $f(x, y) = y - x^3$?

Tip: For problems involving multiple constraints, always check if each constraint is active at the solution boundary to correctly identify and use the relevant gradients in the Lagrangian.