Successive parabolic interpolation

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Successive parabolic interpolation is a technique for finding the extremum (minimum or maximum) of a continuous unimodal function by successively fitting parabolas (polynomials of degree two) to a function of one variable at three unique points or, in general, a function of n variables at 1+n(n+3)/2 points, and at each iteration replacing the "oldest" point with the extremum of the fitted parabola.

Advantages

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Only function values are used, and when this method converges to an extremum, it does so with an order of convergence of approximately 1.325. The superlinear rate of convergence is superior to that of other methods with only linear convergence (such as line search). Moreover, not requiring the computation or approximation of function derivatives makes successive parabolic interpolation a popular alternative to other methods that do require them (such as gradient descent and Newton's method).

Disadvantages

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On the other hand, convergence (even to a local extremum) is not guaranteed when using this method in isolation. For example, if the three points are collinear, the resulting parabola is degenerate and thus does not provide a new candidate point. Furthermore, if function derivatives are available, Newton's method is applicable and exhibits quadratic convergence.

Improvements

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Alternating the parabolic iterations with a more robust method (golden-section search is a popular choice) to choose candidates can greatly increase the probability of convergence without hampering the convergence rate.

References

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Michael Heath (2002). Scientific Computing: An Introductory Survey (2nd ed.). New York: McGraw-Hill. ISBN 0-07-239910-4.

Optimization: Algorithms, methods, and heuristics

Unconstrained nonlinear

Functions

Gradients

Convergence	Trust region Wolfe conditions
Quasi–Newton	Berndt–Hall–Hall–Hausman Broyden–Fletcher–Goldfarb–Shanno and L-BFGS Davidon–Fletcher–Powell Symmetric rank-one (SR1)
Other methods	Conjugate gradient Gauss–Newton Gradient Mirror Levenberg–Marquardt Powell's dog leg method Truncated Newton

Hessians

Newton's method

Constrained nonlinear

General	Barrier methods Penalty methods
Differentiable	Augmented Lagrangian methods Sequential quadratic programming Successive linear programming

Convex optimization

Convex
minimization

Linear and
quadratic

Interior point	Affine scaling Ellipsoid algorithm of Khachiyan Projective algorithm of Karmarkar
Basis-exchange	Simplex algorithm of Dantzig Revised simplex algorithm Criss-cross algorithm Principal pivoting algorithm of Lemke Active-set method