2. Consider the problem of finding the minimum of f(x1, x2) = x² + x2, subject to the con- straints 1 1 and 2x1 + x2 >4. (a) (3 points) Does a minimum exist? Discuss, including a relevant diagram in your dis- cussion. (You may use technology to draw this diagram) (b) (2 points) Write the problem in standard form. (c) (2 points) Show that it is a convex programming problem. Optimisation III 2025: Assignment 5 3 (d) (4 points) Write down the Karush-Kuhn-Tucker conditions for this problem as satis- fied by the minimiser x* = (x1, x2). (e) (4 points) By considering all cases I(x*) = o, {1}, {2}, {1,2}, confirm that the opti- miser for our problem is x* = (}, }).

2. Consider the problem of finding the minimum of f(x1, x2) = x² + x2, subject to the con- straints 1 1 and 2x1 + x2 >4. (a) (3 points) Does a minimum exist? Discuss, including a relevant diagram in your dis- cussion. (You may use technology to draw this diagram) (b) (2 points) Write the problem in standard form. (c) (2 points) Show that it is a convex programming problem. Optimisation III 2025: Assignment 5 3 (d) (4 points) Write down the Karush-Kuhn-Tucker conditions for this problem as satis- fied by the minimiser x* = (x1, x2). (e) (4 points) By considering all cases I(x) = o, {1}, {2}, {1,2}, confirm that the opti- miser for our problem is x = (}, }).

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter4: Polynomial And Rational Functions

Section4.1: Quadratic Functions

Problem 6SC: A company that makes and sells baseball caps has found that the total monthly cost C in dollars of...

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