(7) Consider the following piecewise linear convex problem in R²: minimize: ||ax|| + ||b¹x|| subject to: g(x) = ||x − xo|| - R ≤0 92(x) = ||y — yo|| – R ≤ 0. x Here a and b are two linearly independent vectors in R2. Note that the sets La := {x € R² | aᎢ . 0} and L= { x = R² | 6¹ 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. 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|| + ||b¹x|| subject to: g(x) = ||x − xo|| - R ≤0 92(x) = ||y — yo|| – R ≤ 0. x Here a and b are two linearly independent vectors in R2. Note that the sets La := {x € R² | aᎢ . 0} and L= { x = R² | 6¹ 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. Do the solutions you found in part (c) agree with the Extreme Point Theorem? =

Algebra for College Students

10th Edition

ISBN:9781285195780

Author:Jerome E. Kaufmann, Karen L. Schwitters

Publisher:Jerome E. Kaufmann, Karen L. Schwitters

Chapter12: Algebra Of Matrices

Section12.CR: Review Problem Set

Problem 36CR

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