Talk:Pseudoconvex function

Not same as plurisubharmonic function

The talk of Plurisubharmonic function states that "If $f$ is a plurisubharmonic function and further $f$ is continuous, then $f$ is called a pseudoconvex function." So pseudoconvex is not equivalent to Plurisubharmonic and should not redirect there unless the special case is mentioned in the article. --Xeeron (talk) 10:01, 28 November 2007 (UTC)

Several complex variables has a different use of pseudoconvex functions and pseudoconvex domains, etc. which bewildered even Gian-Carlo Rota (or at least he pretended so). This problem has been addressed with a disambiguation page, by editor SB. Kiefer.Wolfowitz (talk) 22:18, 14 January 2011 (UTC)

Reads like a textbook

The dense use of mathematical notation (including at least two characters that my browser can't even handle) and jargon make this article more closely resemble a (section of a) mathematics textbook rather than an encyclopaedia article. Hrafn^Talk_Stalk^(P) 16:22, 25 December 2010 (UTC)

I've made every effort to state things in words Hrafn, but some symbols are unavoidable as is some jargon. For instance, one certainly needs to be able to say "Euclidean space" without defining that term. I have removed the textbook tag. As for your browser, you clearly have a broken configuration that cannot display unicode symbols properly. Sławomir Biały (talk) 16:26, 25 December 2010 (UTC)

This article is likely to be completely incomprehensible to somebody who does not have at least some university-level algebra, and probably would only be fully comprehensible to somebody who has taken at least two or three years of it at that level. I don't really know what purpose an article that assumes that level of knowledge serves. Hrafn^Talk_Stalk^(P) 16:40, 25 December 2010 (UTC)

There is a certain amount of technical knowledge required to understand most mathematics articles on Wikipedia. I don't see any reason this stub should be held to a higher standard than the rest of our mathematics articles, particularly since the notion is not likely to be relevant to anyone without a substantial quantitative background to start with. Sławomir Biały (talk) 16:51, 25 December 2010 (UTC)

And I don't know why mathematics articles should be held to a lower standard than articles for any other subject -- which are expected to be accessible to the non-specialist. Hrafn^Talk_Stalk^(P) 16:58, 25 December 2010 (UTC)

Well, I didn't say that this article was inaccessible to a non-specialist. I'm not a specialist in convex analysis. It should be accessible to anyone with a good background in university mathematics. While it is certainly the case that our articles are supposed to be made as accessible to as wide an audience as possible, this is always tempered by a realistic view of who the _target audience is. Indeed, most scientific articles on Wikipedia are not accessible to granny (e.g., 1,2-Dimethylcyclopropane). Sławomir Biały (talk) 17:05, 25 December 2010 (UTC)

I would lay claim to "a good background in university mathematics" -- but specialising in statistics (and to a lesser extent calculus), not algebra, and I would estimate that I only understand about half the article. If you need to have specialised in algebra to understand it, then I suspect the article serves little purpose as (i) this limits the readership to a considerable degree & (ii) such specialists are probably capable of reading up the primary literature on the topic in any case. Hrafn^Talk_Stalk^(P) 17:37, 25 December 2010 (UTC)

I don't know where you get the mistaken idea that the article is primarily about algebra. It discusses convex functions and minimization problems, which are both covered in most university level calculus courses. In response to your second point, this argument could be applied to any article in a specialized topic area. However, such articles exist in abundance, are indeed consistent with Wikipedia's mission. Sławomir Biały (talk) 17:54, 25 December 2010 (UTC)

Probably because both the jargon and notation is substantially algebraic. Yes, it can and should "be applied to any article in a specialized topic area". Look at General relativity -- an intensely mathematical topic, but articulated with a minimal usage of mathematical notation, and a very large amount of explanation. Hrafn^Talk_Stalk^(P) 18:50, 25 December 2010 (UTC)

As far as I can tell, there isn't a single notion from abstract algebra in the article. The gradient, local minima, and convexity are all notions of elementary calculus. The comparison with general relativity is specious for a number of reasons. General relativity is a top-level article to a very broad subject area. A better comparison would be with the article Kerr-Newman metric, which is naturally much more technical. Moreover, asking that this article that is just hours old be of comparable quality to a featured article that has existed for ten years with thousands of edits is clearly an unreasonable demand. Progress in Wikipedia is incremental, and demanding that articles start out fully formed featured articles is counterproductive to our general development model. The paucity of articles on basic optimization theory clearly indicates that we need to do more to encourage the growth of articles in this area, rather than jumping the gun on AfD's and unjustified cleanup template messages. Sławomir Biały (talk) 19:36, 25 December 2010 (UTC)

