Elsevier

Journal of Process Control

Volume 118, October 2022, Pages 165-169
Journal of Process Control

Fitting second-order cone constraints to microbial growth data

https://doi.org/10.1016/j.jprocont.2022.08.018Get rights and content

Highlights

  • We formulate an optimization problem for fitting second-order cone constraints to data.

  • We adapt the concave–convex procedure to the optimization problem.

  • We validate our approach on three examples based on microbial growth.

Abstract

Second-order cone programming is a highly tractable convex optimization class. In this paper, we fit general second-order cone constraints to data. This is of use when one must solve large-scale, nonlinear optimization problems, but modeling is either impractical or does not lead to second-order cone or otherwise tractable constraints. Our motivating application is biochemical process optimization, in which we seek to fit second-order cone constraints to microbial growth data. The fitting problem is nonconvex. We solve it using the concave–convex procedure, which takes the form of a sequence of second-order cone programs. We validate our approach on simulated and experimental microbial growth data, and compare its performance with conventional nonlinear least-squares fitting.

Introduction

Second-order cone programming (SOCP) is a highly tractable nonlinear optimization class [1], [2]. In this paper, we fit second-order cone (SOC) constraints to data. This could be useful in several scenarios, e.g., when first principles modeling does not lead to a tractable constraint or is simply not viable, and one must solve large-scale problems that are fundamentally nonlinear.

Our motivating application is the optimization of biochemical processes, for which the Monod [3] and Contois [4] functions are established models of microbial growth. In [5], it was shown that Monod growth with constant biomass and Contois growth can be represented as SOC constraints. These models have been extensively validated over many years, but are not derived from first principles. It is therefore plausible that a general SOC constraint, which has more parameters, could accurately model a broad range of microbial growth. At the same time, by virtue of being an SOC, any such constraint could be used in an optimization problem without sacrificing tractability.

In this paper, we estimate the parameters of a general SOC constraint from data. We formulate this as an optimization problem, which is nonconvex but does have partial SOC structure. We exploit this through the concave–convex procedure (CCP) [6], [7], which here takes the form of a sequence of SOCPs. The problem nonetheless has a number of local minima and a trivial global minimum. We circumvent these difficulties using multiple starting points and by adding constraints on the parameters, as is standard in ellipse fitting [8].

To the best of our knowledge, this is the first attempt to fit general SOC constraints to data. The most relevant existing literature concerns fitting ellipses or conics to data [9], [10]. These papers often focus on problems in computer vision, and therefore in two and three dimensions. We use ideas from these papers to prohibit the trivial solution where all parameters are estimated to be zero [8], and to quantify error in terms of geometric distance [11].

There is an extensive literature on fitting microbial growth rates like the Monod and Contois functions to data [12]. The closest work to ours in this area is [13], in which splines are fit to microbial growth data. While this can better accommodate growth that does not follow a simple mathematical expression, splines are not representable as a tractable or convex constraint in an optimization problem. On the other hand, an empirically fitted SOC constraint can potentially capture a wide range of microbial growth, and is well-suited for use in large scale optimization problems arising in applications like wastewater treatment [14].

Our original contributions are as follows.

  • We formulate an optimization problem for fitting SOC constraints to data. We discuss several constraints for prohibiting trivial solutions and enforcing certain practical requirements, e.g., that microbial growth is zero when the substrate or biomass is zero.

  • We adapt the CCP to the optimization problem, which is implementable as a sequence of SOCPs.

  • We validate our approach on three examples based on microbial growth: (i) Contois growth, which has an exact SOC representation, (ii) general Monod growth and Haldane growth, which do not have SOC representations, and (iii) experimental growth data from [15].

Section snippets

Problem statement

We have a set of data points zkRm, kK, where K is an index set. We want to find the SOC constraint that best fits the data. An SOC constraint is parametrized by the matrices ARn×m, bRn, cR1×m, and dR. Note that the number of rows in A and b, n, is a tunable parameter, which we discuss at the end of Section 3.

In standard form the SOC constraint is written Azk+bczk+d.Note that (1) does not need to be satisfied for all kK. We are rather interested in the surface on which the constraint

Solution via CCP

Problem (2) is a nonconvex optimization. However, given that the number of parameters will be no more than a few dozen, it is reasonable to hope for a global or good local minimum, e.g., by trying multiple starting points. We now show how to exploit the problem’s partial SOC structure in the CCP [6], [7]. We also refer to [18], in which the CCP was applied to a nonlinear model of microbial growth with partial SOC structure.

We first rewrite (2) as minA,b,c,dkKqk2such thatfor all kK,Azk+bqk+

Application to microbial growth

We consider a well-mixed volume with a substrate concentration, s, biomass concentration, x, and growth kinetics, r. Each data point is a measurement, zk=[sk,xk,rk]R3, kK. The kinetics generally take the form r=μ(s,x)x, where μ(s,x) is called the growth rate. Two common cases are the Contois and Monod growth rates, which are respectively given by μC(s,x)=μmaxsKsx+sandμM(s,x)=μmaxsKs+s.Both are parametrized by a maximum specific growth rate, μmax, and the constant, Ks [19].

The next two

Conclusion

We use the CCP to fit SOC constraints to data. Our main motivation is the microbial growth in bioprocesses, wherein certain standard growth rates like the Monod and Haldane functions lead to nonconvex constraints.

In numerical tests on both synthetic and experimental microbial growth data, we found that the fitted SOC surfaces achieved similar or better goodness of fit to standard techniques, and some robustness to noise. Most importantly, unlike most standard growth rates, the resulting SOC

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Funding is acknowledged from the Natural Sciences and Engineering Research Council of Canada and the French LabEx NUMEV , incorporated into the I-Site MUSE.

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