3. Consider the linear programming problemmin c"xs.t. Ax = bx 20where A is a m x n matrix with m = n and A = A", and c = b.Prove or give a counter example to the following claim: "if there exists an such thatAx = b,x 2 0, then i is an optimal solution".

We will use proof by contradiction to show that the given statement is true. For that we will first…

Answered: Consider the linear programming problem…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. Use graphing techniques to solve this linear programming problem. z = 3x₁ + 2x₂ 3x1 + 3x2 < 66 x₁ + 4x₂ ≤ 56 2x₁ + x₂ ≤ 36 X1, X₂ ≥ 0 Maximize: Subject To: 2. Put the following matrix into reduced row echelon form. If this matrix represented the augmented matrix of a system of linear equations in x, y, and z, what would be the solution of the system of linear equations? 1 1 1 1 ³) 3 4 1 4 9. 1 2 3 3. Solve the following linear program by hand using the Simplex Method. Show your work details. Maximize: Subject To: z = 3x₁ + 2x₂ 3x1 + 3xz < 66 x₁ + 4x₂ ≤ 56 2x₁ + x₂ ≤ 36 X₁, X₂ ≥ 0
Solve the following Linear Programming Problem using the two-phase method, showing all the interactions:
Given the following unconstrained nonlinear programming problem, f(x1,X2, X3) = 2x +x,² +x= xX2 - X2X3-I,X3 -4x, +2x2 a. find the optimum point of f b. Determine the Hesse matrix and check whether the optimum point is maximum or minimum.
Problem 2: Consider the following linear program: 2x1 + x2 max x1,x2 ER subject to X1 X1 - 2x1 X1 x2 + X2 + x2 x2 1 - 2 VI VI VI AV 1 остан 5 5 (4) 1. Make a figure that illustrates the feasible set in R². 2. Provide some integers n, m € R and matrices c E R", A € Rmxn, and b € Rm such that it is in standard form maxxER cx s.t. Ax = b and x > 0. 3. On your figure, identify each vertex x € R² and its corresponding basis (possibly more than one) B of the columns of A so that N = 0 where N = {1,...,n} \ B. 4. Solve the linear program using the simplex algorithm in tableau form. Report the iterations on your figure. 5. Use the dual simplex algorithm in order to find a solution if the constraint x₁ < 2 is added.
Complete the following. Minimize g = 8y1 + 7y2 subject to + 4y, 2 24 3y1 Y1 + 4y2 2 16. (a) Form the matrix associated with the given minimization problem. 3 4 24 first constraint 1 4 16 second constraint 8 7 objective function Find the transpose of the above matrix. f (b) Write the dual maximization problem. Be sure to rename the variables x, and x,. Maximize f = subject to <8 X, 2 0, x, 2 0. VI
1. Consider the following Linear Programming (LP) problem: Maximize z = Зх, + 2х2 Subject to 2x, + x2 12 X1, X2 > 0 Using the Two-Phase method, determine the optimal solution (specify the special case(s) involved, if any). 2. Write down the normal dual linear programming (LP) problem to the following non- normal LP problem: Min z = x1 – X2 – 2x3 s.t.-3x1 – x2 + X3 0, x2 < 0, xz urs
3. Convert the following Linear Programming problem into Standard LPP form. Minimize P = 5x1 - 10x2 + x3 Subject to: x1 - x2 + x3 = 120 Where x1, x2, x3 >=0
1. Give the dual of each of the following linear programs: (b) minimize 3x₁ +7x3 + 9x₁ subject to -₁ + 3x₂ - 3 2x1 +52 +8x3 +974 ≥ 1, 10x₁ < -12, x₁ + 4x₂ + 8x32 -4, I1, I₂ ≥ 0, X3 ≤ 0, T4 unrestricted
4. The final simplex matrix for a problem is shown below. Give the solutions to both the maximization problem and the dual minimization problem based on this final matrix. Carefully label each solution. x2 x3 S1 S2 S3 P 1 -0.5 3 -0.7 29 1 -0.8 -2 -0.5 47 -0.6 -1 -2.4 1 53 15 7 13 1 597 O O o-
We are to minimize P = 3a+ 26 subject to the following constraints. a+b 24 3a+b 26 a, b 20 What is the dual problem in matrix form? 11 4 а. 3 1 6 3 20 1 1 4 b. 3 16 1 1 0 133 C. 1 1 2 4 6 0. d. 1 4 3 1 -3 -2 0 е. 13 -3 1 1 -2 4 6 0 1.
Consider the simplex tableau below which has arisen in a maximization problem. Identify the next entering variable and binding row. X2 X3 X4 X5 X6 R.H.S. Basic var. 1 1 -1 3 -2 20 z = 20 1 1 1 1 X6 = 3 X2 = 7 O x3 and row 1; O x4 and row 13; O xs and row 2; O none of the above.
2. Consider the following linear programming problem: min 2-3x12x2 - 513 s.t. x₁ + 2x2 + x3

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Consider the linear programming problem min c"x s.t. Ax = b x2 0 where A is a m x n matrix with m = n and A = AT, and c = b. Prove or give a counter example to the following claim: "if there exists an such that Ax = b,x 2 0, then i is an optimal solution".