A comparison of gradient descent(green) and Newton's method (red) for minimizing a function (with small step sizes). Newton's method uses curvature information to take a more direct route. In calculus, Newton's method is an iterative method for finding the roots of a differentiable function f (i.e. solutions to the equation f(x)=0). In optimization, Newton's method is applied to the derivative f ′ of a twice-differentiable function f to find the roots of the derivative (solutions to f ′(x)=0), also known as the stationary points of f.
The Nelder–Mead method or downhill simplex method or amoeba method is a commonly applied numerical method used to find the minimum or maximum of an objective function in a multidimensional space. It is applied to nonlinear optimization problems for which derivatives may not be known. However, the Nelder–Mead technique is a heuristic search method that can converge to non-stationary points[1] on problems that can be solved by alternative methods.[2]The Nelder–Mead technique was proposed by John Nelder & Roger Mead (1965).[3]
Gradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or of the approximate gradient) of the function at the current point. If instead one takes steps proportional to the positive of the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent.
Gradient descent is also known as steepest descent, or the method of steepest descent. However, gradient descent should not be confused with the method of steepest descent for approximating integrals