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Can something justify this equality to me?

$$\sup_y \{ \langle y,x\rangle + \inf_z\{f(z) - \langle y,z\rangle \} \} = \sup_y \{ \inf_z \{ f(z) - \langle y,x - z\rangle\} \}$$

I don't understand how you can just put the inner product put inside the inf, even though it is a constant. Is it actually true in general that $a + \inf_{\text{ over something not dependent on $a$}} = \inf_{\text{ over something not dependent }} (a)$?

Michael Hardy
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Lemon
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1 Answers1

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If $L$ is the greatest lower bound for a set $S$, then certainly $a+L$ is a lower bound for $S+a = \{ s+ a \mid s\in S\}$, and if any other value $v = (v-a) + a$ is a lower bound for $S+a$ then $v-a$ is a lower bound for $S$ whence $L \geq v-a$ which implies $L + a \geq v$ - i.e., $L+a$ is the greatest lower bound for $S+a$.

user139388
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