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

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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.
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
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]

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