JNTU
JNTU B.Tech III OPTIMIZATION TECHNIQUES Previous Question Paper (February - 2023 - R16)
Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, Answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
25 Marks
Outline the significance of classical optimization techniques.
Answer
Classical optimization techniques are significant because they provide the fundamental mathematical framework for solving constrained and unconstrained optimization problems using differential calculus. They are essential for identifying optimal solutions by setting the first-order derivatives of a function to zero and verifying optimality through second-order conditions, such as the Hessian matrix. These methods serve as the theoretical foundation for more complex numerical algorithms and are particularly effective for problems involving continuous and differentiable functions.
What is the statement of an optimization problem?
Answer
An optimization problem is defined as the process of finding the best solution from a set of available alternatives, typically by minimizing or maximizing a specific objective function subject to a set of constraints. Mathematically, it is stated as:
Subject to the constraints:
Where x represents the vector of decision variables, f(x) is the objective function to be optimized, g_i(x) are the inequality constraints, and h_j(x) are the equality constraints that define the feasible region of the solution space.
Outline the importance of simplex algorithm.
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Label the standard form of a linear programming problem.
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Classify one dimensional minimization methods.
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List the advantages of Powell’s method.
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What are the characteristics of a constrained problem?
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Explain about constrained nonlinear programming.
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What is the concept of sub optimization?
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List the examples illustrating the calculus method of solution.
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Part B
50 Marks
State and explain the necessary and sufficient conditions for a multivariable optimization problem with equality constraints.
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Using Kuhn-Tucker conditions, find the values of β for which the point x1*=1, x2*=2 will be optimal to the problem: max Z = 2x1 + βx2 subject to g1(x1,x2) = x12 + x22 – 5 ≤ 0, g2(x1,x2) = x1 - x2 – 2 ≤ 0.
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Solve the following linear programming problem using simplex method. Maximize Z = x1 - x2 + 3x3 Subject to x1 + x2 + x3 ≤ 10, 2x1 - x3 ≤ 2, 2x1 - 2x2 + 3x3 ≤ 0 and x1, x2, x3 ≥ 0.
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Explain the North – West corner method.
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Solve the following minimization transportation problem.
| P | Q | R | S | Available | |
| A | 4 | 3 | 0 | 5 | 24 |
| B | 1 | 2 | 6 | 1 | 17 |
| C | 3 | 6 | 2 | 3 | 19 |
| Required | 15 | 19 | 18 | 8 |
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Find the minimum of f = λ5 - 5λ3 - 20λ + 5 by the Quadratic interpolation method.
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Solve the following equations using the Steepest descent method (Two iterations only) with the starting point X0 = [0, 0, 1]T: 2x1+ x2= 4, x1+2 x2+x3= 8, x2+3x3=11.
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Consider the problem: Minimize f(x1, x2) = (x1 - 1)2 + (x2 - 2)2 Subject to 2x1 - x2 = 0 And x1 ≤ 10. Construct φk function according to the interior penalty function approach and complete the minimization of φk.
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Explain the basic approach of exterior penalty function method with an example.
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Solve the following problem by dynamic programming approach: Min Z = 3x1 + 5x2, Subject to -3x1+4x2 ≤ 12, -2x1+ x2 ≤ 2, 2x1+ 3x2 ≥ 12, 0 ≤ x1≤ 4, x2 ≥ 2.
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Solve the following L.P.P. by dynamic programming approach: Max Z = 8x1 + 7x2, Subject to 2x1+ x2 ≤ 8, 5x1+ 2x2 ≤ 15, x1, x2 ≥ 0.
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