Math3161 Final 12

THE UNIVERSITY OF NEW SOUTH WALES

SCHOOL OF MATHEMATICS AND STATISTICS

SESSION 1, 2012

MATH3161/MATH5165

Optimization

(1) TIME ALLOWED 2 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 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 CALCULATORS WITH AN AFFIXED UNSW APPROVED STICKERMAY 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.

SESSION 1, 2012 MATH3161/MATH5165 Page 2

1. [25 marks]

i) Consider the following equality constrained optimization problem

(P1) minxR3

x21 x2x3 + 1s.t. 4x21 + x

22 = 16,

2x2 + 3x3 = 25.

Let x = [0, 4, 17/3]T .

a) Write the problem (P1) in standard form.

b) Show that the point x is a regular feasible point for (P1).c) Verify that the point x is a constrained stationary point for (P1).d) Evaluate the Hessian of the Lagrangian at the point x.e)* Using second-order sufficient optimality conditions, determine whether

or not x is a strict local minimizer for (P1).

ii) Consider the following inequality constrained optimization problem

(P2) minxRn

aTx

s.t. 1 xTQx 0,where n 2, Q is a symmetric and positive definite n n matrix,a Rn, a 6= 0 and the constraint function c(x) = 1 xTQx.a) Write down the gradient c(x) and the Hessian 2c(x).b) Stating clearly any results that you use, show that c(x) is a concave

function.

c) Show that (P2) is a convex optimization problem.

d) Show that x =Q1aaTQ1a

is a constrained stationary point for (P2).

e) Determine whether x =Q1aaTQ1a

is a local minimizer, global min-

imizer or neither for the problem (P2).

f) Write down the Wolfe dual maximization problem for (P2).

g) What is the maximum value of the Wolfe dual maximization problemin part f)? Give reasons for your answer. [NOTE: You do not needto solve the dual problem.]

h)* Suppose the constraint in (P2) becomes an equality constraint, sothe new problem (P 2 ) has the single equality constraint c(x) = 0.Establish rigorously whether the new feasible region

:= {x Rn : xTQx 1 = 0}is convex or not.

Please see over . . .


2. [25 marks] Consider minimizing the strictly convex quadratic function

q(x) =1

2xTGx + dTx + c

where G is an (n n) symmetric positive definite constant matrix, d is aconstant n 1 vector and c is a scalar. Let x(1) be the starting point, whereq(x(1)) 6= 0, x(1) 6= x and x is the minimizer of q(x).

i) Consider applying Newtons method to the quadratic function q(x).

a) Write down the Newton direction s(1)N at x

(1).

b) Show that the Newton direction s(1)N is a descent direction at x

(1).

c) Show that Newtons method terminates in one iteration.

ii) Consider applying the steepest descent method with exact line searches tothe quadratic function q(x). Suppose that x(k) and x(k+1) are two consecu-tive points generated by the steepest descent method whereq(x(k)) 6= 0.a) Write down the steepest descent direction s

(k)D at x

(k).

b) Confirm that s(k)D is a descent direction.

c) Write down the line search condition that must be satisfied by the

exact minimizer (k) of `() = q(x(k) + s(k)D ).

d)* Hence or otherwise show that q(x(k+1))Tq(x(k)) = 0.

iii) It is given that the gradient of q(x) is q(x) =[

3x1 x2 2x1 + x2

].

a) Write down the Hessian matrix G of q(x).

b) Find the condition number of G.

c)* Given that x(1) x 1, estimate the least number of iterationsof the steepest descent method with exact line searches required toget x(k+1) x 1010.

d) Consider applying a conjugate gradient method with the Fletcher-Reeves updating formula to the quadratic function q(x). It is giventhat

x(2) =1

17

[26

38

]and s(2) =

30172

[3

7

],

where x(2) is the current iterate and s(2) is the current search direc-tion.

) Show that (2) = 1710

is the exact minimizer of q(x(2) + s(2)).

) Find x(3) using exact arithmetic.

) Is x(3) a minimizer of q(x)? Give reasons for your answer.

Please see over . . .


3. [15 marks]

Consider the optimal control problem

minimize

t10

(x1 + u21) dt

subject to x1 = x1 + u1,x1(0) = 4, x1(t1) = 0,

where u1 Uu, the unrestricted control set and t1 > 0 is free

i) Write down the Hamiltonian function H for this problem. [You mayassume the problem is normal and set the co-state variable z0 = 1.]

ii) Write down the differential equation for the co-state z1 and find z1 as afunction of t.

iii) State clearly the Pontryagin Maximum Principle conditions that an op-timal solution satisfies, including a statement about the value of H alongthe optimal path.

iv)* Assuming that a solution exists, apply the Pontryagin Maximum Prin-ciple conditions to find the optimal state x1(t), the optimal control u

1(t)

and the optimal time t1.

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Math3161 Final 12