MOS A Mathematical Optimization Service James Hubert Merrick Tom as Tinoco De Rubira October 11 2022

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MOS: A Mathematical Optimization Service

James Hubert Merrick∗

, Tom´as Tinoco De Rubira†

October 11, 2022

Abstract

We introduce MOS, a software application designed to facilitate the deployment, integration,

management, and analysis of mathematical optimization models. MOS approaches mathemat-

ical optimization at a higher level of abstraction than existing optimization modeling systems,

enabling its use with all of them. The sole requirement to harness MOS is a simple annotation

of the code specifying the formulation of an optimization model. With this, the model becomes

accessible to humans through the automatic generation of a user interface, and to machines

through an associated API and client libraries. All this is achieved while avoiding the ad hoc

code typically required to obtain such features.

1 Introduction

Whilst known by diﬀerent names in diﬀerent settings, mathematical optimization inﬂuences billions

of dollars in the modern economy, impacting every industry. It consists of the maximization or

minimization of some objective function subject to constraints, and is a natural paradigm for

solving problems in many ﬁelds. In planning applications in operations research and management

science, optimization models act as a decision-support tool for a human decision-maker. In other,

more operational, settings, they enable decisions to be automated, with example problems including

pricing, scheduling, allocation of scarce resources, and routing. In the natural sciences, optimization

models capture the physical laws governing the behavior of natural systems. In machine learning,

they provide the tools for obtaining parameterized models that best ﬁt a particular data set. Boyd

and Vandenberghe (2004) and Luenberger and Ye (2021) provide comprehensive introductions to

the theory and applications of mathematical optimization.

Key tools in mathematical optimization are algebraic modeling systems. Examples of these

are cvxpy (Diamond and Boyd,2016), JuMP (Dunning et al.,2017), Pyomo (Hart et al.,2017),

optmod (Tinoco De Rubira,2020) and GAMS (Bussieck and Meeraus,2004). These tools greatly

facilitate the process of constructing and solving optimization models on computers. They allow

users to construct optimization problems by writing intuitive mathematical expressions, and can

utilize many numerical solvers without the need of custom code that expresses the problem in

solver-speciﬁc data structures and formats.

∗jmerrick@alumni.stanford.edu

†ttinoco@alumni.stanford.edu

arXiv:2210.03813v1 [math.OC] 7 Oct 2022

From the authors’ experience in developing optimization models to support and automate de-

cisions, once a model is formulated using one of the above modeling systems, there is often an

additional non-trivial programming exercise required to facilitate a human or application to in-

teract with the model. In the absence of this code, using the model requires familiarity with the

model’s internal and low-level details, which is seldom documented and user friendly, creating bar-

riers for human users and adding complexity to application pipelines. A custom solution, on the

other hand, typically requires time and resources, including dedicated software engineers.

By approaching the modeling problem at a higher level of abstraction than existing tools, and

by capturing and standardizing common model properties and structure, MOS provides essential

deployment, integration, management and analysis features automatically, removing the need for

custom solutions. The sole requirement is a simple annotation of the ﬁle containing the model

code. This allows a focus, at the development stage, on the core modeling task itself, and at the

production stage, on the model usage itself, reducing the barriers to obtaining value from a model.

Guericke and Cassioli (2019) propose a framework for deploying optimization models based

on microservice architectures, and highlight a gap between solution methods in literature and

solution methods in production environments. MOS also attempts to contribute to the closing of

this gap through a proposed concrete and universal model representation, a modular and ﬂexible

architecture, and an open-source implementation.1

2 Model representation

Boyd and Vandenberghe (2004) introduce an optimization problem as being represented by the

following:

minimize f0(x)

subject to fi(x)≤bi, i = 1, . . . , m, (1)

where x= (x1, . . . , xn) is the vector of variables to be optimized, f0:Rn→Ris the objective

function, and functions fi:Rn→Rtogether with constants bifor i= 1, . . . , m deﬁne the

constraints. MOS considers optimization models as objects that not only consist of optimization

problems having variables, functions, and constraints, as in (1), but also of inputs, outputs, and

intermediate objects. The intermediate or “helper objects” correspond to objects that are created

either in a pre-optimization phase, for facilitating the construction of problem variables, functions

and constraints, or in a post-optimization phase, for facilitating the construction of outputs. This

model representation is helpful for establishing a layer of abstraction that enables the deﬁnition

and implementation of tools for interacting with, and analyzing models. Figure 1 illustrates the

MOS optimization model representation.

3 Design

Figure 2 shows the architecture of MOS, which includes the following components:

•Backend: Manages model data and access, and provides a REST API for interacting with

models.

1MOS is available at https://github.com/Fuinn.

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MOS:AMathematicalOptimizationServiceJamesHubertMerrick*,TomasTinocoDeRubiraOctober11,2022AbstractWeintroduceMOS,asoftwareapplicationdesignedtofacilitatethedeployment,integration,management,andanalysisofmathematicaloptimizationmodels.MOSapproachesmathemat-icaloptimizationatahigherlevelofabstraction...

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MOS A Mathematical Optimization Service James Hubert Merrick Tom as Tinoco De Rubira October 11 2022

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