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Given a smooth manifold $M$ and another $S$, consider a smooth function $\psi: S \times M \rightarrow \mathbb{R}$, and use this to define $\psi_s:M\rightarrow \mathbb{R}$ by $\phi_s(p):= \psi(s,p)$.

Further assume that a dense set of $s \in S$ yield $\psi_s$ with no local optima on $M$, only a single global maximum, and a single global minimum. Then, is it true that for any $s$ for which there are critical points of $\psi_s$ besides the global max and min, these points can never be local optima with negative semi definite hessian?

Assuming compactness of $M$ is ok if needed, as is assuming all functions involved are bounded above and below and analytic (to exclude constant regions).

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  • $\begingroup$ The lack of local optima does not prevent critical points which are not saddle (think x^3 near 0). Moreover, speaking of saddle points you seem to assume some differentiability, but you don't precise the regularity of your functions. $\endgroup$
    – Benoît Kloeckner
    Commented Sep 1, 2017 at 20:29
  • $\begingroup$ No, you can build 1D counterexamples, start out with an increasing $\phi$ and deform this into something that is constant on an interval, and then go back to the initial $\phi$. $\endgroup$
    – Christian Remling
    Commented Sep 1, 2017 at 22:10
  • $\begingroup$ That's not a counterexample to the question, perhaps I wasn't clear enough. $\endgroup$
    – Benjamin
    Commented Sep 1, 2017 at 22:28
  • $\begingroup$ I think I should have said, local optimum with negative semi definte hessian. $\endgroup$
    – Benjamin
    Commented Sep 1, 2017 at 22:49
  • $\begingroup$ Edited to reflect this $\endgroup$
    – Benjamin
    Commented Sep 1, 2017 at 22:50

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