
# step size rule based on the Armijo rule

Determines a step size $s>0$ such that

where $\eta$ is a descent direction, e.g. $\eta = -\nabla F(x)$. where the search is performed by $s^\rho$, $% $ and $c>0$ is a constant. For further details see Definition 4.2.2 in [1]

### Optional Parameters

Gradient ($\eta$)
if this value is not given, we assume that $\eta=-\nabla F$ is the negative gradient. Otherwise you have to spezify a gradient here
InitialStepSize: ($1$)
initial step size $s$ as starting point for the line search
rho (0.5)
decrease exponent for line search, i.e. we update $s\rightarrow s^\rho$
c ($0.0001$)
constant in front of the inner product.

### Matlab Documentation

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 % stepSizeAsrmijo(M,F,x,descentDir) compute the step size by Armijo's rule % % INPUT % M : a manifold % F : a functional @(x) % x : the current point % descentDir : a descent direction % % OPTIONAL % Gradient : (descentDir) gradient direction (if descentDir is not -grad) % InitialStepSize : (1) initial step size as starting point for search % rho : (0.5) decrease factor x_next = (x)^rho % c : (0.0001) the factor in front of the norm % --- % MVIRT | R. Bergmann | 2018-03-15

### References

1. Absil, P-A, Mahony, R and Sepulchre, R (2008). Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton and Oxford