# What is the proof for boundedness of the following LP problem?

$$\begin{array}{ll} \text{maximize} & Ax + By \\ \text{subject to}& Cz + Dw = 1 \\ & A’x + B’y \le C’z + D’w \\ &x, y, z, w \ge 0. \end{array}$$

where A, B, C, D, A’, B’, C’, D’ are constant positive 1×n matrix and x, y, z, w are n×1 matrix.

What is the proof for boundedness of the above LP model? Is it bounded or not? If not, with what conditions the model is bounded?

• Welcome to MSE! What have you tried so far? What does it mean if a LP is bounded? Please show your attempt and specify your question. – UnbelieveTable Aug 27 '19 at 6:51
• In maximization problem, it means having an upper bound for the objective function. – Hessam Nedaei Aug 28 '19 at 2:21

The dual is: $$\begin{array}{ll} \text{minimize} & \theta \\ \text{subject to}& \lambda A' \ge A \\ & \lambda B' \ge B \\ & -\lambda C' + \theta C \ge 0 \\ & -\lambda D' + \theta D \ge 0 \\ &\lambda \ge 0, \theta free. \end{array}$$