Adding Constraints

A fair number of problems in operations research, decision science, machine learning, physics, and econometrics boil down to constraint optimization.

An objective function needs to be optimized, either by minimizing a cost or maximizing a reward, in relation to specific variables that have certain constraints. Real-life examples include production planning, scheduling, Sudoku, or Crossword puzzles. 

Adding constraints prunes the search space. With unlimited resources, you might look at every possible solution. With limited resources, you’re focused on a few things you can do.

Constraints can even turn unsolvable problems into solvable ones. Consider the grade school example of a system of linear systems. An equation with two unknowns, e.g., x + y = 4, has many solutions (e.g., (1,3) or (2,2)). But if you add another constraint, that x = 3y. You can solve for the unique solution (3,1).