duality theorems

Duality theorems in mathematics and optimization reveal a fundamental relationship between two related problems: the primal and the dual. Essentially, they show that solving one problem provides valuable insights or bounds for the other. For example, in linear programming, the best solution to the primal problem (e.g., maximizing profit) is closely connected to the solution of its dual (e.g., minimizing cost). When certain conditions hold, the solutions are equal, enabling more efficient problem-solving and deeper understanding of the constraints and objectives involved.