Consider the following linear programming problem (P) in standard form:(P) min{c"x : Ax = b, x > 0},where A E R"mxn, b E R™, and c e R".Let ī E R" be a vertex of the feasible region of (P).(4.1) Prove the following proposition:I E R" is the unique optimal solution of (P) if and only if c™d > 0 for all feasibledirections d E R" at i such that d + 0.

⇒part: Suppose that x¯∈ℝn, a feasible solution, is unique optimal. That means for any y∈P such that…

Answered: Consider the following linear…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

1. Suppose that x₁ = 0, x2 = 2, x3 = 0 is an optimal solution to the linear programming problem Maximize zi + 3x2 + x3 subject to 21 ≥ 0 x1 + x2 + 2x3 ≤ 3 1 + 2x2 + x3 ≤ 4 2x1 + x₂ + x3 ≤5 " 12 ≥0, 23 20. Using the principle of complementary slackness and the duality theorem, find an optimal solution to the dual problem. What value will the objective function of the dual problem have at this optimal solution?
Use the simplex method to solve the linear proglamniy Maximize z = 9x, +8X, subject to x, + 2x2 s 2, 3x1 +2x, 8, and 2x, +3x2 s 10 with x, 20 and x, 2 0. O A. Maximum is 17 when x, = 1, X, = 1 O B. Maximum is 45 when x, = 5, x2 = 0 O C. Maximum is 16 when x, = 0, X2 = 2 O D. Maximum is 18 when x, = 2, x, = 0
(a) Consider the following linear program: minimise x1 + 2x2 673 HILING subject to 4x1 8x2 + x3 = 10, 73 3x1 9.12 ≤20, 2.r1 + 10.x2 + 2.73 30, I1 ≤ 40, I1, I₂ > 0. T3 unrestricted (i) Convert program (1) to standard inequality form, and give the value of the constraint matrix A, the vector b (which gives the right hand side of the constraints), and the vector c (which gives the coefficients for the objective function). (ii) Give the dual of program (1). (b) Consider a linear program in standard inequality form: maximise cx subject to Ax ≤ b, X>0 Let t > 0) be some constant and suppose we modify the program by multiplying the right-hand side of each constraint by t maximise cTx subject to Ax ≤ tb, x>0 (i) Show that x is a feasible solution of program (2) if and only if tx is a feasible solution of program (3). (ii) Show that if program (2) is unbounded then the program (3) is unbounded. (iii) Show that if tx is an optimal solution of program (3) then x is an optimal solution of program…
Consider the linear programming problem shown in the image: a) Define the auxiliary problem that is determined in phase 1. Do not solve, just define the auxiliary problem.
Consider the following LP model and it is graphical solution to answer questions from a) to g). Maximize z= 4x,+5x2 Subject to: X:5 .(1) 2x:+X;512 ..(2) X,+2x;s12 ..(3) X1, X220.. .(4) 12 11 10 D A *9 10 11 12 n a) What is the optimal solution x1, X2 and z? b) Determine the optimality range of the objective function coefficient c1. c) Determine the feasibility range of the right-hand side for constraint 1, 2 and 3. d) How many basic feasible solutions in this problem? e) Is there any binding constraint(s)? f) Dose any constraint has a slack? which one and by how much? g) Dose any constraint has a surplus? which one and by how much?
Consider the linear programming problem: Maximize/minimize z = -3x - y subject to: -10x + 5y ≤ 22 2x-3y≤6 x, y ≥ 0 The minimum or maximum (only one of these exist) of this problem occurs at a corner point of the feasible region with an x-coordinate equal to: Answer: K
Given the Following nonlinear programming problem, F Cx) = 1 x1 9x + 1 3 2. a Determine the type oF critical point F(X), Local or global maximum (minimum) Get the minimum value Of F(X) in the Interval OSIE2
(33) Show that the feasible set X = {r : g1(x) < 0,..., ge(x) <0} is a convex set, if all the functions g; are convex.
Consider the linear program minimize z=2x1-3x2 subject to 4x1+3x2<=12 x1-2x2<=2 x1, x2>=0 Represent the point x=(1,1)T as a convex comination of extreme points. (i) Draw graph of feasible region; show location of x (ii) Find 2 different convex combinations using 3 extreme points. (iii) Find a 3rd convex combination using all 4 extreme points. [Hint: find 3 different combinations of extreme points that surround the point you are trying to represent; remember that the coefficients must sum to one and be > 0]
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
C. Consider the linear program max Xx1+ Bx2 s.t. X1 – x2 0, x2 > 0 where a E R and BER are real parameters. Determine all values of a and B for which this linear program (a) has a unique optimal solution; (b) has more than one optimal solution; (c) is unbounded.
Consider the simplex tableau that is the final one in a problem to maximize x + 2y + 3z. xyzu v M 210-380 21 3015205 100651 19 What are the values of x, y and z when the maximum value of x + 2y + 3z occurs? A) x = 2, y = 5, z = 0. B) x = 19, y = 2, z = 5. C) x = 2, y = 1, z = 0. D) x = 0, y = 2, z = 5. E) none of these

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS