# non linear programming homework help

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• Q1:An electronics repair shop has 3 technicians, and can handle 4 types of repairs (jobs) only. Not all the technicians have the skills to perform all repairs, and their wages are different. However, repairs can be split and finished independently by those capable of performing them. Data are given in the tables below. 1. The objective is to minimize the cost of all incoming repairs (60 items). Formulate the problem as an optimization problem, identifying all variables and constraints properly and clearly. Explain as necessary . 2. Write the set of AMPL statements which model this problem (by hand or otherwise)See Answer
• Q2:The following data are obtained during an experiment involving 2 independent variables, y being the measured response. It is assumed that the intercept 𝛽0 = 0 and the relationship is linear with 2 parameters y = 𝛽1x1 + 𝛽2x2. 1. What is the associated least squares optimization problem? 2. Determine the model parameters. 3. What is the predicted value for x₁ = 3, x₂ = 1? 4. How would the problem be modified to include an interaction between the 2 predictors?See Answer
• Q3:Q1: Consider the function f: R2 → R given by f(x1, x₂) (a) Find all stationary points of f. mum/maximum or a saddle point. x²₁x₁x₂ + x² + x₁-x2. Classify each point, i.e., determine if it is a local mini- (b) Starting at x = (0,0), find the local minimum of f using Newton's method (algorithm below). Stop when ||Vf(x, ak)|| <0.01. For the optimal step size use Matlab or similar to find the minimum of g(a) = f(x*+adk). You may also use Matlab or similar to work with the gradient and Hessian of f. [15 marks] Newton's Method 1: Start from an initial point º, set k=0. 2: If (Vf(x) < ) then exit. If V2f(a) is positive definite then dk-[V2 f(x)]-¹Vf(x) else dk = -f(x). 3: Calculate ak = minazo f(x + adk). 4: Set æk+1=k+ad and k=k+1 then return to step 2.See Answer
• Q4:Q2: Consider the minimisation of the following function of two variables: f(x1, x2)=-In(1+x₁) = x₂. Subject to the linear constraints: 2x1 + x₂ ≤3; 21, x₂ ≥ 0. (a) Prove that this is a convex minimisation problem. (b) Write down the Karush-Kuhn-Tucker conditions for this problem. (c) Find all solutions of the above KKT conditions. (d) Are the solutions you found a local or a global minimum (maximum)? Justify your answer. [15 marks]See Answer
• Q5:Q3: Choose one of the following two problems: 1 (a) Show that the following equality is true: d² dt2 m A¹h(x(t)) = x(t)¹A₁V²h₂(x(t)) [EXX³h (@(0))] # i=1 (t) + A¹Vh(x(t))ä(t) [15 marks] Where A Rmx1, h: Rnx1 → Rmx1, x R → Rx1 and t E R. The bold notation Vh denotes the Jacobian matrix of h. This calculation is used in week 10 lectures notes (along with some similar versions). (b) Independently investigate non-linear least squares. Provide a concise derivation and con- clude by showing that each iteration of the Gauss-Newton algorithm is equivalent to solving a linear system. [20 marks]See Answer
• Q6:1. The Simple Linear Regression model can be written as Y₁ = B₂+ B₁2, +4 i=1,2,..., n where 3, and 3, are unknown parameters and e,' s are independent and identically dis- tributed random variables with E(e) = 0 and Var(e) = 0². Also, B, and 3₁ denotes the least-squares estimates for the parameters 3, and 3₁. (a) Derive the formulae for 8, and 3, directly from the sum of squared error (SSE) formula. [3 Marks] (b) Derive the formulae for the variances of the estimators , and 3₁. Express the results in terms of o2. [4 Marks] 2. Data on horse racing were collected. The following table gives the attendance (in thousands of people) at a racetrack and the amount of money (in millions) that was bet on n = 20 randomly selected days. The data is also available in the file Gambling.xlsx. (See Test4_supplementary.txt for instructions to import the Excel data into R.) Attendance Amount Attendance Amount (thousands (millions) (thousands) (millions) 36.5 149.8 50.6 200.2 234.0 174.3 216.1 250.1 201.8 229.4 257.7 311.3 301.7 308.9 339.4 317.4 51.4 50.9 48.8 58.2 61.5 68.5 66.7 71.2 77.0 307.8 349.2 404.5 Page 2 43.9 57.9 54.5 63.2 64.4 67.9 70.9 83.8 79.7 340.4 455.6 410.3See Answer
• Q7:Use R for computations in this question. The linear model function 1m() is ***NOT*** allowed. • Question 2 ***MUST*** be answered properly in a .pdf file. (That is, the PDF file should include answers to Question 1 and Question 2) • In addition, all computations in Question 2 ***MUST*** be properly recorded in the file Test4_supplementary. txt and ***MUST*** be reproducible by simply copying a block of annotated R codes and running them in the R Console. (a) Fit a simple linear regression model E(Y) = 3, + Biz to the data (amount of money as a function of attendance). (b) What is the expected attendance if the amount bet is 250 millions? (c) Conduct a test of hypothesis [2 marks] * END ** [1 Marks] H.: 3₁0 versus Ha: 31 0. Do the data present sufficient evidence to indicate that the slope 3₁ differs from 0 at a significance level a = 0.05? Describe the process leading to the findings. [2 marks] (d) If appropriate (justify your answer), calculate a 95% confidence interval for the slope 3₁ of the regression line. Provide a proper conclusion. [3 Marks]See Answer
