I was asked (not as a homework problem), to find the dual program of the following feasibility program.
$$
\text{find } P \succ 0
$$
$$
\text{subject to } A_{i}P + PA_{i}^{T} \prec 0 \text{ for all i}
$$
However, I am a little bit confused with the definition of a dual program in this case. Does it mean that when primal is feasible then dual is infeasible and when primal is infeasible then dual is feasible? Or do we only need to require one direction (i.e. the primal is infeasible implies dual is feasible)?
Just to add, I think maybe the dual program is like the one in Generalized Farkas Lemma where one and only one of the programs is satisfied.
Best Answer
Just to answer this question. I confirmed with my professor that the dual program by definition is a program that is infeasible when the primal is feasible and feasible when the primal is infeasible so we would need both directions.
Though not directly what this question asks, such a dual program is basically finding whether there exists a hyperplane that separates two sets. I will not provide the exact detail here as it is a little more complicated and not what this question is asking.