(7) Consider the following piecewise linear convex problem in R²: minimize: ||ax|| + ||bx|| subject to: 9₁(x) = ||x − xo|| - 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 | ax=0} and Lb := {x € R² | b¹ 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 instead of b. This means you can change a to -a or b to -b if you want. of a or (c) Graph and solve this problem graphically in R². Hint: consider 3 cases depending on the the direction of the vector a + b. Do the solutions you found in part (c) agree with the Extreme Point Theorem?
(7) Consider the following piecewise linear convex problem in R²: minimize: ||ax|| + ||bx|| subject to: 9₁(x) = ||x − xo|| - 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 | ax=0} and Lb := {x € R² | b¹ 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 instead of b. This means you can change a to -a or b to -b if you want. of a or (c) Graph and solve this problem graphically in R². Hint: consider 3 cases depending on the the direction of the vector a + b. 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
Transcribed Image Text:(7) Consider the following piecewise linear convex problem in R²:
minimize: ||ax|| + ||bx||
subject to: 9₁(x) = ||x − xo|| - 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 |
ax=0} and Lb := {x € R² | b¹ 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
instead of b. This means you can change a to -a or b to -b if you want.
of a or
(c) Graph and solve this problem graphically in R². Hint: consider 3 cases depending on
the the direction of the vector a + b.
Do the solutions you found in part (c) agree with the Extreme Point Theorem?
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