A variable in primal relates to a constraint in dual, in the most basic sense (so basically it's a hyperplane).  Not sure if it helps Mahdi?  Can explain more tomorrow when we talk...

In Duality Theory, a Dual variable is defined for each of the Primal constraints and conversely for each of the Primal decision variables, a Dual constraint is constructed. Theoretically, a relationship exists between Primal and Dual decision variables.

For e.g. from the viewpoint of Resource allocation problems, a Primal decision variable (say x) represents economic activities, whereas the associated Dual decision variable (say y) represents dual prices which is simply the Worth per unit of resource or the Imputed  cost per unit of resource.

Accordingly, we can say that the required Dual decision variables (y) or the Imputed cost / Worth of the utilized resources required for each of the Primal decision variables(x) have to be defined (should be related) to yield the required objective values per unit of Primal decision variable(x). 

In Economic terms, the system is defined to be Stable when the Objective values (output) yielded for each of the Primal decision variables equals the input or worth of the resources required per unit of primal variable (i.e., the value of Dual decision variables). 

It'll be highly appreciated if you find the above information helpful. Nonetheless, this topic demands thoughtful discussion. Thanks.

The previous answers gave good replies. You may also encounter other names for the dual variables -- shadow price, or dual prices. Also, there is potential for confusion if the software you use uses a sign convention that is not what you are expecting. The transport problem in LP has an especially nice relationship between primal (Xij flow variable) and the dual values associated with the source (Ui) and sink (Vj) constraints.  For Xij > 0 in an optimal solution we know that Vj - Ui = Cij. There is also a dual feasibility test that says Vj - Ui 0 (ignoring possible degeneracy here). The Vj and Ui are sometime interpreted as the value of the commodity at the sink (Vj) and source (Ui) respectively. (By the way Cij is transport cost per unit of flow.)

Thanks for your responses. Let me ask my question differently.

The main body of my question is that: is it possible to represent a single decision variable of primal problem with respect to what we have in the dual space? 

For instance, assume that we have max-min (maximizing the minimum) problem. In the minimization problem, we would like to optimize two objective functions simultaneously (bi-objective problem) where the second objective function is x_i. In order to solve such a problem, I do not consider the second objective of the minimization problem (x_i). Then I dualize the minimization problem which results in a maximization problem and finally I get a max-max problem which is equivalent to a normal maximization problem. Now I would like to plug the second objective function (x_i) into the problem but it is not possible because we are in the dual space and therefore, I need to represent x_i in the dual space.

Thanks.

At best I'd say that you could hope for some nice analytical translation or identity to match the DUAL variables (which you could plot or understand) to their PRIMAL counterparts. It does not make much sense to seek a primal way to report a dualized objective? In other words, when you translate the min problem to a max problem in the dual variables, the answer of that max problem is in terms of those dual variables. As I mentioned in the previous answer, there can be some ways to move from the DUAL to the PRIMAL in problem-specific ways.

An interrelationship between the primal and dual optimal solutions is the so called complementary slackness conditions under feasibility for both primal and dual solutions.These relations imply (in the absence of degeneracy and multiple optimal solutions) for each constraint, either the resource is fully used with the corresponding nonzero dual variable, or the dual variable is zero with the corresponding unused resources being nonzero.

Alternatively, for each variable (again without degeneracy) at optimality, either the primal variable is non-zero with zero reduced cost or the variable is set to zero with a

non-zero reduced cost.

It means that your question might be answered in terms of (un)used resources and reduced costs.

Thanks Boris for your answer.

I have not found any answer to my question so far. Can you please let me know more in detail, considering a general problem, how (un)used resources and reduced costs can be exploited in order to find any answer to my question. 

For each Primal Constraint, the corresponding Non-zero Dual variable represents the Worth per unit of resources used or the Reduced cost per unit of resources fully utilized. A zero value of a Dual variable indicates that its associated resources are abundant (unused).

The resultant value of all non-zero Dual variables i.e., the Total Worth or the Reduced costs of all the resources fully used equals the Input value required to yield a unit of Primal Decision variable. So, the summation of these Input values for each of the Dual variables can be exploited to find out the Output yielded for each of the Non-zero Primal decision variables, i.e., the Objective value required per unit of Primal variable.

Alternately, for each of the Dual Constraints, a Primal decision variable can be defined. A non-zero Primal variable represents the full utilization of corresponding resources (not Zero reduced cost per unit of resource). This can be explained by a problem specific situation. In this way, you can move from Dual solution to Primal answers.

Thanks.