3
$\begingroup$

For a function $f: \mathbb{R}^n \to \mathbb{R}$, I am looking for the definition of $f$ to be unimodal.

  1. From Wikipedia

    $f$ is unimodal if there is a one to one differentiable mapping $X = G(Z)$ such that $f(G(Z))$ is convex.

  2. Also from Wikipedia:

    the super-level sets $L(f, t)$ of $f$, defined by $$ L(f, t) = \{ x \in \mathbb{R}^{n} | f(x) \geq t \}, $$ are convex subsets of $\mathbb{R}^n$ for every $t ≥ 0$. (This property is sometimes referred to as being unimodal.)

  3. Added: By the comment below, a common definition is that a function is unimodal, if it has exactly one local maximum.

    I wonder if this common definition allows existence of more than one local minimum? For example a "W" shape function defined on $\mathbb{R}$ goes to $\infty$ when approaching $\pm \infty$ in domain.

    If this definition does not allow existence of any local minimum, then for any line through the mode $m \in \mathbb{R}^n$ of $f$ in its domain, does the restriction of $f$ to this line increase on one side of $m$ and decrease on the other side of $m$?

I wonder if the first two definitions agree with each other? If not, when?

What is your definition, if possible?

Thanks and regards!

  • 1
    I suspect that Definition 1 has a typo: "$f(G(Z))$ is concave" would be more appropriate in the context of unimodality. Either way, the definitions are rather different. Personally, I prefer to use "unimodal" in the sense "has exactly one point of local maximum". But you are free to use it in any sense you wish *as long as you precisely state your definition*. And if your definition of "unimodal" is uncommon, you should pay it extra.2012-05-25
  • 2
    I would concur with @LeonidKovalev; albeit the sense I have seen it used (line search) was having exactly one local minimum.2012-05-25
  • 0
    @copper.hat: Thanks! (1) I wonder if you mean having exactly one local maximum, instead of having exactly one local minimum, for definition of unimodality? (2) If it is maximum in (1), does having exactly one local maximum allows having more than one local minimum?2012-05-25
  • 0
    My only exposure to the term was with line search algorithms for optimization. It was a convenient assumption (one local min.) to prove some convergence properties.2012-05-25
  • 0
    @copper.hat: Does that definition you were exposed to allow existence of local maximum(s)?2012-05-25
  • 0
    Tim: Yes, it did. The main theoretical concern was to ensure a descent for a 'bracketing' line search.2012-05-25
  • 0
    @copper.hat: I searched in Bertsekas's and Bazarar's books on nolinear optimization. The definitions they used for unimodality need montonicity on the two intervals divided by the mode. So I wonder in what references you saw that unimodality is defined only by single local minimum without requiring monotonicity?2012-05-25
  • 0
    Tim: Sorry, this was my recollection from notes taken in the mid-80's from my advisor's (Polak) lecture. I think I may have disposed of the notes a long time ago, I will check when I get a chance. No reference, though...2012-05-25
  • 0
    @copper.hat: Thanks! I found in [Polak's book](http://books.google.com/books?id=gcYR884tdCEC&lpg=PP1&hl=pt-PT&pg=PA146#v=onepage&q&f=false) that his unimodality is defined for a continuously differentiable function such that there is only one point whose derivative is zero and it is the unique global minimizer. It seems that it doesn't allow existence of local maximums.2012-05-25
  • 0
    Wow! So much for memory :-(.2012-05-25
  • 0
    @copper.hat: In Polak's definition, is a unimodal function strictly decreasing before the mode and strictly increasing after the mode? PS: I kind of envy you to have someone like him to teach you.2012-05-25
  • 0
    Hold on, I'm busy on a trip down memory lane :-). Yes, it must be strictly decreasing before (because $\phi'(\lambda) <0$) and strictly increasing after (similar reason). I found one of my old barrier algorithms on page 247. This book came out long after I graduated. He is an amazing person.2012-05-25

1 Answers 1