1. Use Taylor series arguments to show that for a problem with all linear constraints the optimality conditions (16.2.3), (16.2.4) and (16.2.8) ensure that if 5:1: is such that if c1($* | 5:3) 2 -- - = Cg(f* + 5:8) 2 0 then F($* | 5:8) :3 F(:I:*). (16.2.10) If the constraints are all quadratic show that conditions (16.2.3), (16.2.4) and (16.2.9) will cause (16.2.10) to hold when ci(:t* + 61:) : :c;(:1:*+5:1:) :0. 2. Show that, if G is positive denite, the problem 1 Minimize EatTGic | hTat subject to 11:31.1: 2 1 has a solution given by :c = ()\\I | G)_1h for some scalar A. How does this result relate to trust region methods (Chapter 9)? 3. Show that, if the constraints (16.2.2) are all divided by a constant factor k, the solution of the modied nonlinear programming problem is unchanged except that the new Lagrange multipliers are given by HG, . . . , 113A? 4. If (16.2.1), (16.2.2) has a solution 3:* where the constraint normals are not linearly independent show that A?\" . . ,Af are not uniquely dened