Class Test (optimisation), S1 2020
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STUDENT NAME:
ENGG7302 Advanced Computational Techniques in Engineering
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STUDENT NUMBER:
THE UNIVERSITY OF QUEENSLAND
School of Information Technology & Electrical Engineering
Class Test (Optimisation), S1 2020
Advanced Computational Techniques in Engineering
(MEngSc) OPEN BOOK
TIME: NINETY minutes for working
ANSWER ALL QUESTIONS IN THIS FILE
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Class Test (optimisation), S1 2020 ENGG7302 Advanced Computational Techniques in Engineering
Question 1. (70 marks: 1.1~1.6: 8 marks each; 1.7~1.8: 11 marks each)
For each question, please select the correct option (only one option is correct among the four
1.1 Consider the function𝑓(𝑥, 𝑦) = 𝑥! − 𝑦”, the function may have the following properties: [1] the point (𝑥 = 0,𝑦 = 0) is not a critical point; [2] the point (𝑥 = 0,𝑦 = 0) is a stationary point; [3] the point (𝑥 = 0, 𝑦 = 0) is a saddle point; [4] it is not unimodal function in the domain | 𝑥| < 1, |𝑦| < 1. 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 Gradient Decent Method (Steepest Decent Method) may have the following property [1] it requires an initial guess; [2] If it is required to perform line search, golden search method can be used to determine the step size; [3] it does not require to calculate the second order derivatives of the objective function; [4] the solution will always be the same if one uses either a fixed step size or not. Then it is in general correct to say that
(a) [1] ~ [4] are all correct
(b) ONLY [1], [3] and [4] are correct
(c) ONLY [1], [2] and [4] are correct
(d) ONLY [1], [2] and [3] are correct
1.3 Multidimensional Newton’s method (method 1) and its revised version — Conjugate gradient method (method 2) may have the following properties [1] both methods requires an initial guess; [2] both methods can avoid calculating the inverse of the Hessian matrix; [3] both methods requires calculating the gradient of the objective function once in each iteration. [4] both methods generally faster than the gradient descent method. Then it is in general correct to say that
(a) [1] ~ [4] are all correct
(b) ONLY [1],[2] and [4] are correct
(c) ONLY [1],[3] and [4] are correct
(d) ONLY [1],[4] are correct
1.4 Non-classical optimization methods (simulated annealing (SA) and genetic algorithm (GA)) may have the following properties: [1] both SA and GA methods involve random search during optimisation; [2] Given the following two individuals in a GA operation: X1=[1 0 1 0 1]; X2=[0 0 0 0]. If the new children individuals are X1=[1 0 0 0]; X2=[0 0 1 0 1], then ‘cross-over’ might be operated. [3] in SA, if the initial temperature is set up too low, it may lead to a sub-optimal solution. [4] If two users to run the same SA algorithm for the given optimisation problem, their final solution may be different from each other. Then it is in general correct to say that
(a) ONLY [3] is incorrect
(b) ONLY [1], [3], [4] are incorrect
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Class Test (optimisation), S1 2020 ENGG7302 Advanced Computational Techniques in Engineering
(c) [1] ~ [4] are correct
(d) ONLY [1], [4] are correct
1.5 Consider the linear programming problem, min = −8x1 −11x2 − 20x3 , subject to linear x
inequality/equality constraints where
5x1 + 4x2 + 5x3 = 40, − 3x1 + 2x2 − 5x3 = 14, x1 ≥ 0, x2 ≥ 0, x3 ≥ 0 . To solve it using the
simplex method, we need to convert it to standard form. During this operation, how many slack variables should be used?
1.6 Consider using the matlab function linprog(f, A, b, Aeq, beq, LB, UB) to solve the following linear programming problem:
maximise subject to
You may set up the input parameters using some of the following ways.
[1] f=[-10; 6]; [2] A=[2 0; -3 -2]; [4] LB=[-inf;0]; [5] UB=[0;inf]; Then it is in general correct to say that
(a) ONLY [6] must be incorrect;
(b) ONLY [3] and [6] must be incorrect;
(c) ONLY [3], [4] and [6] must be incorrect; (d) ONLY [1],[2],[3],[4],[6] must be incorrect.
[3] A=[-3 -2]; [6] UB=[4; inf];
𝑧 = 10𝑥 − 6𝑦
3𝑥 + 2𝑦 ≥ 18 3𝑦 = 18
1.7 Use Lagrange multipliers to find all the critical points of function ƒ (x; y) = xy on the given surface 3x2 + y2=6. Then it is in general correct to say that
(a) There are one critical points;
(b) There are two critical points;
(c) There are three critical points;
(d) There are four critical points.
1.8 In an experiment, three particular variables 𝑎, 𝑏, 𝑐 were measured three times (𝑛=1,2,3). It’s known that the three variables have the following relationship:
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Class Test (optimisation), S1 2020 ENGG7302 Advanced Computational Techniques in Engineering
𝑦 = −𝑎(𝑛)𝑥 + 𝑐(𝑛) 𝑏(𝑛) 𝑏(𝑛)
where 𝑥,𝑦 are unknown constants. Suppose that the measurement was carried out in a noisy environment. And it’s also known that
[𝑎(1) 𝑎(2) 𝑎(3)][𝑎(1) 𝑎(2) 𝑎(3)]# = 20 [𝑏(1) 𝑏(2) 𝑏(3)][𝑏(1) 𝑏(2) 𝑏(3)]# =20
(d) >1 1? <𝑦= = <30= 11
10= <𝑥= = <30= 20 𝑦 30
20= <𝑥= = <20= 11𝑥 10
20 30 𝑦 30
𝑎(1) 𝑎(2) 𝑎(3) 𝑏(1) 10
8𝑐(1) 𝑐(2) 𝑐(3)9 :𝑏(2); = <30= 𝑏(3)
If one uses the least-squares (LS) method to estimate the unknown constants 𝑥, 𝑦 , the normal equation will be:
(a) <30 10
(b) <20 10
10= <𝑥= = <20= 30 𝑦 20
Question 2. (30 marks)
A UQ shop sells two types of fresh juice (type A & type B), The type A juice sells for $6 each and the type B juice sells for $8 each. The type A juice is made from 3 oranges, 2 apples and one mango. The type B juice is made from one melon, 3 apples and one orange. Each day the shop has 50 oranges, 40 apples, 10 melons and 15 mangos, and the shop sells all of the drinks they can make. Please maximize the revenue of the shop.
2.1 Please set up an appropriate linear programming (LP) model and write down the simplex tableau (do not solve it).
2.2 Solve the above LP problem using the graphical approach.
2.3 Consider using the Matlab function intlinprog( ) to solve the above LP problem. Write down the
objective function vector 𝑓 , index vector for integer variables, matrices A, Aeq , and vectors b,beq,LB,UB (inMatlabformat).
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Class Test (optimisation), S1 2020 ENGG7302 Advanced Computational Techniques in Engineering 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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