II Semester M.E. (Power and Energy Systems) Degree Examination, January 2015 (Semester Scheme) PS – 211 : LINEAR AND NON LINEAR OPTIMIZATION Time : 3 Hours
Max. Marks : 100
Instruction : Answer any five full questions. 1. a) State five engineering applications of optimization.
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b) Minimize f (x1 x2) = 3 x12 + x 22 + 2x1 x2 + 6x1 + 2x2 subject to 2x1 – x2 = 4 and x1, x2 ≥ 0 using the method of Lagrangian multipliers.
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c) Verify whether the following functions are convex or concave i) f (x ) = 7 x12 + 10 x 22 + 7 x 23 − 4 x1 x 2 + 2 x1 x 3 − 4 x 2 x 3 ii) f (x ) = 6 x 12 + 4 x 2 + x1 x 2 − x12 − x 22 iii) f (x ) = 4 x12 + 3 x 22 + 5 x 23 + 6 x1 x 2 + x1 x 3 − 3 x1 − 2 x 2 + 15
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2. a) Use the Kuhn-Tucker conditions to solve the following N/L PP. Maximize Z = x12 − x 22 − x 23 + 4 x 1 + 6 x 2 Subject to x1 + x2 ≤ 2 2x1 + 3x2 ≤ 12 and x1 x 2 ≤ 0.
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P.T.O.
PED – 123
*PED123*
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b) Four products are processed sequentially on three machines. The following table gives the pertinent data.
Manufacturing time (hr) per unit
m/c
Cost per/hr ($)
Product 1
Product 2
Product 3
Product 4
Capacity (hr)
1
10
2
3
4
2
500
2
5
3
2
1
2
380
3
4
7
3
2
1
450
75
70
55
45
Unit-selling price ($)
Formulate the problem as an LP model and find the optimum solution using graphical method.
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3. a) Minimize f = x – 4y Subject to x – y ≥ – 4 4x + 5y ≤ 45 5x – 2y ≤ 20 5x + 2y ≤ 10 and x, y ≥ 0.
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b) Solve the following LP problem by revised simplex method. Maximize f = x1 + 2x2 + x3 Subject to 2x1 + x 2 – x3 ≤ 2 – 2x1 + x2 – 5x3 ≥ – 6 4x1 + x2 + x3 ≤ 6 and x1 x2 x3 ≥ 0.
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*PED123*
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PED – 123
4. a) Use dual simplex method to Minimize Z = 2x1 + 2x2 + 4x3 Subject to
2x1 + 3x2 + 5x3 ≥ 2 3x1 + x2 + 7x3 ≤ 3 x1 + 4x2 + 6x3 ≤ 5 and x1 x 2 x 3 ≥ 0.
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b) Use Big-M method to Maximize Z = 5x1 – 2x2 + 3x3 Subject to 2x1 + 2x2 – x 3 ≥ 2 3x1 – 4x2 ≤ 3 x1 + 3x3 ≤ 5 and x1 x 2 x3 ≥ 0. 5. a) What is a unimodal function ? What is the difference between Newton and Quasi Newton methods ?
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b) Explain univariate method.
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⎧− 1⎫ c) Minimize f (x) = 100 (x2 – x12 )2 + (1 – x1)2 from the starting point X1 = ⎨ 1⎬ ⎩ ⎭ using Newton’s method. Perform two iterations.
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6. a) Why is Powell’s method called a pattern search method, explain. b) The fuel cost of three thermal plants in $/hr is given by
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F1 = 400 + 8.2 P1 + 0.003 P12 F2 = 480 + 8.1 P 2 + 0.0025 P22 F3 = 600 + 6.8 P3 + 0.003 P32 Where P1, P2 and P3 are in MW. Find the economic schedule for a load of 1000 MW using Newton’s method. Given P10 = 400 MW, P20 = 200 MW and P30 = 400 MW.
7. a) Explain how sensitivity analysis is carried out on optimal solution.
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b) Discuss the method of optimization using Genetic algorithms.
and x1 x2 x3 ⥠0. 10. 5. a) What is a unimodal function ? What is the difference between Newton and. Quasi Newton methods ? 6. b) Explain univariate method.
x1 ⥠0, x2 ⥠0, x3 ⥠0 and x4 ⥠0. 10. 5. a) Explain the following methods with the help of flow-chart. i) Powell's method. ii) Steepest Descent Method. 10.
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Non-degenerate Piecewise Linear Systems: A. Finite Newton Algorithm and Applications in. Machine Learning. Xiao-Tong Yuan [email protected]. Department of Statistics, Rutgers University, NJ 08854, U.S.A., and Department of. Electrical and Comput
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