This might be trivial but I am struggling to justify the following simplification. from: h

Pattab 2022-07-02 Answered
This might be trivial but I am struggling to justify the following simplification.from:
h ( x + d ) h ( x ) j G j f ( x ) d j + γ j G | d j |
to:
h ( x + d ) h ( x ) max j G | j f ( x ) | j G | d j | + γ j G | d j |

Specifically, why is there a negative in front of the maximization?

Note:
I can get behind the fact that
j G j f ( x ) d j max j G | j f ( x ) | j G | d j |
provided the jacobian is nonnegative element-wise. But then why add the negative sign?
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Answers (2)

Tamia Padilla
Answered 2022-07-03 Author has 16 answers
j G j f ( x ) d j max j G | j f ( x ) | j G | d j |
is not true. For example, let G = , f ( x ) = x, and d 1 = 1. Then
j G j f ( x ) d j = 1 1 = 1
but
max j G | j f ( x ) | j G | d j | = | 1 | | 1 | = 1
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Jonathan Miles
Answered 2022-07-04 Author has 3 answers
Using a | a | for all real number a, we have
j G j f ( x ) d j j G | j f ( x ) | | d j |
j G ( max k G | k f ( x ) | ) | d j | = ( max k G | k f ( x ) | ) ( j G | d j | ) .
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