# δMOEA: Multi-Objective Grid Search Algorithm

(C) 2017 DecisionVis LLC

## Executive Summary

δMOEA is an optimization library that helps people make better decisions using computer models of a problem domain. δMOEA searches the model inputs for combinations that produce optimal model outputs with respect to multiple objectives. We call δMOEA a "Grid Search" algorithm because it samples the model inputs on a grid rather than attempting to optimize continuous values.

This version of δMOEA is written in Python.

The Python version of δMOEA carries the 3-clause BSD
license. See `COPYING.md`

for details.

## Getting Started With δMOEA

- Read the rest of this document for an overview of what δMOEA does.
- Read doc/getting_started.md for a quick guide to downloading and using δMOEA.
- Refer to doc/api.md for documentation on δMOEA's library functions.
- Examine the examples in examples/ for more ways to use δMOEA.
- Read the documents under doc/technotes/ for the design rationale.
- For a list of technical questions and answers about δMOEA, read doc/design_questions.md.

## Computer Models

Computer models of a problem domain express value judgments about it. If nothing else, what the author has chosen to model is a statement about what is important. Deciding to use optimization, however, often implies an extra layer of value judgment on top of the domain model. We call this extra layer the "optimization model" to distinguish it from the domain model. We call the inputs to the optimization model "decisions" and the outputs "objectives", "constraints", and "tagalongs".

Figure 1 shows how an optimization model wraps a domain model.

*Decisions*are translated by the optimization model into inputs for the domain model.- Some domain model inputs may be held constant and not subjected to optimization.
- Some domain model outputs are translated into
*objectives*for optimization. Objectives are numbers that δMOEA will attempt to minimize or maximize. - Other domain model outputs are translated into
*constraints*for optimization. Constraints are numbers that δMOEA will preemptively try to drive to zero, before considering the objectives. - Still other domain model outputs are captured as
*tagalongs*. Tagalongs have no role in optimization but may be of use for decision-making. In addition, tagalongs may preserve domain model inputs and outputs so that old domain model evaluations may be reused with a new optimization model. - Some domain model outputs may be ignored and discarded by the optimization model.

## Multi-Objective Optimization

Multi-Objective Optimization differs from conventional (single-objective) optimization in that it seeks to approximate a "Pareto Set" representing the tradeoffs among multiple objectives, rather than to approximate a single optimal value. Figure 2 illustrates the difference between the progress of single-objective and multi-objective optimization.

Under single-objective optimization, the objective value is continually improved over time (minimized in this example). This is the top plot in Figure 2. Multi-objective optimization, on the other hand, attempts to optimize two or more objectives at once (minimizing two objectives, in this example.) The bottom left plot shows both objectives over time. Rather than a single value, both objectives develop a range of values that trade off against each other. The final tradeoff is shown in the bottom right plot.

Figure 3 is an animated version of Figure 2 that illustrates the progress of the optimization runs over time, as both optimization runs converge towards approximations of the optimal values.

## Why δMOEA?

Optimization is a powerful tool for understanding a problem. It pushes a domain model to its limits and identifies gaps in our conception of a problem. Multi-objective optimization is particularly powerful: compared to single-objective optimization, adding even one more objective allows you to make much more nuanced decisions and avoid blowing past the point of diminishing returns. (See Figures 2 and 3.)

Also, compared to not doing optimization at all, and
simply sampling the entire decision space on a grid, δMOEA
saves a vast amount of computer time. As an optimization
algorithm, δMOEA focuses its sampling on the *interesting*
part of the decision space, where interest is defined by
the user in terms of objectives and constraints.

Compared with other multi-objective optimization
algorithms, δMOEA scales well to large numbers of
objectives (it has been tested up to 20 objectives)
and makes more efficient use of expensive computational
resources. Furthermore, δMOEA has been designed to
integrate with existing parallel evaluation approaches --
it is readily parallelized but does not impose any one
approach on its users. δMOEA will work with anything from
Python's `multiprocessing`

library, to MPI, to homegrown
ØMQ job queues.

Its design also makes δMOEA easy to use as a library rather than as an application. Where most optimization routines want to take control of your model, δMOEA decouples sampling and evaluation to put the user in control. This makes it possible to embed δMOEA in model code rather than shoehorning model code into an optimization program.

Finally, because δMOEA is already grid-oriented, it will avoid a great deal of unnecessary sampling on mixed-integer problems compared to other MOEAs.

## About The Name

δMOEA uses an evolutionary optimization heuristic to improve its Pareto approximation. This is the origin of the "MOEA" in its name: it is a Multi-Objective Evolutionary Algorithm. The δ alludes to the sampling grid in the decision space, where δ is the grid spacing. The name δMOEA also pays homage to Deb et al.'s εMOEA, an influential algorithm that applies a grid on the objective space rather than the decision space.

## Python Compatibility

δMOEA is compatible with both Python 2.7.13 (or later) and Python 3.6 (or later). It is likely compatible with earlier versions of Python 2.7 and Python 3, but that has not been tested. Furthermore, while performance between Python versions is equivalent, results are similar rather than identical due to differences in their random number generators.

In addition, δMOEA has no library dependencies beyond the Python standard library, so it should work on any platform and with any interpreter. (Reports of incompatibility are encouraged.)

## Open Source

δMOEA is open source because we believe in multi-objective optimization and we want people to use it. As a business, DecisionVis LLC has found that licensing MOEAs gets in the way of consulting relationships and produces minimal revenue. The degree to which MOEAs need to integrate with domain models and parallelization environments also makes closed-source releases unreasonably expensive to support and forces us to spend time fighting uninteresting integration problems. So we decided to develop and release an open source MOEA to let us work with our customers on interesting problems instead.

## What Next?

Refer to doc/getting_started.md for an overview of how to get and use δMOEA.

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