Theorem of duality

The theorem of duality states that many mathematical problems, especially in optimization, have a dual problem that mirrors the original (primal) problem. Solving one provides valuable insights or bounds for the other. For example, in linear programming, the dual problem relates to the same situation but from a different perspective—like measuring the value of resources versus the worth of products. This duality helps in understanding the problem better, verifying solutions, and sometimes simplifying complex calculations by focusing on the easier of the two. It highlights a deep symmetry between seemingly different formulations of the same problem.