Hrafn, I challenge you to find one thing from algebra in the article. — Kallikanzarid^talk 18:23, 26 December 2010 (UTC)

It doesn't read like a textbook, it reads like an encyclopedia article on a mathematical subject, indeed it doesn't try to teach the subject, but presents it, as it should. This article, on a notable mathematical subject, is entirely approppriate, and looks similar to thousands of other math articles (just click on some of the links in the article, e.g. Convex function). This is not the place to argue for a change in inclusion criteria for scientific/technical articles. Editors try to make articles as accessible as possible, but it can't be done at the expense of a proper encyclopedic presentation of the subject, which demands to faithfully render what the sources say (per WP:V and WP:NOR), so imposes a minimum of rigor, symbols and jargon. Cenarium (talk) 22:31, 26 December 2010 (UTC)

Hrafn, I see from your userpage that you spend a lot of time with evolution related articles. I can't tell whether you have much background in mathematics. Speaking as somebody who does, I can say that an appropriate prerequisite for this article is multivariable calculus, which is (here in America) usually a second-year college course. Lots of science and engineering majors are required to take this course, so there are really quite a lot of people out there who might understand the article. The article is much less intimidating than some of our really specialized mathematical articles (e.g., spectral sequence is not likely to be understood by anyone who is not at least a math graduate student). However, I think that even very advanced articles are appropriate for Wikipedia. Such topics are covered, for example, in specialist encyclopedias such as the Springer Encyclopedia of Mathematics., so there's no reason why they can't also be covered here. Ozob (talk) 01:32, 27 December 2010 (UTC)

Unicode symbols

Oh, and according to Miscellaneous Technical (Unicode block), the symbols I'm missing are 'KEYBOARD' (2328) & 'LEFT-POINTING ANGLE BRACKET' (2329). Why these would appear in a mathematical article, or be expected in a standard OS installation or web browser, I don't know. Hrafn^Talk_Stalk^(P) 16:44, 25 December 2010 (UTC)

Exactly what the status of unicode symbols is in mathematics articles is a perennial debate, but our current Manual of Style does give its blessing to unicode symbols that correspond to named html entities. The left and right angle brackets are ⟨ and ⟩. But as you probably know, the appropriate way to lobby for this sort of change would be to start a threat at WT:MOSMATH. (However, this is a discussion that has happened before many times already.) Sławomir Biały (talk) 16:56, 25 December 2010 (UTC)

Merge from Pseudolinear function

It has been proposed that the content in Pseudolinear function be merged to this page. Any objections? Isheden (talk) 10:48, 14 January 2011 (UTC)

Agree. Kiefer.Wolfowitz (talk) 22:19, 14 January 2011 (UTC)

I see that the content has already been merged, aside from a trivial consequence of the definition and the references below:

• Chew, Kim Lin; Choo, Eng Ung (1984). "Pseudolinearity and efficiency". Mathematical Programming. 28 (2): 226–239. doi:10.1007/BF02612363. {{cite journal}}: Invalid |ref=harv (help)
• Mishra, Shashi Kant; Giorgi, Giorgio (2008). "η-Pseudolinearity: Invexity and Generalized Monotonicity". Invexity and optimization. Nonconvex optimization and its applications. Vol. 88. Springer. ISBN 978-3-540-78562-0. {{cite book}}: Invalid |ref=harv (help); Unknown parameter |isbn10= ignored (help)
• Kaul, R. N.; Lyall, Vinod; Kaur, Surjeet (1988). "Semilocal pseudolinearity and efficiency". European Journal of Operational Research. 36 (3). Elsevier Science B.V.: 402–409. doi:10.1016/0377-2217(88)90133-6. {{cite journal}}: Invalid |ref=harv (help); Unknown parameter |month= ignored (help)
• Jeyakumar, V.; Yang, X. Q. (1995). "On characterizing the solution sets of pseudolinear programs". Journal of Optimization Theory and Applications. 87 (3): 747–755. doi:10.1007/BF02192142. {{cite journal}}: Invalid |ref=harv (help); Unknown parameter |month= ignored (help)
• Komlósi, S. (1993-06-11). "First and second order characterizations of pseudolinear functions". European Journal of Operational Research. 67 (2). Elsevier Science B.V.: 278–286. doi:10.1016/0377-2217(93)90069-Y. {{cite journal}}: Invalid |ref=harv (help)
• Ansari, Qamrul Hasan; Schaible, Siegfried; Yao, Jen-Chih (1999). "η-Pseudolinearity". Decisions in Economics and Finance. 22 (1–2): 31–39. doi:10.1007/BF02912349. {{cite journal}}: Invalid |ref=harv (help); Unknown parameter |month= ignored (help)
• Giorgi, Giorgio; Rueda, Norma G. (2009). "η-Pseudolinearity and Efficiency" (PDF). International Journal of Optimization: Theory, Methods and Applications. 1 (2): 155–159. ISSN 2070-5565. {{cite journal}}: Invalid |ref=harv (help)
• Cambini, Alberto; Martein, Laura (2009). "Section 3.3: Quasilinearity and Pseudolinearity". Lecture Notes in Economics and Mathematical Systems. Vol. 616. Springer. pp. 50–57. doi:10.1007/978-3-540-70876-6. ISBN 978-3-540-70875-9. {{cite book}}: Invalid |ref=harv (help); Missing or empty |title= (help); Unknown parameter |booktitle= ignored (help); Unknown parameter |isbn10= ignored (help)

