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Class Test (optimisation), S2 2019
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THE UNIVERSITY OF QUEENSLAND
School of Information Technology & Electrical Engineering
Class Test (Optimisation), S2 2019
ENGG7302
Advanced Computational Techniques in Engineering
(MEngSc) CLOSED BOOK
TIME: NINETY minutes for working
FIVE minutes for perusal before examination begins
ANSWER ALL QUESTIONS ON SHEET PROVIDED QUESTIONS CARRY THE NUMBER OF MARKS INDICATED
Drawing instruments and one battery-operated or solar-powered electronic calculator may be used but NO pre-programmed material or calculator instruction booklets are allowed in the examination room.
STUDENT NAME:
ENGG7302 Advanced Computational Techniques in Engineering
STUDENT NUMBER:
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Class Test (optimisation), S2 2019 ENGG7302 Advanced Computational Techniques in Engineering
Question 1. (40 marks)
For each question, please select the correct option (only one option is correct among the
four options)
1.1 Consider the function𝑓(𝑥, 𝑦) = 𝑥! − 𝑦!, the function may have the following properties: [1]thepoint(𝑥=0,𝑦=0)isacriticalpoint;[2]thepoint (𝑥=0,𝑦=0)isastationary point; [3] the point (𝑥 = 0, 𝑦 = 0) is a saddle point; [4] it is not a coercive function. Then it is in general correct to say that
(a) [1]~[4] are all correct
(b) ONLY [1], [3] are correct
(c) ONLY [2], [3] are correct
(d) ONLY [1], [2], [3] are correct
1.2 Multidimensional Newton’s method may have the following property [1] it is necessary to have an initial guess; [2] it requires performing line search; [3] it requires calculating the Hessian matrix; [4] it requires to calculate the first-order derivative of the objective function. [5] it definitely requires calculating inverse of Hessian matrix. Then it is in general correct to say that
(a) [1] ~ [5] are all correct
(b) ONLY [1], [3], [4] are incorrect
(c) ONLY [2], [4] are incorrect
(d) ONLY [2], [5] are incorrect
1.3 Non-classical optimization methods (simulated annealing (SA) and genetic algorithm (GA)) may have the following property: [1] in SA, during the simulation of cooling, the selection of state involves random search; [2] in GA, the reproduction involves random search; [3] Given the following two individuals in a GA operation: X1=[1 0 1 0 1 0 1 0]; X2=[0000 0111].IfthenewchildindividualsareX1=[1010111];X2=[0000010 1 0], then ‘cross-over’ might be operated. [4] in SA, the higher initial temperature, the better the final solution. Then it is in general correct to say that
(a) ONLY [4] is incorrect
(b) ONLY [3], [4] are incorrect
(c) [1] ~ [4] are correct
(d) ONLY [3] is incorrect
1.4 Conjugate gradient method may have the following properties: [1] it calculates gradients of objective function once in each iteration; [2] it involves one-dimensional optimisation in each iteration; [3] it does not calculate 2nd-order derivatives; [4] it involves gradient decent method based operation; [5] it is effective for unconstrained optimization. [6] it involves Newton’s method-based operation. Then it is in general correct to say that
(a) [1] ~ [6] are correct
(b) ONLY [2]~[5] are correct
(c) ONLY [1],[5], and [6] are incorrect
(d) ONLY [2]~[4] are correct
1.5 Consider using the matlab function linprog(f, A, b, Aeq, beq, LB, UB) to solve the following linear programming problem:
maximise 𝑧 = 10𝑥 − 6𝑦 subject to
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Class Test (optimisation), S2 2019 ENGG7302 Advanced Computational Techniques in Engineering
2𝑥 ≤ 8
3𝑥 + 2𝑦 ≤ 18
3𝑦 ≤ 18
𝑥, 𝑦 ≥ 0
You may set up the input parameters using some of the following ways.
