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Monotonic grey box direct search optimization

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Abstract

We are interested in blackbox optimization for which the user is aware of monotonic behaviour of some constraints defining the problem. That is, when increasing a variable, the user is able to predict if a function increases or decreases, but is unable to quantify the amount by which it varies. We refer to this type of problems as “monotonic grey box” optimization problems. Our objective is to develop an algorithmic mechanism that exploits this monotonic information to find a feasible solution as quickly as possible. With this goal in mind, we have built a theoretical foundation through a thorough study of monotonicity on cones of multivariate functions. We introduce a trend matrix and a trend direction to guide the Mesh Adaptive Direct Search (Mads) algorithm when optimizing a monotonic grey box optimization problem. Different strategies are tested on a some analytical test problems, and on a real hydroelectric dam optimization problem.

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Acknowledgements

Thanks to NSERC CRD grant (#RDCPJ 490744-15) with Hydro-Québec and Rio Tinto.

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Authors and Affiliations

  1. GERAD, Département de mathématiques et génie industriel, Polytechnique Montréal, C.P. 6079, Succ. Centre-ville, Montreal, QC, H3C 3A7, Canada

    Charles Audet, Catherine Poissant & Christophe Tribes

  2. Rio Tinto, Énergie Électrique, Rio Tinto Aluminium, 1954 Rue Davis, Jonquière, QC, G7S 3B6, Canada

    Pascal Côté

  3. Giro, 5800, Rue Saint-Denis, Montréal, QC, H2S 3L5, Canada

    Catherine Poissant

Authors
  1. Charles Audet
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  2. Pascal Côté
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  3. Catherine Poissant
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  4. Christophe Tribes
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Correspondence to Christophe Tribes.

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Appendix

Appendix

The trend matrices for the analytical problems from Sect. 4.1 and the MDO problem from Sect.  4.2 are given as follows:

CHENWANG_F2 [14] (\(n=8,\, m=6\))

$$\begin{aligned} T= \left[ \begin{array}{rrrrrrrr} 0 &{}\quad 0 &{}\quad 0 &{}\quad {\text{ N }/A} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad {\text{ N }/A} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad {\text{ N }/A} \\ 1 &{}\quad -1 &{}\quad 0 &{}\quad 1 &{}\quad {\text{ N }/A} &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad -1 &{}\quad 0 &{}\quad 1 &{}\quad {\text{ N }/A}\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad -1 \\ \end{array}\right] \end{aligned}$$

CHENWANG_F3 [14] (\(n=10,\, m=8\))

$$\begin{aligned} T= \left[ \begin{array}{rrrrrrrrrr} 1 &{}\quad 1 &{}\quad -1 &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad -1 \\ 1 &{}\quad -1 &{}\quad 1 &{}\quad {\text{ N }/A} &{}\quad 1 &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad {\text{ N }/A} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad -1 &{}\quad 0 \\ -1 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad {\text{ N }/A} \\ 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 \\ \end{array}\right] \end{aligned}$$

HS83 [16] (\(n=5,\, m=6\))

$$\begin{aligned} T= \left[ \begin{array}{rrrrrrr} -1 &{}\quad 1 &{}\quad -1 &{}\quad 1 &{}\quad -1 &{}\quad 1 \\ -1 &{}\quad 1 &{}\quad -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad -1 &{}\quad -1 &{}\quad -1 &{}\quad -1 &{}\quad 1 \\ -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 1 \\ {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad -1 &{}\quad 1 &{}\quad -1 &{}\quad 1 \\ \end{array}\right] \end{aligned}$$

HS114 [19] (\(n=9,\, m=6\))

$$\begin{aligned} T= \left[ \begin{array}{rrrrrrr} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad -1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad -1 &{}\quad 1 &{}\quad -1 \\ \end{array}\right] \end{aligned}$$

MAD6 [19] (\(n=5, \, m=7\))

$$\begin{aligned} T= \left[ \begin{array}{rrrrrrrr} -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 1 &{}\quad -1 &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 1 &{}\quad 0 \\ \end{array}\right] \end{aligned}$$

PIGACHE [21] (\(n=4, \, m=11\))

$$\begin{aligned} T= \left[ \begin{array}{rrrrrrrrrrrr} -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad {\text{ N }/A} &{}\quad 0 &{}\quad 0 &{}\quad {\text{ N }/A} &{}\quad 1 &{}\quad -1 &{}\quad -1 \\ 1 &{}\quad -1 &{}\quad 1 &{}\quad -1 &{}\quad {\text{ N }/A} &{}\quad 1 &{}\quad -1 &{}\quad -1 &{}\quad -1 &{}\quad -1 &{}\quad 1 \\ -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad -1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad -1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad -1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad -1 &{}\quad 0 \\ \end{array}\right] \end{aligned}$$

TAOWANG_F2 [26] (\(n=7,\, m=4\))

$$\begin{aligned} T= \left[ \begin{array}{rrrrr} {\text{ N }/A} &{}\quad 1 &{}\quad 1 &{}\quad {\text{ N }/A} \\ {\text{ N }/A} &{}\quad 1 &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} \\ {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 0 &{}\quad {\text{ N }/A} \\ {\text{ N }/A} &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ {\text{ N }/A} &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad {\text{ N }/A} &{}\quad 1 \\ 0 &{}\quad 0 &{}\quad -1 &{}\quad -1 \\ \end{array}\right] \end{aligned}$$

ZHAOWANG_F5 [28] (\(n=13,\, m=9\))

$$\begin{aligned} T= \left[ \begin{array}{rrrrrrrrrr} 1 &{}\quad 1 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 &{}\quad -1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 \\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ \end{array}\right] \end{aligned}$$

MDO [25] (\(n=10,\, m=10\))

$$\begin{aligned} T= \left[ \begin{array}{rrrrrrrrrr} 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 0 &{}\quad 0 \\ {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 0 &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ -1 &{}\quad -1 &{}\quad -1 &{}\quad -1 &{}\quad -1 &{}\quad 0 &{}\quad 1 &{}\quad -1 &{}\quad 1 &{}\quad 1 \\ {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 1 &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 0 &{}\quad 0 \\ -1 &{}\quad -1 &{}\quad -1 &{}\quad -1 &{}\quad -1 &{}\quad 0 &{}\quad 1 &{}\quad -1 &{}\quad 1 &{}\quad -1 \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad -1 &{}\quad 1 &{}\quad -1 &{}\quad 1 \\ {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 0 &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 0 &{}\quad 0 \\ {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 0 &{}\quad {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 0 &{}\quad 0 \\ {\text{ N }/A} &{}\quad {\text{ N }/A} &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ \end{array}\right] . \end{aligned}$$

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Audet, C., Côté, P., Poissant, C. et al. Monotonic grey box direct search optimization. Optim Lett 14, 3–18 (2020). https://doi.org/10.1007/s11590-019-01497-8

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