(7) Consider the following piecewise linear convex problem in R²: T minimize: ||ax|| + ||bx|| - - subject to: 91(x) = ||x − x0|| – R≤ 0 92(x) = ||y — Yo|| – R ≤ 0. - - = Here a and b are two linearly independent vectors in R². Note that the sets La {x = R² | a²x = 0} and Lɩ := {x = R² | bª x = 0} are two lines through the origin. You can assume the the square 91(x), 92 ≤ 0 (centred at (xo, yo) with sides 2R) does not intersect either of these lines (so it is contained “between the lines”). (a) Show that the function f(x) = ||ax|| is a convex function on R². (b) Convert this problem into an equivalent LPP without introducing new variables. Hint: it may be convenient to note that the problem does not change if we use -a instead of a or -b instead of b. This means you can change a to -a or b to -b if you want. (c) Graph and solve this problem graphically in R². Hint: consider 3 cases depending on the the direction of the vector a + b. (d) Do the solutions you found in part (c) agree with the Extreme Point Theorem?

(7) Consider the following piecewise linear convex problem in R²: T minimize: ||ax|| + ||bx|| - - subject to: 91(x) = ||x − x0|| – R≤ 0 92(x) = ||y — Yo|| – R ≤ 0. - - = Here a and b are two linearly independent vectors in R². Note that the sets La {x = R² | a²x = 0} and Lɩ := {x = R² | bª x = 0} are two lines through the origin. You can assume the the square 91(x), 92 ≤ 0 (centred at (xo, yo) with sides 2R) does not intersect either of these lines (so it is contained “between the lines”). (a) Show that the function f(x) = ||ax|| is a convex function on R². (b) Convert this problem into an equivalent LPP without introducing new variables. Hint: it may be convenient to note that the problem does not change if we use -a instead of a or -b instead of b. This means you can change a to -a or b to -b if you want. (c) Graph and solve this problem graphically in R². Hint: consider 3 cases depending on the the direction of the vector a + b. (d) Do the solutions you found in part (c) agree with the Extreme Point Theorem?

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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