Consider the NLP min 21x1 - 322 - x1x2 + 1x2 - 522 s.t. 1x122 + 1x222 … 4 0 … x1 … 2, x2 Ú 0 with optimal solution x* = 11.088, 1.6782.

(a) Use unsquared penalty functions to reduce this problem to an unconstrained penalty model.

(b) Explain why local minima of the unconstrained model in part

(a) must be global minima for all m Ú 0.

(c) Determine whether the penalty objective of part

(a) is differentiable. Explain.

(d) Determine whether there will be a penalty multiplier m large enough that the unconstrained optimum in part

(a) is optimal in the original model. Explain.

(e) Suppose that we are solving the given constrained NLP by the sequential unconstrained penalty Algorithm 17A. Explain why it is reasonable to begin with multiplier m = 0.5 and increase it by a factor b = 2 after each unconstrained optimization.

(f) Use class optimization software to apply Algorithm 17A, starting at x = 13, 52 and managing the penalty multiplier as in part (e).