The primal and dual problems in linear programming are interconnected yet distinct in their formulation. The primal problem is the original linear programming problem that typically aims to maximize or minimize an objective function subject to certain constraints. In contrast, the dual problem is derived from the primal, reformulating the objectives and constraints, where each constraint in the primal corresponds to a variable in the dual and vice versa.

In the primal problem, the objective is usually to optimize a linear function based on decision variables, while in the dual, the objective function is formed from the primal’s constraints, often involving a switch between maximization and minimization. The number of variables in the dual matches the number of constraints in the primal, highlighting their symmetrical relationship.

The optimal solution of the primal provides a bound on the dual's optimal solution and vice versa, as established by the strong duality theorem, which states that if one problem has an optimal solution, so does the other, with their objective function values being equal. Additionally, the primal problem typically represents a direct application of resource allocation, while the dual offers insights into the value of those constraints, revealing shadow prices of resources.

Finally, the feasibility of the two problems is linked: if the primal problem is feasible and bounded, the dual will also be feasible and bounded, and the same holds true in reverse.