Consider the objective function zx1 ellipsis xn cTB B1 b si

Consider the objective function: z(x_1, ellipsis, x_n) = c^T_B B^-1 b+ sigma_j elementof F (c_j - c_B^T B^-1 A_j) x_j and the derivative: Give an L.P examples in R^2 to illustrate what happens for positive derivatives and negative derivatives.

Solution

Consider the objective function A(x,y) = xy given that 2x+2y=100

2x+2y=100 => y = 50-x

And so, A(x) = x(50-x)

Consider derivative :

dA/dx = 50-2x =0

=> x=25

Now for x<25, dA/dx is positive. Hence the function A(x) increases or objective function increases for positive derivative

If x>25, dA/dx is negative . Hence the function A(x) decreases or objective function decreases for negative derivative.