• Q8:12:25 PM Fri Jun 30 SIE340 SUM23 Lecture14 Bieten item 1 2 3 4 5 volume, cubic feet 0.2 0.5 0.2 0.2 0.3 A hiker's knapsack will hold no more than 0.8 cubic feet of canned goods. Some food items that are being considered for inclusion in the knapsack are shown in the table. The hiker wants to maximize the protein content of the items that are selected for inclusion in the knapsack. Quiz grams of protein 12 40 15 20 10 [... O (i) If the hiker choose item 1, he must choose item 2. (ii) If the hiker choose item 3, he cannot choose item 4. (a) Write the decision variables and give the meaning of each decision variable. (b) Write the whole LP model. (objective function, constraints, sign restrictions) (c) Add a constraint to satisfy each of the following cases: 76% Done 25/25See Answer
• Q9:1.1 Identify degrees of freedom for static and dynamic analysis of this frame. Figure P1.1 A gable frame for Problem 1.1See Answer
• Q10:1.2 Identify degree of freedom and derive the relationship between structural resisting force and displacement of this structure. ΕΙ m L Figure P1.2 A simple beam for Problems 1.2 to 1.4See Answer
• Q11:1.3 Formulate equation of motion of the structure in Figure P1.2 when the mass is subjected to externally applied force p(t).See Answer
• Q12:1.4 Formulate the equation of motion of structure in Figure P1.2 when the structure is subjected to vertical ground excitation (1).See Answer
• Q13:The following data are obtained during an experiment involving 2 independent variables, y being the measured response. It is assumed that the intercept 𝛽0 = 0 and the relationship is linear with 2 parameters y = 𝛽1x1 + 𝛽2x2. 1. What is the associated least squares optimization problem? 2. Determine the model parameters. 3. What is the predicted value for x₁ = 3, x₂ = 1? 4. How would the problem be modified to include an interaction between the 2 predictors?See Answer
• Q14:An electronics repair shop has 3 technicians, and can handle 4 types of repairs (jobs) only. Not all the technicians have the skills to perform all repairs, and their wages are different. However, repairs can be split and finished independently by those capable of performing them. Data are given in the tables below. 1. The objective is to minimize the cost of all incoming repairs (60 items). Formulate the problem as an optimization problem, identifying all variables and constraints properly and clearly. Explain as necessary . 2. Write the set of AMPL statements which model this problem (by hand or otherwise)See Answer
• Q15:Let x, y and z be the sides of a rectangular body. 1. Formulate the problem of finding the tetrahedral body of maximum volume with diagonal of unit length. 2. Write the KKT conditions for this problem. 3. Is the cube an optimal solution? Justify your answer.See Answer
• Q17: 1. Consider the model defined by x^{\prime}(t)=x(2-0.4 x-0.3 y) y^{\prime}(t)=y(1-0.1 y-0.3 x) nd and classify the equilibrium points of this system. You can use alculator/computer to help with the computations! (b) Find the x and y nullclines for this system. You can use a calcula-tor/computer to help with the computations! (c) Now, use "pplane.jar" and print the phase plane with the nullclines. Inorder to see the nullclines, you might have to change the values of x-max,y-max, x-min, y-min in the “PPlane Equation Window”, based on your answers to the previous part. To show the nullclines, go up to "Solution"then "Show Nullclines". Label then equilibrium points and print the phase plane with nullclines. (d) Using your phase plane, determine what happens to these two species in the long run under the initial conditions x(0)1.5 and y(0) = 3.5.= (e) Using your phase plane, determine what happens to these two species in the long run under the initial conditions x(0) = 1 and y(0) = 1.See Answer
• Q18: 2. Consider the model defined by x^{\prime}(t)=x(1-0.1 x-0.05 y) y^{\prime}(t)=y(1.7-0.1 y-0.15 x) (a) Find and classify the equilibrium points of this system. You can use a calculator/computer to help with the computations! (b) Find the x and y nullclines for this system. You can use a calculator/computer to help with the computations! (c) Now, use "pplane.jar” and print the phase plane with the nullclines. In-order to see the nullclines, you might have to change the values of x-max,y-max, x-min, y-min in the “PPlane Equation Window", based on your answers to the previous part. To show the nullclines, go up to "Solution"then “Show Nullclines". Label then equilibrium points and print the phase plane with nullclines. (d) Using your phase plane, determine what happens to these two species in-the long run under the initial conditions x(0) = 1 and y(0) = 1. (e) Using your phase plane, determine what happens to these two species in the long run under the initial conditions x(0) = 4 and y(0) = 10.See Answer

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