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I am dealing with a function, which is a product of two strongly convex functions, and trying to determine the number of its local minimum. For example, I have $$H=f(x)\cdot g(x)$$, in which both $f$ and $g$ are strongly convex functions and thus both have only one minimum themselves.

I wonder if there is any theory or conclusion about the number of its local minima. Intuitively, I suspect it to be 2. (I know in general $H$ is not convex.)

Thanks!

Matteo
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Mafen
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  • Suppose $f(x)=e^x$ and $g(x)=e^{-x}$, so that $h(x)=1$. Now consider replacing $f(x)$ with $(1+\epsilon\sin x)e^x$... –  Jan 24 '16 at 15:38
  • Comment aside: optima is already a plural in Latin. No s required. – Bernard Jan 24 '16 at 15:44
  • @Rahul Wow, excellent example. Thanks a lot! – Mafen Jan 24 '16 at 15:57
  • @Bernard Got that, thanks! – Mafen Jan 24 '16 at 15:58
  • You might want to consider posting and accosting answer yourself if you figured it out – Stella Biderman Jan 24 '16 at 16:00
  • @Rahul What if $f$ and $g$ are both quadratic functions? Actually the variable $x$ in my problem is matrix and both two functions are quadratic. – Mafen Jan 24 '16 at 16:08
  • Consider these three possibilities: $$f(x) = x^2 \qquad g(x) = x^2$$ $$f(x) = (x-1)^2 \quad g(x)=(x+1)^2$$ $$f(x)=(x-1)^2+1 \qquad g(x)=(x+1)^2-1$$ – Michael Grant Jan 24 '16 at 18:48
  • For multivariable functions you can more than two local minima, for example when $f(x,y)=10x^2+(y-1)^2$ and $g(x,y)=(x-1)^2+10y^2$ then $f(x,y)g(x,y)$ has minima near $(0,1)$, $(1,0)$, and $(0,0)$. Probably lots of local minima are possible if $f$ and $g$ can be less than zero. –  Jan 25 '16 at 00:57
  • @MichaelGrant So for these cases there are at most two local minima, right ? – Mafen Jan 25 '16 at 04:40
  • @Rahul Yeah, you are right. Thanks! Actually I am dealing with a function $f=trace( RR^H WW^H-RW+RR^H )$, where $R$ and $W$ are matrices and the superscript means hermitian transpose. Do you have any idea about its number of local minima? – Mafen Jan 25 '16 at 06:35
  • Sorry to resurrect an old question, but what is the answer? And does it hold also if the functions are strictly (but not strongly) convex? Could anyone please provide a reference?Thank you. – Matteo Mar 08 '18 at 13:09

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