>Consider the following linear fractional program (LFP):
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 your solution in part (c) satisfy the first order necessary conditions (including regularity)?
Homework Answers
Consider the mathematical program
Consider the mathematical program$$ \begin{array}{ll} \max & 3 x_{1}+x_{2}+3 x_{3} \\ \text { s.t. } & 2 x_{1}+x_{2}+x_{3}+x_{4}=2 \\ & x_{1}+2 x_{2}+3 x_{3}+2 x_{5}=5 \\ & 2 x_{1}+2 x_{2}+x_{3}+3 x_{6}=6 \\ & x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6} \geq 0 \end{array} $$Three feasible solutions ((a) through (c)) are listed below.(a) \(x^{(0)}=(0.2,0.1,0.3,1.2,1.85,1.7)\)(b) \(\mathbf{x}^{(0)}=(0.9,0,0,0.2,2.05,1.4)\)(c) \(\mathbf{x}^{(0)}=(0.3,0.1,0.4,0.9,1.65,1.6)\)Please choose one appropriate interior point from the list, and use the Karmarkar's Method at the interior point and determine the optimal solution.
Problem on Linear programming and Simplex method
Problem on Linear programming and Simplex methodThe \(\ell_{1}\) 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...
Solve the following linear fractional program: max ? = ?1+2?2+?3+6/ 3?1+?3+5 s.t. ?1 + ?2 +...
Solve the following linear fractional program: max ? = ?1+2?2+?3+6/ 3?1+?3+5 s.t. ?1 + ?2 + 3?3 ≤ 10 2?1 + 3?2 ≤ 7 ?1,?2,?3 ≥ 0 Let ? = 1/ 3?1+?3+5 and ?1 = ??1,?2 = ??2, ?3 = ??3.
Indicate which of the following is an all-integer linear program and which is a mixed-integer linear...
Indicate which of the following is an all-integer linear program and which is a mixed-integer linear program. Write the LP Relaxation for the problem but do not attempt to solve. Min 9x1 + 10x2 s.t. (1) 8x1 + 10x2 2 8 (2) 8X1 + 12x2 = 12 X1, X2 = 0 and integer Is this linear program an all-integer linear program or a mixed-integer linear program? This is an all-integer linear program. This is a mixed-integer linear program. Write the...
Consider the linear system A.r = b where A = 1
Consider the linear system \(A x=b\) where \(A=\left[\begin{array}{rr}2 & -1 \\ -1 & 2\end{array}\right], b=\left[\begin{array}{l}1 \\ 1\end{array}\right], x=\left[\begin{array}{l}1 \\ 1\end{array}\right]\).We showed in class, using the eigenvlaues and eigenvectors of the iteration matrix \(M_{G S}\), that for \(x^{(0)}=\left[\begin{array}{ll}0 & 0\end{array}\right]^{T}\) the error at the \(k^{t h}\) step of the Gauss-Seidel iteration is given by$$ e^{(k)}=\left(\frac{1}{4}\right)^{k}\left[\begin{array}{l} 2 \\ 1 \end{array}\right] $$for \(k \geq 1\). Following the same procedure, derive an analogous expression for the error in Jacobi's method for the same system.
Solve the following linear program using the graphical solution procedure: (1) What are the value of...
Solve the following linear program using the graphical solution procedure: (1) What are the value of A and B at each extreme point and (2) an optimal solution? Max 5 A + 7 B s.t (1) A <= 6 (2) 2 A + 3 B >= 18 (3) A + B <= 8 (4) A and B are Non-Negative
1. (20 pts.) Consider the following linear program: max 4x4 +xz+5x3 +3x4 s.t. *1 -X2 -X3...
1. (20 pts.) Consider the following linear program: max 4x4 +xz+5x3 +3x4 s.t. *1 -X2 -X3 +3X, 51 5x +xz+3X3 +8X555 -X2 +2x2+3x3 -5x53 It is claimed that the solution x* = (0,14,0,5) is an optimal solution to the problem. Give a proof of the claim. Do not use the simplex method to solve this problem.
Let (IP) be the following integer program: max (0, -1). s.t. 1-1 -107 -10 10 -11</-0.5...
