Pseudo-convex functions and global minima

[fading notch]

I want to confirm whether my reasoning is correct regarding the following problem:

Suppose g is a pseudo convex function on R^n (see screenshot for definition). Let x* be a stationary point satisfying the first order necessary condition i.e it is a local minimizer with grad(g(x *)) = 0. Then for any point z in R^n we have grad(g(x*))(z-x) = 0 => grad(g(x*))(z-x) >=0 which by pseudo convexity of g implies g(z) >= g(x*) so x* is a global minimum. Is that right? Sounds too simple to be true.

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[eternal ravine]

I don't see any problem with it

since the domain is R^n

[fading notch]

It is indeed unbounded

Thank you for your confirmation!

[fervent trail]

this part looks silly to me

what is x in your argument?

$$\left\langle \nabla g(x^), z-x^ \right\rangle = 0 \Rightarrow g(z) \geqslant g(x^*)$$

[tepid rivet]

aL

[fervent trail]

is what you mean

since z is arbitrary. x* must be a global minimiser

@fading notch

[fading notch]

I meant to write x* but that was probably not picked up by the escape character on discord

Thanks for pointing it out but yes I meant exactly what you wrote

@fervent trail

[fading notch]

+close

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