1.Consider problemmin xí + x2s.t. ri + x < 5X1 + 2x2 = 4X1, X2 > 0.Verify that the KKT necessary and sufficient conditions are satisfied atx* = (). Illustrate the problem and its optimal solution graphically.

We are given a non-linear optimization problem. min x12+x22subject…

Answered: 1. Consider problem min xí + x5 s.t. ai…

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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4. Consider the following optimization problem Min f(x) = x,-2*2+523-ни s.t. 2x₁-3x₂=-5 -x3+2x473 -22₂ + 3x3 + x47-6 43 3x₁-4₂+7x4-1 トレ x₁^3 Z. (a) Convert the problem to conventional form. (b) Convert the problem to standard form.
11. Carry out a single-variable search to minimize the function 12 f(x) = 3x² + 5 on the intervaleres X using (a) golden section. (b) interval halving. Each search method is to use four functional evaluations only. Compare the final search intervals obtained by the above methods.
5. Company A plans to market a new product. The marketing department has determined that the demand for these products is p = - 0.05x + 620,0sx< 30,000, where p denotes the product's unit price (in dollars) and x denotes the quantity demanded. Also, the company estimates that the total cost (in dollars) incurred in manufacturing x products per year of production will be C(x) = 100x +710,000. Find the marginal revenue function and the marginal profit function.
5) By false position method the equation f (x) = e* – 3x² has a root between 0.5 and 1.0. O F
What is S'?
5. Sketch the quadratic f(x) = (x-4)² + 1 by moving the 5 main points from the parent function, y = x² in the appropriate manner. 10-
3. Consider the function f(x) = r - : 1+x 1-x a. Explain why, when evaluated in a floating point system, this expression is subject to a large increase of its relative error (catastrophic cancelation) when x is a very small number. b. Find a formula that is equivalent to the one above in exact mathematics and that does not suffer of catastrophic cancelation.
1) Perform 3 iterations to Maximize f(x) = -1.5x6 - 2x¹ + 12x a) Using Golden-Section Search (x₁ = 0, xu = 2) b) Using Newton's Method (xo = 2) L
2. Determine all critical points for the following. Identify the maxima/minima. 2/3 a) y = 6x - 4x + 1 b) y = (x + 1)* – 0.5x

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1. Consider problem min xí + x5 s.t. ai + x3 < 5 x1 + 2x2 = 4 X1, X2 > 0. Verify that the KKT necessary and sufficient conditions are satisfied at æ* = (). Illustrate the problem and its optimal solution graphically.