Under what conditions does ax ≥ a0 imply cx ≥ c0 if x ≥ 0? How do the conditions change if ≤ x ≤ u? Consider the linear programming problem min 4x1 + 4x2 + 3x3 x1 + 2x2 + x3 ≥ 2 2x1 + x2 + x3 ≥ 3 x1,...

Under what conditions does ax ≥ a0 imply cx ≥ c0 if x ≥ 0? How do

the conditions change if ≤ x ≤ u?

Consider the linear programming problem

min 4x1 + 4x2 + 3x3

x1 + 2x2 + x3 ≥ 2

2x1 + x2 + x3 ≥ 3

x1, x2, x3 ≥ 0

Solve the classical dual by hand and use the solution to obtain the surrogate

that provides the tightest bound on the optimal value of the primal. What is

the optimal value? (There is no need to solve the primal directly.) Now use

complementary slackness to find an optimal solution of the primal by solving

two simultaneous equations.