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Powell's method, strictly Powell's conjugate direction method, is an algorithm proposed by Michael J. D. Powell for finding a local minimum of a function. The function need not be differentiable, and no derivatives are taken. The function must be a real-valued function of a fixed number of real-valued inputs. The caller passes in the initial point.
The golden-section search is a technique for finding an extremum (minimum or maximum) of a function inside a specified interval. For a strictly unimodal function with an extremum inside the interval, it will find that extremum, while for an interval containing multiple extrema (possibly including the interval boundaries), it will converge to one of them.
At each iteration, there is a set of "working points" in which we know the value of f (and possibly also its derivative). Based on these points, we can compute a polynomial that fits the known values, and find its minimum analytically. The minimum point becomes a new working point, and we proceed to the next iteration: [1]: sec.5
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
Newton's method can be used to find a minimum or maximum of a function f(x). The derivative is zero at a minimum or maximum, so local minima and maxima can be found by applying Newton's method to the derivative. [39] The iteration becomes: + = ′ ″ ().
Similarly, the function has a global (or absolute) minimum point at x ∗, if f(x ∗) ≤ f(x) for all x in X. The value of the function at a maximum point is called the maximum value of the function, denoted (()), and the value of the function at a minimum point is called the minimum value of the function, (denoted (()) for clarity ...
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The gradient descent can take many iterations to compute a local minimum with a required accuracy, if the curvature in different directions is very different for the given function. For such functions, preconditioning, which changes the geometry of the space to shape the function level sets like concentric circles, cures the slow convergence ...