[1] f=[-10; 6]; A=[2 0; 3 2; 0 3]; b=[8; 18;18]; Aeq=[]; beq=[]; LB=[0;0]; UB=[inf;inf]; [2] f=[-10; 6]; A=[2 0; 3 2]; b=[8;18]; Aeq=[]; beq=[];LB=[0;0]; UB=[inf;6];
[3] f=[-10; 6]; A=[3 2]; b=[18]; Aeq=[]; beq=[];LB=[0;0]; UB=[4;6];
[4] f=[-10; 6]; A=[2 0; 3 2; 0 3]; b=[8;18;18]; Aeq=[]; beq=[]; LB=zeros(2,1);
Then it is in general correct to say that
(a) [1], [2], [3] and [4] are all correct
(b) [1], [2], [3] and [4] are all incorrect
(c) ONLY [1], [2], [3] are correct.
(d) ONLY [1], [2] are correct.
Question 2. (10 marks)
Use Gauss-Newton method to fit a nonlinear model function 𝑓(𝑡, 𝑥1, 𝑥2) to data
/𝑡(𝑖), 𝑦(𝑖)1, 𝑖 = 1,2, … , 𝐼, where 𝐼 is the max number of sampling points.
If we take the initial guess (𝑥1, 𝑥2) = (𝑥1(0), 𝑥2(0)), then form the following linear system
equation to iteratively find the fitting parameter (𝑥1, 𝑥2)
𝐴𝑥 = 𝑏, [Eq. 2.1]
Where A is the Jacobian matrix, 𝑥 is the step size, 𝑏 is the initial residual function between model result and the data.
2.1 Please use normal equation approach to find the first iterative solution (that is, the step-1 solution).
2.2 If Eq. [2.1] is inadequate to find a solution, revise Eq. [2.1] and find an alternative solution. please describe with equation(s) and briefly explain your approach.
Question 3. (20 marks)
Use Lagrange multipliers to find all the critical points of function ƒ (x; y; z) = x2 – y2 on the given surface x2 + 2y2 + 3z2 = 1.
Determine the maxima and minima of ƒ on the surface (or curve) by evaluating ƒ at the critical values.
Question 4. (20 marks)
Each week a UQ food store owner can spend at most $100,00 on food and soft drinks. A box of food costs the store owner $100 and a box of soft drink costs him $200. Each box of food is sold for a profit of $40 while each box of soft drink is sold for a profit of $70. The store owner estimates that at least 10 boxes of food but no more than 80 are sold each week. She
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Class Test (optimisation), S2 2019 ENGG7302 Advanced Computational Techniques in Engineering
also estimates that the number of boxes of food sold is at most double the drinks. How many boxes of food and soft drinks should be sold in order to maximize the profit?
4.1 Please set up an appropriate linear programming (LP) model and write down the simplex tableau (please do not solve it).
4.2 Solve the above LP problem using the graphical approach.
Question 5. (10 marks)
We have the following optimization problem,
min (max(𝑒”))
where 𝑥 ≥ 1.
Set up the linear programing model to solve it.
Appendix:
linprog is a matlab function for solving the linear programming problem:
minfT*x subjectto: A*x<=b
x LB<=x <=UB
x = linprog( f, A, b, Aeq, beq, LB, UB )
x- the design variables
f- linear objective function vector
A- matrix for linear inequality constraints
b- vector for linear inequality constraints
Aeq- matrix for linear equality constraints, it is an empty matrix for this case beq- vector for linear equality constraints, it is empty for this case
LB- vector of lower bounds UB- vector of upper bounds
intlinprog is a matlab function for solving the mixed-integer linear programming problem:
minfT*x subjectto: A*x<=b
x LB<=x <=UB
x = linprog( f, index, A, b, Aeq, beq, LB, UB ) x- the design variables
f- linear objective function vector
index- index for integer variables
A- matrix for linear inequality constraints
b- vector for linear inequality constraints
Aeq- matrix for linear equality constraints, it is an empty matrix for this case beq- vector for linear equality constraints, it is empty for this case
LB- vector of lower bounds
UB- vector of upper bounds
END OF EXAMINATION
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