Tuesday, June 4, 2019

Logarithmic mean temperature difference requires yet another cone?

The logarithmic mean temperature difference
$$LMTD(x,y) = \frac{x-y}{\ln(x/y)}$$

can be used to model heat exchange, e.g., when optimizing the reuse of excess process heat onsite at industrial facilities. In the heat exchanger network synthesis problem due to Yee and Grossmann (link goes to a recent paper by Mistry and Misener), we find that its reciprocal raised to some power, that is
$$RecLMTD^\beta(x,y) = \left(\frac{\ln(x/y)}{x-y}\right)^\beta,$$

can be extracted as a separately contributing term in the objective function. Capitalizing on the convexity of this term on $(x,y) \in \mathbb{R}_+^2$, for all considered $\beta \geq 0$, this leads to better performance when solving the otherwise nonconvex problem as argued in the paper.

A challenge to find the conic reformulation of this function was posed under the Oberwolfach Workshop on Mixed-Integer Nonlinear Optimization (2019) and we accepted. Of course, this is trivial if no restrictions are put on the set of cones as one may just define
$$\mathcal{K} = \mathrm{cl}\{(t,s,x,y) \in \mathbb{R}_{++}^4 : t \geq s \cdot RecLMTD^\beta(x/s, y/s) \}$$
and call it a day. This cone is nonempty, closed and convex and hence obeys $K = (K^{*})^{*}$ as well as all the usual properties of conic duality. Computationally, however, the cone is not particularly desirable and we can do better with a bit of reformulation:
$$\begin{array}{lll} t \geq \left(\frac{\ln(x/y)}{x-y}\right)^\beta,\\t \geq \left(\frac{\ln(u/y + 1)}{u}\right)^\beta,& u = x-y,\\ y \geq \frac{u}{\exp(ut^{1/\beta})-1},& u = x-y,\\ y \geq \frac{u}{\exp(u/s)-1},& u = x-y, & s \geq t^{-1/\beta},\\ \end{array}$$
where I substitute in the first step, rewrite assuming either $u > 0$ or $u <0$ (both leads to the same) in the second, and extract a power cone representable subexpression in the third. This means that the representation problem of $RecLMTD^\beta$ have been reduced to the representation problem of
$$\mathcal{K} = \mathrm{cl}\left\{(t,s,x) \in \mathbb{R}_{+}^2 \times \mathbb{R}_{++} : t \geq \frac{x}{\exp(x/s)-1} \right\},$$
which, just like the quadratic, power and exponential cones, is defined as the epigraph of the perspective of a univarite convex function; in this case $\frac{x}{\exp(x)-1}$. Whether this cone can be written in terms of the others, or has potential for computationally efficient implementations itself, remains open. We invite anyone interested in barrier functions and interior-point algorithms to take a crack at it.

What is possible right now?
The logarithmic mean is bounded from above and below by, respectively, the arithmetic and the geometric mean, both of which are representable in MOSEK. There is also Chen's approximation which corresponds to a power cone. Finally it can be observed that $\log(\exp(-x)+1)$ is a fairly good underestimator of $\frac{x}{\exp(x)-1}$. This means that the set described by the inequality from above,
$$y \geq \frac{u}{\exp(u/s)-1},$$

can be outer approximated by
$$y \geq s \log(\exp(-u/s)+1),$$

which is representable in MOSEK as the homogenization of the softplus function. Beware that the usefulness of any of these alternatives is unknown to the author.