Let (IP) be the following integer program: max (0, -1). s.t. 1-1 -107 -10 10 -11</-0.5 1-10 / 0 z integer Let (LP) be the LP relaxation of (IP). We draw the feasible region of (LP) here. 2. . . . . . . to (LP), and prove that it is optimal by providing a (a) Determine an optimal solution certificate of optimality.
Consider the following linear program: Max 2A + 10B s.t. 3A ≤ 15 B ≤ 6...
Consider the following linear program: Max 2A + 10B s.t. 3A ≤ 15 B ≤ 6 4A + 4B = 28 A, B ≥ 0 a. draw graph that shows the feasible region for the problem b. What are the extreme points of the feasible region c. Draw graph that shows the optimal solution for the problem
Problem 3: Complentary slackness From Deterministic Operations Research, David J. Rader, Jr. Supp...
Problem 3: Complentary slackness From Deterministic Operations Research, David J. Rader, Jr. Suppose that xı -2, r2 0, T3 -4 is an optimal solution to the linear program: max 4x1 + 2x2 + 3x3 s.t. 2x1+3x2 +x3 12 1 4r2 2x3 10 3zi 23 10 2x1 +3x2 +t3 12 21, 32,c3 20 (a) Using only the primal solution, complementary slackness, and the strong duality theorem, find an optimal solution to the dual problem. (b) Write the dual LP of this...
Indicate which of the following is an all-integer linear program and which is a mixed-integer linear program. Write the LP Relaxation for the problem but do not attempt to solve. Min 9x1 + 10x2 s.t. (1) 8x1 + 10x2 2 8 (2) 8X1 + 12x2 = 12 X1, X2 = 0 and integer Is this linear program an all-integer linear program or a mixed-integer linear program? This is an all-integer linear program. This is a mixed-integer linear program. Write the...
1. (20 pts.) Consider the following linear program: max 4x4 +xz+5x3 +3x4 s.t. *1 -X2 -X3 +3X, 51 5x +xz+3X3 +8X555 -X2 +2x2+3x3 -5x53 It is claimed that the solution x* = (0,14,0,5) is an optimal solution to the problem. Give a proof of the claim. Do not use the simplex method to solve this problem.
Let (IP) be the following integer program: max (0, -1). s.t. 1-1 -107 -10 10 -11</-0.5 1-10 / 0 z integer Let (LP) be the LP relaxation of (IP). We draw the feasible region of (LP) here. 2. . . . . . . to (LP), and prove that it is optimal by providing a (a) Determine an optimal solution certificate of optimality.
Problem 3: Complentary slackness From Deterministic Operations Research, David J. Rader, Jr. Suppose that xı -2, r2 0, T3 -4 is an optimal solution to the linear program: max 4x1 + 2x2 + 3x3 s.t. 2x1+3x2 +x3 12 1 4r2 2x3 10 3zi 23 10 2x1 +3x2 +t3 12 21, 32,c3 20 (a) Using only the primal solution, complementary slackness, and the strong duality theorem, find an optimal solution to the dual problem. (b) Write the dual LP of this...
Most questions answered within 3 hours.
-
Consider the following equilibrium system: COCl2(g) CO(g) +
Cl2(g) A 10.00 L evacuated flask is filled...
asked 8 minutes ago -
1) What are the two distinct steps that one needs to perform
when developing a data...
asked 12 minutes ago -
2) Write a C++ program that uses a class called “Degree” to
obtain the trigonometric
values...
asked 1 hour ago -
1. In eukaryotic cells the genomes of
&
asked 1 hour ago -
The standard enthalpy of propanol (C3H7OH) is -303.0 kJ/mol.
Compute both of the
gross and net...
asked 1 hour ago -
Why PWM using H-bridge for control motor speed is more power
effiecient than the linear amplifier...
asked 1 hour ago -
In 1999, Carly Fiorina famously said,"I hope that we are at a
point that everyone is...
asked 1 hour ago -
Individuals in a species of moth vary in wing color from white to
black, but all...
asked 1 hour ago -
The following standards for variable manufacturing overhead have
been established for a company that makes only...
asked 1 hour ago -
The United States government wanted to determine what proportion
of Americans approve of the current president,...
asked 2 hours ago -
2-N stone is dropped from the top of a cliff. The stone hits the
ground 3...
asked 2 hours ago -
Vitamins participate in metabolic reactions in their coenzyme
forms . Match the following reactions to the respective...
asked 2 hours ago