Solve a constrained problem using KKT conditions

Use Karush–Kuhn–Tucker (KKT) conditions to solve the following constrained optimization problem:

Minimize

$\min_{x,y}\; f(x,y)=x^2+y^2$

subject to

• $x+y \ge 1$
• $x \ge 0$
• $y \ge 0$

Tasks:

1. Write the Lagrangian and KKT conditions (stationarity, primal feasibility, dual feasibility, complementary slackness).
2. Identify which constraints are active at the optimum.
3. Solve for $(x^*,y^*)$ and the optimal value.