Problem on Linear programming and Simplex method
The \(\ell_{1}\) norm of a vector \(v \in \mathbb{R}\) is defined by
$$ \|v\|_{1}:=\sum_{i=1}^{n}\left|v_{i}\right| $$
Problems of the form Minimize \(\|v\|_{1}\) subject to \(v \in \mathbb{R}^{n}\) and \(A v=b\) arise very frequently in applied math, particularly in the field of compressed sensing.
Consider the special case of this problem whith \(n=3\),
$$ A=\left(\begin{array}{lll} 1 & 1 & 0 \\ 3 & 0 & 1 \end{array}\right) \quad \text { and } \quad b=\left(\begin{array}{l} 3 \\ 8 \end{array}\right) $$
(a) (3 points) Explain how to transform this into the following equivalent linear program in standard form (no need for a complete proof of equivalence)
Minimize \(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\)
under the constraints
$$ \begin{array}{r} x_{1}-x_{2}+x_{3}-x_{4}=3 \\ 3 x_{1}-3 x_{2}+x_{5}-x_{6}=8 \\ x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} \geq 0 \end{array} $$
(b) (6 points) Solve the linear program of part (a) using the simplex method.
(c) ( 2 points) What \(v \in \mathbb{R}^{3}\) minimizes the original problem?
Problem settingConsider the linear transformation \(\phi(\cdot): \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) on the standard vector space of dimension two over the field of real numbers defined as:$$ \phi\left(\left(\begin{array}{l} x_{0} \\ x_{1} \end{array}\right)\right)=\left(\begin{array}{r} 3 x_{0}-x_{1} \\ -7 x_{0}+2 x_{1} \end{array}\right) $$Problem taskFind \(\mathcal{R}_{G \rightarrow E}(\) id \()\) that is the change of basis matrix from basis \(G\) to the standard basis \(E\) where the standard basis vectors are:$$ \begin{array}{l} \vec{e}_{0}=\left(\begin{array}{l} 1 \\ 0 \end{array}\right) \\ \vec{e}_{1}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right) \end{array} $$given that...
Consider the following linear fractional program (LFP):$$ \begin{array}{ll} \max f\left(x_{1}, x_{2}\right)= & \frac{10 x_{1}+20 x_{2}+10}{3 x_{1}+4 x_{2}+20} \\ \text { s.t. } \quad & x_{1}+3 x_{2} \leq 50 \\ & 3 x_{1}+2 x_{2} \leq 80 \\ & x_{1}, x_{2} \geq 0 \end{array} $$(a) Transform this problem into an equivalent linear program.(b) Use Matlab (or other software) to solve the LP you created in part (a).(c) Use your answer from part (a) to find a solution to the original LFP.(d) Does...
3. (3pts) Consider the \(3 \times 3\) matrices \(B=\left[\begin{array}{ccc}1 & 1 & 2 \\ -1 & 0 & 4 \\ 0 & 0 & 1\end{array}\right]\) and \(A=\left[\begin{array}{lll}\mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3}\end{array}\right]\), where \(\mathbf{a}_{1}\), \(\mathbf{a}_{2}\), and \(\mathrm{a}_{9}\) are the columns of \(A\). Let \(A B=\left[\begin{array}{lll}v_{1} & v_{2} & v_{3}\end{array}\right]\), where \(v_{1}, v_{2}\), and \(v_{3}\) are the columns of the product. Express a as a linear combination of \(\mathbf{v}_{1}, \mathbf{v}_{2}\), and \(\mathbf{v}_{3}\).4. (3pts) Let \(T(x)=A x\) be the linear transformation given by$$...
1. Suppose that \(T\) is the matrix transformation defined by the matrix \(A\) and \(S\) the matrix transformation defined by \(B\) where$$ A=\left[\begin{array}{rrr} 3 & -1 & 0 \\ 1 & 2 & 2 \\ -1 & 3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 1 & -1 & 0 \\ 2 & 1 & 2 \end{array}\right] $$a. If \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\), what are the values of \(m\) and \(n ?\) What values of \(m\) and \(n\) are appropriate for the...
This problem uses least squares to find the curve \(y=a x+b x^{2}\) that best fits these 4 points in the plane:$$ \left(x_{1}, y_{1}\right)=(-2,2), \quad\left(x_{2}, y_{2}\right)=(-1,1), \quad\left(x_{1}, y_{3}\right)=(1,0), \quad\left(x_{4}, y_{4}\right)=(2,2) . $$a. Write down 4 equations \(a x_{i}+b x_{i}^{2}=y_{i}, i=1,2,3,4\), that would be true if the line actually went through a11 four points.b. Now write those four equations in the form \(\mathbf{A}\left[\begin{array}{l}a \\ b\end{array}\right]=\mathbf{y}\)c. Now find \(\left[\begin{array}{l}\hat{a} \\ \hat{b}\end{array}\right]\) that minimizes \(\left\|A\left[\begin{array}{l}a \\ b\end{array}\right]-\mathbf{y}\right\|^{2}\).
True or false: $$ V=\left\{\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \in \mathbb{R}^{3}: x \geq 0\right\} $$is a subspace of R3. True False Question 10 (1 point) True or false: $$ V=\left\{\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \in \mathbb{R}^{3}: x-y=z+1\right\} $$is a subspace of R3. True False
Suppose that the functions \(f: \mathbb{R}^{3} \rightarrow \mathbb{R}, g: \mathbb{R}^{3} \rightarrow \mathbb{R}\), and \(h: \mathbb{R}^{3} \rightarrow \mathbb{R}\) are continuously differentiable and let \(\left(x_{0}, y_{0}, z_{0}\right)\) be a point in \(\mathbb{R}^{3}\) at which$$ f\left(x_{0}, y_{0}, z_{0}\right)=g\left(x_{0}, y_{0}, z_{0}\right)=h\left(x_{0}, y_{0}, z_{0}\right)=0 $$and$$ \left\langle\nabla f\left(x_{0}, y_{0}, z_{0}\right), \nabla g\left(x_{0}, y_{0}, z_{0}\right) \times \nabla h\left(x_{0}, y_{0}, z_{0}\right)\right\rangle \neq 0 $$By considering the set of solutions of this system as consisting of the intersection of a surface with a path, explain why that in a...
Let \(A=\left[\begin{array}{ccc}2 & 0 & -1 \\ 1 & -5 & 1 \\ 2 & -7 & 1\end{array}\right]\)a) Compute \(A^{-1} .\)b) Use \(A^{-1}\) to solve the following system of linear exuations:$$ \begin{array}{r} 2 x_{1}+-x_{3}=3 \\ x_{1}-5 x_{2}+x_{3}=1 \\ 2 x_{1}-7 x_{2}+x_{3}=4 \end{array} $$
Let \(T: R^{3} \rightarrow R^{2}\) defined by \(T\left(\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right]\right)=\left[\begin{array}{c}2 x_{1}+x_{3} \\ -x_{2}\end{array}\right]\).a. Find the matrix \(A\) such that \(T(x)=A x\)b. Demonstrate that \(T\) is a linear transformation.
Solve the linear programming problem by simplex method. . Minimize C= -x - 2y + z. subject to 2x + y +2 < 14 4x + 2y + 3z < 28 2x + 5y + 5z < 30 x = 0, y>02 > 0