Why study duality in optimization?

Question

Is the formulation of dual problem easier than the primal problem to solve, so that duality provides convenience for solving the optimization problem?

Or, does the dual formulation provide a useful theoretical instrument to prove theorems or show properties of optimization problems?

I am not clear about the motivation to study the duality in optimization. Some examples can be very helpful! Thanks!

Answer 1

Duality makes the problem easier most of the times it is being used for example an optimization problem with 3 variables and 10 constraints has a duality with 3 constraints and 10 variables. This is because the complexity of an optimization problem mostly depends on the number of constraints not variables.

Answer 2

One major reason is that the dual problem helps you to derive lower bounds (in the case of minimization) on the achievable objective. This can be exploited when developing solvers.