Inequality Constraints

1. Inequality Constraints Lecture 7

2. Inequality Contraints (I) • A Review of Lagrange Multipliers • As we discussed last time, the first order necessary conditions for a constrained optimum include that the projected gradient has to equal to zero

3. Inequality Contraints (II) • An alternative condition based on lagrange multipliers is

4. Inequality Contraints (III) • Linking the lagrange and the null space

5. Inequality Contraints (IV) • A numerical example: • In the constrained optimization problem which we discussed in the last lecture, we derived an optimal solution of

6. Inequality Contraints (V) • The constraint matrix for the problem was

7. Inequality Contraints (V) • The Lagrangian multiplier condition for this problem then becomes

8. Inequality Contraints (VI) • Solving for l1 and l2 then involves solving for the reduced form of the linear system:

9. Inequality Contraints (VII) • Which yields a solution of

10. Inequality Contraints (VIII) • In a maximization problem: • If the constraint cuts the frontier below the global maximum the lagrange multiplier will take on a positive value implying that an increase in the right hand side of the constraint will increase the objective function value. • Similarly, if the constraint cuts the frontier above the global maximum the lagrange multiplier will take on a negative value implying that an increase in the right hand side of the constraint will cause the objective function to decline.

11. Inequality Contraints (IX) • Thus, in the portfolio problem, income is being constrained below its optimum. • In general, the optimality conditions for a inequality contrained optimum are little different than the conditions for an equality constrained optimum. The primary difference involves restricting the sign of the Lagrange multipliers.

12. Inequality Contraints (IX) • Conditions for an inequality constrained maximum or is negative semidefinite

13. Inequality Contraints (IX) • Given the same conditions, we see that the solution to the previous problem is not optimum since l2 = -15.675 (the conditions for a greater than constraint under minimization are the same as the conditions for less than constraints under maximization except that under minimization the projected Hessian must be positive definite.

14. Inequality Contraints (X) • Taking a new problem from the Constrained2.ma notebook, assume that we want to maximize utility defined as

15. Inequality Contraints (XI) • We see that the unconstrained problem has a global maximum in the positive quadrant. The exact maximum The income required to reach this optimum vector is 33.06

16. Inequality Contraints (XII) • Thus, we want to evaluate two scenarios. Under the first scenario the level of income is set at 40.0 which is beyond the level required for the global maximum. • Under this scenario we solve

17. Inequality Contraints (XIII) which yields an optimum of and an optimal l of -.14383 which does not satisfy the condition for an optimum that l > 0.

18. Inequality Contraints (XIV) • The second scenario then involves constraining y to be less than 30. • Under this scenario, the optimal solution becomes With a l of 0.065537. Thus, the Lagrange multiplier condition is met