I put them here because it's not obvious to me that they should be included. I'll let someone else decide. RockMagnetist (talk) 21:56, 21 March 2013 (UTC)

Generalization to nondifferentiable functions

The two definitions given in this section do not coincide:

Consider the function $f:\mathbb {R} \to \mathbb {R}$ with $f(x)=-x$ whenever $x\in D=\{x={\frac {n}{2^{p}}}|\;p\in \mathbb {N} \cup \{0\},\,n\in \mathbb {N} \}$ and $f(x)=|x|$, elsewhere. For the upper Dini derivative it holds $f^{+}(x,1)=-1$ for all $x<0$, $f^{+}(x,1)=+1$ for all $x\geq 0$ with $x\notin D$ and $f^{+}(x,1)=+\infty$ for all $x\in D$.

Likewise, $f^{+}(x,-1)=+1$ for all $x\leq 0$, $f^{+}(x,-1)=-1$ for all $x>0$ with $x\notin D$ and $f^{+}(x,-1)=+\infty$ for all $x\in D$.

Now set $x_{0}=0$. Then $f^{+}(x_{0},1)=+1$, but $f$ is not increasing in this direction.

On the other hand, given the definition of the subdifferential as cited from https://en.wikipedia.org/wiki/Subderivative,

$\partial f(x)=\{x^{*}\in \mathbb {R} |\;\forall y\in \mathbb {R} :\;f(y)-f(x)\geq x^{*}(y-x)\}$,

thus $\partial f(x)=\{-1\}$, whenever $x\leq 0$ or $x\in D$ and $\partial f(x)=\emptyset$, elsewhere.

Especially, if $-1(y-x)\geq 0$ for some $x\leq 0$ or $x\in D$, then $f(y)\geq f(x)$, so $f$ is pseudoconvex in the sense of the second definition.

From what I know, pseudoconvexity in a point can be defined with respect to a specified derivative, that is given a derivative $f'(x,\cdot )$, then $f$ is pseudoconvex in a point $x_{0}$, iff for all $x\in X$, $f(x)<f(x_{0})$ implies $f'(x_{0},x-x_{0})<0$ and $f$ is pseudoconvex with respect to this derivative, iff the same holds true everywhere.

Or it can be defined with respect to some subdifferential as is done above.

Defining

$\partial f(x_{0})=\{x^{*}\in X^{*}|\;\forall x\in X:\,x^{*}(x-x_{0})\leq f'(x_{0},x-x_{0})\}$

pseudoconvexity wrt to der derivatie implies pseudoconvexity wrt the corresponding subdifferential, I'm not sure about the reverse direction.

Agreed??

Catrinski (talk) 11:19, 27 February 2015 (UTC)

Wrong example of pseudoconvex function in Properties section?

I might just be stupid, but doesn't the derivative of $x\mapsto x+x^{3}$ vanish at $x=0$, without this being a local minimum of the function? — Preceding unsigned comment added by 83.216.94.239 (talk) 12:23, 30 June 2016 (UTC)

Actually it doesn't. The derivative is: $1+3x^{2}$; which at $x=0$ is equal to 1. More generally, the function doesn't have critical points. — Preceding unsigned comment added by Matiasvd (talk • contribs) 01:41, 24 March 2021 (UTC)

Add a graph

A graph would clarify the topic : one of a pseudoconvex function and one that is not, both showing the vectors used in the definition. 89.86.44.221 (talk) 04:40, 3 April 2018 (UTC)