## Multi-Objective Optimization - 2D Decision Space

As promised in this previous post here is a quick look at a pareto-optimum solution for 2 objective functions when there are 2 decision variables instead of just 1. As in my previous example I model MAR ratio and Sharpe ratio as my objective functions, only this time they are functions of two variables x and y instead of just x. Let's take a look at them.

## Multiple Objectives

Prompted by an interesting topic raised by "Sluggo" of the "Trading Blox Trader's Roundtable", I took a side trip into the topic of optimizing for multiple objectives. In the context of trading system design the issue is driven by the desire to have as many measures of "goodness" as high as possible.

Imagine we only care about Sharpe Ratio and MAR ratio. Is the parameter set that maximizes MAR ratio better than the set that maximizes Sharpe ratio? What about a compromise? If I have a set of parameters and make changes to it and find both MAR and Sharpe improve, it is obvious which set of parameters is better. If a change to the parameter set improves MAR at the cost of Sharpe or vice versa, what should I do?

The first step is to identify the set of parameter sets where a trade-off is necessary, the so-called "pareto-optimum" set. Our ultimate solution must be from the pareto-optimum set. Any parameter set outside of the pareto optimum can ALWAYS be changed in some way so that at least one of our goodness measures (objective functions) can be improved and none of the others are made worse - so you would never settle for this solution.

This post addresses the first steps in finding the pareto optimum solution to a multi-objective optimization.
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