heref is the objective functionand {gi and {hi are given functions so that hi ( and gi ( represent the equality and inequality constraintsrespectively the region where the objective function is defined and where all the constraints are satisfied is called the feasible region an optimization problem without constraints is said to be an unconstrained problem for an unconstrained continuous optimization problemthe search space is often taken to be ( subset ofrn and is assumed to be function for sufficiently high (typically or suffices)that isits -th order derivative is continuous for function the standard approach to minimizing (xis to solve the equation ( ( where (xis the gradient of at the solutions xto ( are called stationary points stationary points can be local/global minimizerslocal/global maximizersor saddle points (which are neitherifin additionthe function is the condition ( ( ) feasible region for all stationary points saddle points ( ensures that the stationary point xis local minimizer of the condition ( states that the hessian matrix of at xis positive definite recall that we write to indicate that matrix is positive definite in figure we have multiextremal objective function on there are four stationary pointstwo are local minimizersone is local maximizerand one is neither minimizer nor maximizerbut saddle point local maximum (xsaddle point local minimum global minimum figure multiextremal objective function in one dimension convexity and optimization an important class of optimization problems is related to the notion of convexity set is said to be convex if for all it holds that ( ax for all in additionthe objective function is convex function provided that for each in the interior of there exists vector such that (yf ( ( )vy ( convex function