With inequality constraints, the problem can be characterized in terms of the [[geometric optimality conditions]], [[Fritz John conditions]] and [[Karush–Kuhn–Tucker conditions]], under which simple problems may be solvable.
====Linear programming====
If the objective function and all of the hard constraints are linear and some hard constraints are inequalities, then the problem is a [[linear programming]] problem. This can be solved by the [[simplex method]], which usually works in [[polynomial time]] in the problem size but is not guaranteed to, or by [[interior point method]]s which are guaranteed to work in polynomial time.
====Quadratic programming====
If all the hard constraints are linear and some are inequalities, but the objective function is quadratic, the problem is a [[quadratic programming]] problem. It can still be solved in polynomial time by the [[ellipsoid method]] if the objective function is [[Convex function|convex]]; otherwise the problem is [[NP hard]].
