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THE UNIVERSITY OF NEW SOUTH WALESSCHOOL OF MATHEMATICS AND STATISTICS

SESSION 1, JUNE 2007

~ A l T I I 3 1 6 1 / ~ A l T I I 5 1 6 5Optimization

(1) TIME ALLOWED - Two hours(2) TOTAL NUMBER OF QUESTIONS - 3(3) ANSWER ALL QUESTIONS(4) THE QUESTIONS ARE NOT OF EQUAL VALUE(5) ALL STUDENTS MAY ATTEMPT ALL QUESTIONS. MARKS GAINED

ON ANY QUESTION WILL BE COUNTED. GRADES OF PASS ANDCREDIT CAN BE GAINED BY SATISFACTORY PERFORMANCE ONUNSTARRED QUESTIONS.GRADES OF DISTINCTION AND HIGH DISTINCTION WILL REQUIRESATISFACTORY PERFORMANCE ON ALL QUESTIONS, INCLUDINGSTARRED QUESTIONS

(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE(7) ONLY THE PROVIDED ELECTRONIC CALCULATORS MAY BE USED

All answers must be written in ink. Except where they are expressly required pencilsmay only be used for drawing, sketching or graphical work.

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SESSION 1, JUNE 2007 MATH3161/MATH5165 Page 2

1. [23 marks]

Minimizesubject to

(NLP)

i) Consider the following nonlinear programming problem2 2 9Xl + x2 - 2" XI - 4X2

X2 - xi 2 0,3XI + 2X2 - 9 = O.

a) Sketch the feasible region of (N LP).b) Is the problem (N LP) a convex programming problem? Give reasons

for your answer.c) Show that the point x* = [ ~ is a regular point for (N LP).d) Verify that the point x* = [ ~ J T i s a constrained stationary point

for (NLP).e) Determine whether or no t the point x* = [ ~ is global minimizer

of (N LP).

ii) Consider the following optimization problem(P) Minimize xi + x + x

subject to Xl + 2X2 + X3 = 1.a) Can you guarantee that a global minimizer exists for this problem?

Give reasons for your answer.b) Formulate the (quadratic) penalty function problem for (P) with

parameter /-L > O.c) Let x(/-L) = /-L [1 2 If. Verify that, for each /-L > 0, x(/-L) is a6/-L + 1global minimizer of the penalty function problem of part b).d) Find the limit point x* of the vectors x(/-L) as /-L -7 00.e) Using second-order sufficient optimality conditions, show that the

limit point x* of part cl) is a strict local minimizer of (P).

Please see over . . .

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SESSION 1, JUNE 2007 MATH3161/MATH5165 Page 3

2. [22 marks] Consider the quadratic function

on lR.n , where d is a constant n x 1 vector, G is a constant n x n symmetricmatrix and c is a scalar.

i) a) Write down the gradient \7q(x) and the Hessian \7 2q(x) of q.b) Suppose that q(x) is a convex function. State what can then be

deduced about G.c) Stating clearly any theorems that you use, show that x* is a mini

mizer for q(x) if and only if G is positive semi-definite and Gx* +d = O.

d)* If q(x) is a st rict ly convex function then show that G is positivedefinite.

e)* Show that x* is the unique minimizer for q(x) if and only if G ispositive definite and Gx* + d = O.

ii) Consider applying Newton's method to q(x) starting at x(1) when G ispositive definite. Let x* be the minimizer of q(x) and x(1) # x*.a) Write down the Newton direction at x(1) and show that it is a descent

direction.b) How many iterations will Newton's method take to reach the mini

mizer x* of q(x). Give reasons for your answer.iii) Consider applying the method of steepest descent with exact line

searches to q(x) starting at x(1) when G is positive definite. Let x*be the minimizer of q(x) and x(1) # x*.a) Show that the initial search direction is S(l) = G(x* - x(l)).b) I f the next iterate X(2) = x* then show that the step length 0'(1) is an

eigenvalue of G- 1 .c) Let

Find the minimizer x* of q(x) and estimate the least number ofiterations that would be sufficient to reach a point within a distanceof 0.5 x 10-6 of the minimizer x*.

Please see over . . .

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SESSION 1, JUNE 2007 MATH3161/MATH5165 Page 4

3. [20 marks] Consider the following optimal control problemMinimize f

0

6 x 2 dtuEU In

subject to x= ux(O) = 1, x(6) = 4with bounded controls U := {u I 0 ::; u(t) ::; 1, Vt E [0, 6]}.

i) Write down the Hamiltonian function H for this problem. [You mayassume the problem is normal and set Zo = -1.]

ii) Write down the differential equation for the costate variable z.iii) State clearly the Pontryagin Maximum Principle conditions that an optimal solution satisfies.

iv) From the conditions in part iii) find the optimal control u* as a functionof z.

v) Using a phase-space diagram find the optimal control u*(t) as a functionoft.

vi) Find the formula for the optimal trajectory x* (t).