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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.
====Nonlinear programming====
If the objective function or some of the constraints are nonlinear(equalities or inequalities), then the problem is a [[nonlinear programming]] problem.
====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 is one type of nonlinear programming. 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]].
====KKT conditions====
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