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I have recently learned about the logarithmic mean

$$\frac{x-y}{\ln(x)-\ln(y)},\quad x,y > 0.$$

It is used a lot in chemical engineering optimization models e.g. see slide 15 of Developing spatial branch & bound solvers.

But I wonder if anyone else has come across the logarithmic mean elsewhere in optimization?

Rao and Dey (2014)1 mention a relation to the Lambert $W$ function that I find interesting. Borwein and Lindstrom (2016)2 discuss about the usage of the Lambert $W$ function in optimization.

PS. The blog post Logarithmic mean temperature difference requires yet another cone? discusses whether you can build a conic representation.


References

[1] Rao, M., Dey, A. (2014). Scope of the Logarithmic Mean. The Australian Journal of Mathematical Analysis and Applications. 11(1):1-10.

[2] Lindstrom, S. B., Borwein, J. M. (2016). Meetings with Lambert $W$ and Other Special Functions in Optimization and Analysis.

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  • $\begingroup$ I just heard Logarithmic Mean Divisia Index (LMDI). I am not sure if it is completely related.. $\endgroup$ – independentvariable Jun 7 '19 at 15:24
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    $\begingroup$ Also a very old paper: downloads.hindawi.com/journals/ijmms/1982/718951.pdf This paper gives some relevant references [2-5] $\endgroup$ – independentvariable Jun 7 '19 at 15:26
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    $\begingroup$ It could come up again. It has a way pf appearing in probability, statistics, and other places - see stats.stackexchange.com/search?q=lambert+W . And it's pretty cool that it is conic representable. I suspect the OP included it because Lambert W, and how it is conic representable, has some similarity with and might have some bearing on logarithmic mean. If capital letters are not allowed in tags, I guess lambert-w is best. $\endgroup$ – Mark L. Stone Jun 27 '19 at 3:23
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    $\begingroup$ The Lambert is an up and coming function. I have now added a link to by late Borwein about the Lambert function and optimization. $\endgroup$ – ErlingMOSEK Jun 27 '19 at 4:59
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    $\begingroup$ I'm actually just adding lambertw (i.e. automatic expcone conversion) to YALMIP as we need it in a small project. $\endgroup$ – Johan Löfberg Jun 27 '19 at 16:18
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But I wonder if anyone else has come across the logarithmic mean elsewhere in optimization?

Optimization is simply the selection of a best element (with regard to some criterion) from some set of available alternatives. (From: "The Nature of Mathematical Programming", by George B. Dantzig).

The logarithmic mean has a relationship with other means as follows:

From Long and Chu (2011)1,

Let $A(a,b)=(a+b)/2$, $I(a,b)=1/e(b^b/a^a)^{1/(b−a)}$, $L(a,b)=(b−a)/(\ln b−\ln a)$, $G(a,b)=\sqrt{ab}$ and $H(a,b)=2ab/(a+b)$ be the arithmetic, identric, logarithmic, geometric, and harmonic means of two positive numbers $a$ and $b$ with $a\ne b$, respectively. Then \begin{align}\min\{a, b\}&< H(a, b)< G(a, b) = L_{−2}(a, b)< L(a, b) = L_{-1}(a, b)\\&< I(a, b) = L_0(a, b)< A(a, b) = L_1(a, b)<\max\{a, b\}.\end{align}

See also Pittenger (1987)2 for a discussion of logarithmic means (page 286), and the Glivenko-Cantelli theorem (p. 287). Further resources include the Dvoretzky–Kiefer–Wolfowitz inequality and building CDF bands from Wikipedia.

A logarithmic relationship occurs when the logs of two variables plotted against each other create a straight line. A semilogarithmic relationship is when there is linearity when only one of the variables is scaled as a logarithm.

For further reading on the (weighted) logarithmic mean, see p. 885-886 of Neuman (1994)3.


In logistic regression the right-hand predictor side of the equation must be linear with the left-hand outcome side of the equation. You must test for linearity in the logit (in logistic regression the logit is the outcome side). This is commonly done with the Box-Tidwell transformation4.

  • Add to the logistic model interaction terms which are the cross product of each independent times its natural logarithm $X\ln X$. If these terms are significant, then there is nonlinearity in the logit.

"In the logistic model, the log-odds (the logarithm of the odds) for the value labeled "1" is a linear combination of one or more independent variables ("predictors"); the independent variables can each be a binary variable (two classes, coded by an indicator variable) or a continuous variable (any real value). The corresponding probability of the value labeled "1" can vary between 0 (certainly the value "0") and 1 (certainly the value "1"), hence the labeling; the function that converts log-odds to probability is the logistic function, hence the name. The unit of measurement for the log-odds scale is called a logit, from logistic unit, hence the alternative names. Analogous models with a different sigmoid function instead of the logistic function can also be used, such as the probit model; the defining characteristic of the logistic model is that increasing one of the independent variables multiplicatively scales the odds of the given outcome at a constant rate, with each dependent variable having its own parameter; for a binary independent variable this generalizes the odds ratio.".

See also Logistic regression: Definition of the logistic function from Wikipedia.


References

[1] Long, B-Y., Chu, Y-M. (2011). Optimal generalized logarithmic mean bounds for the geometric combination of arithmetic and harmonic means. Journal of the Indonesian Mathematical Society. 17(2):85–96.

[2] Pittenger, A. O. (1987). Limit theorems for logarithmic means. Journal of Mathematical Analysis and Applications. 123(1):281-291.

[3] Neuman, E. (1994). The Weighted Logarthmic Mean. Journal of Mathematical Analysis and Applications. 188(3):885-900.

[4] Joyce T., Donovan, J., Murphy, E. (2006). The application of the box-tidwell transformation in reliability modeling. Annual Reliability and Maintainability Symposium.

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