A linear program in 2D

Consider the optimization problem

displaystylemax_x : 3x_1 + 1.5 x_2 ~:~ -1 le x_1 le 2, ;; 0 le x_2 le 3 .

The problem can be put in standard form:

displaystylemin_x : -3x_1 - 1.5 x_2 ~:~ x_1 le 2, ;; -x_1 le 1, ;; x_2 le 3, ;; -x_2 le 0.

This is an LP, since the objective and constraint functions are all affine.

Geometric view of the toy linear program above. The level curves (curves of constant value) of the objective function are shown: these are straight lines orthogonal to the objective vector, c = (3,1.5) . The problem amounts to find the largest value of such that t = c^Tx for some feasible . The optimal point is x^ast = (2,3) .