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

$h({x}^{\ast}+d)-h({x}^{\ast})\ge \sum _{j\text{\u29f8}\in G}{\mathrm{\nabla}}_{j}f({x}^{\ast})\ast {d}_{j}+\gamma \sum _{j\text{\u29f8}\in G}|{d}_{j}|$

to:

$h({x}^{\ast}+d)-h({x}^{\ast})\ge -\underset{j\text{\u29f8}\in G}{max}|{\mathrm{\nabla}}_{j}f({x}^{\ast})|\sum _{j\text{\u29f8}\in G}|{d}_{j}|+\gamma \sum _{j\text{\u29f8}\in G}|{d}_{j}|$

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

Note:

I can get behind the fact that

$\sum _{j\text{\u29f8}\in G}{\mathrm{\nabla}}_{j}f({x}^{\ast})\ast {d}_{j}\ge \underset{j\text{\u29f8}\in G}{max}|{\mathrm{\nabla}}_{j}f({x}^{\ast})|\sum _{j\text{\u29f8}\in G}|{d}_{j}|$

provided the jacobian is nonnegative element-wise. But then why add the negative sign?

$h({x}^{\ast}+d)-h({x}^{\ast})\ge \sum _{j\text{\u29f8}\in G}{\mathrm{\nabla}}_{j}f({x}^{\ast})\ast {d}_{j}+\gamma \sum _{j\text{\u29f8}\in G}|{d}_{j}|$

to:

$h({x}^{\ast}+d)-h({x}^{\ast})\ge -\underset{j\text{\u29f8}\in G}{max}|{\mathrm{\nabla}}_{j}f({x}^{\ast})|\sum _{j\text{\u29f8}\in G}|{d}_{j}|+\gamma \sum _{j\text{\u29f8}\in G}|{d}_{j}|$

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

Note:

I can get behind the fact that

$\sum _{j\text{\u29f8}\in G}{\mathrm{\nabla}}_{j}f({x}^{\ast})\ast {d}_{j}\ge \underset{j\text{\u29f8}\in G}{max}|{\mathrm{\nabla}}_{j}f({x}^{\ast})|\sum _{j\text{\u29f8}\in G}|{d}_{j}|$

provided the jacobian is nonnegative element-wise. But then why add the negative sign?