The steepest descent algorithm follows the downhill gradient of the function at each step. When a downhill step is successful the step-size is increased by a factor of two. If the downhill step leads to a higher function value then the algorithm backtracks and the step size is decreased using the parameter tol.

A suitable value of tol for most applications is 0. The steepest descent method is inefficient and is included only for demonstration purposes. The algorithms described in this section use only the value of the function at each evaluation point. These methods use the Simplex algorithm of Nelder and Mead. Starting from the initial vector , the algorithm constructs an additional vectors using the step size vector as follows:. These vectors form the vertices of a simplex in dimensions.

On each iteration the algorithm uses simple geometrical transformations to update the vector corresponding to the highest function value. The geometric transformations are reflection, reflection followed by expansion, contraction and multiple contraction. Using these transformations the simplex moves through the space towards the minimum, where it contracts itself.

After each iteration, the best vertex is returned. Note, that due to the nature of the algorithm not every step improves the current best parameter vector. Usually several iterations are required. The minimizer-specific characteristic size is calculated as the average distance from the geometrical center of the simplex to all its vertices. This size can be used as a stopping criteria, as the simplex contracts itself near the minimum. It uses the same underlying algorithm, but the simplex updates are computed more efficiently for high-dimensional problems.

In addition, the size of simplex is calculated as the RMS distance of each vertex from the center rather than the mean distance, allowing a linear update of this quantity on each step. The memory usage is for both algorithms. This example program finds the minimum of the paraboloid function defined earlier. The location of the minimum is offset from the origin in and , and the function value at the minimum is non-zero. The main program is given below, it requires the example function given earlier in this chapter.

The initial step-size is chosen as 0. The program terminates when the norm of the gradient has been reduced below 0. The output of the program is shown below,. Note that the algorithm gradually increases the step size as it successfully moves downhill, as can be seen by plotting the successive points in Fig. The conjugate gradient algorithm finds the minimum on its second direction because the function is purely quadratic.

Additional iterations would be needed for a more complicated function. Here is another example using the Nelder-Mead Simplex algorithm to minimize the same example object function, as above.

The simplex size first increases, while the simplex moves towards the minimum. After a while the size begins to decrease as the simplex contracts around the minimum. A brief description of multidimensional minimization algorithms and more recent references can be found in,. Nelder and R. Mead, A simplex method for function minimization , Computer Journal vol. GSL 2.

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A user-specified maximum number of iterations has been reached. An error has occurred. Also provides theoretical background which provides insights into how methods are derived. This edition offers revised coverage of basic theory and standard techniques, with updated discussions of line search methods, Newton and quasi-Newton methods, and conjugate direction methods, as well as a comprehensive treatment of restricted step or trust region methods not commonly found in the literature.

Also includes recent developments in hybrid methods for nonlinear least squares; an extended discussion of linear programming, with new methods for stable updating of LU factors; and a completely new section on network programming. The accuracy of the line minimization is specified by the parameter tol. The minimum along this line occurs when the function gradient g and the search direction p are orthogonal.

### Optimization

The line minimization terminates when. The search direction is updated using the Fletcher-Reeves formula where , and the line minimization is then repeated for the new search direction. This is the Polak-Ribiere conjugate gradient algorithm. It is similar to the Fletcher-Reeves method, differing only in the choice of the coefficient. Both methods work well when the evaluation point is close enough to the minimum of the objective function that it is well approximated by a quadratic hypersurface.

This is a quasi-Newton method which builds up an approximation to the second derivatives of the function using the difference between successive gradient vectors. By combining the first and second derivatives the algorithm is able to take Newton-type steps towards the function minimum, assuming quadratic behavior in that region.

It supersedes the original bfgs routine and requires substantially fewer function and gradient evaluations. The user-supplied tolerance tol corresponds to the parameter used by Fletcher.

A value of 0. The steepest descent algorithm follows the downhill gradient of the function at each step. When a downhill step is successful the step-size is increased by a factor of two. If the downhill step leads to a higher function value then the algorithm backtracks and the step size is decreased using the parameter tol. A suitable value of tol for most applications is 0.

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The steepest descent method is inefficient and is included only for demonstration purposes. The algorithms described in this section use only the value of the function at each evaluation point. These methods use the Simplex algorithm of Nelder and Mead. Starting from the initial vector , the algorithm constructs an additional vectors using the step size vector as follows:. These vectors form the vertices of a simplex in dimensions.

On each iteration the algorithm uses simple geometrical transformations to update the vector corresponding to the highest function value. The geometric transformations are reflection, reflection followed by expansion, contraction and multiple contraction. Using these transformations the simplex moves through the space towards the minimum, where it contracts itself. After each iteration, the best vertex is returned. Note, that due to the nature of the algorithm not every step improves the current best parameter vector. Usually several iterations are required.

The minimizer-specific characteristic size is calculated as the average distance from the geometrical center of the simplex to all its vertices. This size can be used as a stopping criteria, as the simplex contracts itself near the minimum. It uses the same underlying algorithm, but the simplex updates are computed more efficiently for high-dimensional problems.

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In addition, the size of simplex is calculated as the RMS distance of each vertex from the center rather than the mean distance, allowing a linear update of this quantity on each step. The memory usage is for both algorithms. This example program finds the minimum of the paraboloid function defined earlier. The location of the minimum is offset from the origin in and , and the function value at the minimum is non-zero. The main program is given below, it requires the example function given earlier in this chapter.

The initial step-size is chosen as 0. The program terminates when the norm of the gradient has been reduced below 0.

The output of the program is shown below,.