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As shown in the figure, the search direction is orthogonal to the level set through and touches the level set for a smaller function value in .
The convergence of the sequence , generated by the the method of steepest descent, can be shown under fairly general assumptions. It is sufficient that is bounded from below and is Lipschitz continuous in a neighborhood of the set , i.e.,
To ensure convergence of the algorithm, it is not necessary to find the exact one-dimensional minimium in the direction of . It is sufficient to use a non-optimal decent direction and obtain merely a reduction of the current function value which is proportional to .
|automatically generated 6/14/2016