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THE UNIVERSITY OF NEW SOUTH WALESSCHOOL OF JVIATHEMATICS

SESSION 1, JUNE 2006

MATH3161/MATH5165Optimization

(1) TIME ALLOWED - Two hours(2) TOTAL NUMBER OF QUESTIONS - 3(3) ANSWER ALL Q U E S T I O N g ~ : :(4) THE QUESTIONS ARE NOT OF EQUAL VALUE(5) ALL STUDENTS MAY ATTEMPT ALL QUESTIONS. MARKS GAINEDON 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 2006

1. [25 marks]

MATH3161/MATH5165 Page 2

i) Consider the problem of minimizing the functionf(x) = xi + x - 2Xl + 1

[_;] .tarting from x(1) =a) Calculate the gradient \ l f(x) and Hessian \72 f(x) of f.b) Is the function f convex, concave or neither on ]R2? Give reasons foryour answer.c) Calculate the steepest descent direction sl at x(1).d) Show that the Newton direction at x(ll is

e) Is the Newton direction sW a descent direction for f at x(ll? Givereasons for your answer.f) Given that the sequence, generated by the Newton method, is {x(k)},where x(k) = [ 1 ] , find the limit point x* of the sequence {x(k)}.

( ~ ) kg) Show that x* is a global minimizer of f.h) Show that the rate of convergence of the sequence is linear, but not

quadratic. Explain why the rate of convergence is only linear.ii)* Consider the following equality constrained problem

(EP) minimizexElRnsubject to

where b E ]R, 0 =!=- a E ]Rn and n > 1.a) Show that the feasible set n := {x E ]Rn I aTx = b} is a convex set.b) Show that the problem (EP) is a convex programming problem.c) Using first-order necessary optimality conditions, find the global min

imizer and global minimum of (EP).

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SESSION 1, JUNE 2006

2. [25 marks]

MATH3161/MATH5165 Page 3

i) Consider the following optimization problem(P) minimize 3xi + 2 x + 2XIX2 - 20XI - 16x2

subject to 5 - X l - X2 0, X l 2: 0, X2 2: O.a) Sketch th e feasible region of the problem (P).b) Can you guarantee that a global minimizer exists for this problem?

Give reasons for your answer.c) Formulate th e (exterior) penalty function problem for (P) with

parameter J.L.

d) Let x(J.L) = [ 7 i ~ : : 5 2 ] .Verify that x(J.L) is a stationary point of th e8J.!+143/-,+5penalty function problem.e) Find the limit point x* of the vectors x(J.L) as J.L -t 00.f) Verify that the limit point x* is a constrained stationary point for(P).g) Determine whether or not x* is a global minimizer of (P).

ii) A conjugate gradient method with th e Fletcher-Reeves updating formulais being used to minimize a function j on JR2. It is given that

[3X I - X2 - 2] 1[26]g(x) = , X(2) = 17

X2 - X l 3830 [3]nd 8(2) = =--172 '7

where x(2) is th e current iterate, S(2) is the current search direction andg(x) = V j (x) is the gradient of f.a) Find the exact minimizer a(2) of j(x(2) + as(2) and X(3) using exact

arithmetic.b) Is X(3) a minimizer of j7 Give reasons for your answer.

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SESSION 1, JUNE 2006 MATH3161/MATH5165 Page 4

3. [15 marks]Consider the following optimal control problem

minimize fo\ u - 2x) dtuEU )0sub ject to x= x + ux(O) = 1,

with bounded controls U := {u I 0 :::; u(t) :::; 1, Vt E [0, 2]} and a final valuex(2), which is not specified.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 necessary conditions that an optimal solutionsatisfies, including any transversality condition and statement about thevalue of H along the optimal path.iv) From the conditions in (iii) find the optimal control u* as a function of z.v)* Using the differential equation for z in (ii) and the transversality condi-

tion,a) find the optimal costate variable z*(t) as a function of t.b) find z*(O).c) sketch z*(t) over the interval [0, 2].

vi)* Find the optimal control u*(t) as a function of t and the value of anyswitching time. Sketch u*(t) over the interval [0, 2].