why can we know when constraint set is non-empty,then duality gap must be zero

Question

The primal problem

$max \sum\limits_{i=1}^{M}\sum\limits_{k=1}^{K} \tau log_2(1+\frac{|g_{k,i}|^2P_{k,i}}{\sigma^2_n})$,${P_{k,i} \ge 0,\forall k,i}$

subjected to

1.$\tau P_{k,1} \le w_{k,1}^{*(l-1)}E_0^k,k=1,2,...,K$

2.$\tau P_{k,i} \le w_{k,i}^{*(l-1)}[E_0^k+\sum\limits^{M}_{j=1}E_H^{k,j}],k=1,2,...,K,i=1,2,...,M$

3.$\sum\limits_{j=1}^{i}\tau P_{k,j} \le E_0^k+\sum\limits^{i-1}_{j=1}(1-w_{k,i}^{*(l-1)})E_H^{k,j},k=1,2,...,K,i=1,2,...,M$

And the paper said the primal problem is convex with affine constraint,and the constraint set is non-empty,hence the duality gap is zero and the KKT stationarity conditions are necessary and sufficient for optimality.

I want to ask

1.how do i know the constraint set is non-empty?

2.why can we know when constraint set is non-empty,then duality gap must be zero

3.If duality gap is zero,what does it mean?does it mean there must some optimal solution?

4.I know the duality gap is a information in primal and master problem,so can i say that the primal and master problem is actually fix one of variable,then calculate the other variable,after calculating other variable,then i use other variable to update the variable i fix in the beginning?

i mean,if i want to know the optimal value about A and B, the primal and master problem is actually fix the A first,then use some method to calculate the optimal B,then use this optimal B to calculate the optimal A ?

Answer 1

Here is my answer:

1- I assume that $P_{k,i}$, $E_0^k$, and $E_{H}^{k,j}$ are the decision variables. It is easy to see that if I set all the decision variables to zero, the corresponding candidate solution is feasible in the problem, and thus, the feasible set is non-empty!

2- In the theory of the strong duality, whenever the feasible set is a polytope (all constraints are linear inequalities or equalities), the only condition to guarantee the strong duality holds is that the feasible set is non-empty. Since the feasible set in your problem is non-empty, you have strong duality. See the second condition in Proposition 5.3.1. of "Bertsekas, Dimitri P. Convex optimization theory. Belmont: Athena Scientific, 2009."

3- No. It means that if you right the dual problem, the optimal value remains the same.

4- I don't have a clue what you mean by master problem.