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M 2. (a) Consider the nonlinear programming problem (where c is a positive constam x+2y

M 2. (a) Consider the nonlinear programming problem (where c is a positive constam x+2y

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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PROBLEMS FOR SECTION 3.5 1. Solve the problem max 1 - x² – y² subject to x 2 2 and y > 3 by a direct argument, and then see what the Kuhn–Tucker conditions have to say about the problem. GM 2. (a) Consider the nonlinear programming problem (where c is a positive constant) x+2y <c maximize In(x + 1) + In(y + 1) subject to x + y<2 Write down the necessary Kuhn-Tucker conditions for a point (x, y) to be a solution of the problem. (b) Solve the problem for c = 5/2. (Theorem 3.6.1 will secure that the optimum is attained.) (c) Let V(c) denote the value function. Find the value of V'(5/2). 3. Solve the following problem (assuming it has a solution) minimize 4 In(x² + 2) + y² subject to x²+ y 2 2, (Hint: Reformulate it as a standard Kuhn-Tucker maximization problem.) GM 4. Solve the problem max–-(x – a)² – (y - b)? subject to x< 1, y s 2, for all possible values of the constants a and b. (A good check of the results is to use a geometric interpretation of the problem. See Example 1.) 5. Consider the problem max ƒ(x, y) = xy subject to g(x, y) = (x + y – 2)² < 0. Explain why the solution is (x, y) = (1, 1). Verify that the Kuhn-Tucker conditions are not satisfied for any 1, and that the CQ đoes not hold at (1, 1). G6. (a) Find the only possible solution to the nonlinear programming problem maximize x - y subject to (b) Solve the problem by using iterated optimization: Find first the maximun value f(x) in the problem of maximizing x – y' subject to x < y, where x is fixed and y varies. Then S A single vector a is linearly independent if and only if it is not the zero vector. 20215 OF202 quiz w.R JNJearnings.csv panoway
Transcribed Image Text:PROBLEMS FOR SECTION 3.5 1. Solve the problem max 1 - x² – y² subject to x 2 2 and y > 3 by a direct argument, and then see what the Kuhn–Tucker conditions have to say about the problem. GM 2. (a) Consider the nonlinear programming problem (where c is a positive constant) x+2y <c maximize In(x + 1) + In(y + 1) subject to x + y<2 Write down the necessary Kuhn-Tucker conditions for a point (x, y) to be a solution of the problem. (b) Solve the problem for c = 5/2. (Theorem 3.6.1 will secure that the optimum is attained.) (c) Let V(c) denote the value function. Find the value of V'(5/2). 3. Solve the following problem (assuming it has a solution) minimize 4 In(x² + 2) + y² subject to x²+ y 2 2, (Hint: Reformulate it as a standard Kuhn-Tucker maximization problem.) GM 4. Solve the problem max–-(x – a)² – (y - b)? subject to x< 1, y s 2, for all possible values of the constants a and b. (A good check of the results is to use a geometric interpretation of the problem. See Example 1.) 5. Consider the problem max ƒ(x, y) = xy subject to g(x, y) = (x + y – 2)² < 0. Explain why the solution is (x, y) = (1, 1). Verify that the Kuhn-Tucker conditions are not satisfied for any 1, and that the CQ đoes not hold at (1, 1). G6. (a) Find the only possible solution to the nonlinear programming problem maximize x - y subject to (b) Solve the problem by using iterated optimization: Find first the maximun value f(x) in the problem of maximizing x – y' subject to x < y, where x is fixed and y varies. Then S A single vector a is linearly independent if and only if it is not the zero vector. 20215 OF202 quiz w.R JNJearnings.csv panoway
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