Usages of logarithmic mean in optimization

Question

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.

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_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.}

Answer 1

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.

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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.

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_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.}