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1-D minimization using Charalambous' method

`[a,gX,perf,retcode,delta,tol] = srchcha(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)`

`srchcha` is a linear search routine. It searches
in a given direction to locate the minimum of the performance function
in that direction. It uses a technique based on Charalambous'
method.

`[a,gX,perf,retcode,delta,tol] = srchcha(net,X,Pd,Tl,Ai,Q,TS,dX,gX,perf,dperf,delta,tol,ch_perf)` takes
these inputs,

net | Neural network |

X | Vector containing current values of weights and biases |

Pd | Delayed input vectors |

Tl | Layer target vectors |

Ai | Initial input delay conditions |

Q | Batch size |

TS | Time steps |

dX | Search direction vector |

gX | Gradient vector |

perf | Performance value at current |

dperf | Slope of performance value at current |

delta | Initial step size |

tol | Tolerance on search |

ch_perf | Change in performance on previous step |

and returns

a | Step size that minimizes performance |

gX | Gradient at new minimum point |

perf | Performance value at new minimum point |

retcode | Return code that has three elements. The first two elements correspond to the number of function evaluations in the two stages of the search. The third element is a return code. These have different meanings for different search algorithms. Some might not be used in this function. |

0 Normal | |

1 Minimum
step taken | |

2 Maximum
step taken | |

3 Beta
condition not met | |

delta | New initial step size, based on the current step size |

tol | New tolerance on search |

Parameters used for the Charalambous algorithm are

alpha | Scale factor that determines sufficient reduction in |

beta | Scale factor that determines sufficiently large step size |

gama | Parameter to avoid small reductions in performance, usually set to 0.1 |

scale_tol | Parameter that relates the tolerance |

The defaults for these parameters are set in the training function
that calls them. See `traincgf`, `traincgb`, `traincgp`, `trainbfg`, and `trainoss`.

Dimensions for these variables are

Pd |
| Each element |

Tl |
| Each element |

Ai |
| Each element |

where

Ni | = | net.numInputs |

Nl | = | net.numLayers |

LD | = | net.numLayerDelays |

Ri | = | net.inputs{i}.size |

Si | = | net.layers{i}.size |

Vi | = | net.targets{i}.size |

Dij | = | Ri * length(net.inputWeights{i,j}.delays) |

Here is a problem consisting of inputs `p` and
targets `t` to be solved with a network.

p = [0 1 2 3 4 5]; t = [0 0 0 1 1 1];

A two-layer feed-forward network is created. The network's
input ranges from `[0 to 10]`. The first layer has
two `tansig` neurons, and the second layer has one `logsig` neuron.
The `traincgf` network training function and the `srchcha` search
function are to be used.

net = newff([0 5],[2 1],{'tansig','logsig'},'traincgf'); a = sim(net,p)

net.trainParam.searchFcn = 'srchcha'; net.trainParam.epochs = 50; net.trainParam.show = 10; net.trainParam.goal = 0.1; net = train(net,p,t); a = sim(net,p)

You can create a standard network that uses `srchcha` with `newff`, `newcf`,
or `newelm`.

To prepare a custom network to be trained with `traincgf`,
using the line search function `srchcha`,

The `srchcha` function can be used with any
of the following training functions: `traincgf`, `traincgb`, `traincgp`, `trainbfg`, `trainoss`.

The method of Charalambous, `srchcha`,
was designed to be used in combination with a conjugate gradient algorithm
for neural network training. Like `srchbre` and `srchhyb`, it is a hybrid search. It uses
a cubic interpolation together with a type of sectioning.

See [Char92] for a description of Charalambous' search. This routine is used as the default search for most of the conjugate gradient algorithms because it appears to produce excellent results for many different problems. It does require the computation of the derivatives (backpropagation) in addition to the computation of performance, but it overcomes this limitation by locating the minimum with fewer steps. This is not true for all problems, and you might want to experiment with other line searches.

Charalambous, C., "Conjugate gradient algorithm for efficient
training of artificial neural networks," *IEEE Proceedings*,
Vol. 139, No. 3, June, 1992, pp. 301–310.

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