(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
Related questions
Question
solve parts c and d for this question.
Transcribed Image Text:(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?
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