Note on Multi-objective optimization (preference-based MO)

A book of Kalyanmoy Deb gives me some interesting points for the beginner of this MOEAs (Multi-objective Optimization using Evolutionary Algorithms) area.

Tracing back to when I studied Decision Making System, I leant about optimizing multi-constraint or multi-goal. However, I sense that somehow what I've learn are definitely different from Multi-objective optimization. The book of Deb leads me well to inside knowledge.

According to what I've learned, the solution need to be a maximum or minimum value of the problem using many constraints or objectives. However, these constraints and goals are the weighted sum of the objectives where a weight for an objective is proportional to the preference factor assigned to that particular objective. This is a so-called the scalarized method, scalarizing an objective vector into a single composite objective function converts the multi-objective optimization problem into a single-objective optimization problem. When such a composite objective function is optimized, in most cases it is possible to obtain one particular trade-off solution.

This procedure of handling multi-objective optimization problems is much simpler, yet still being more subjective than the ideal procedure. We call this procedure a preference-based multi-objective optimization.

A schematic of this procedure is described as follows.

"Based on the higher-level information, a preference vector w is first chosen. Thereafter, the preference vector is used to construct the composite function, which is then optimized to ind a single trade-off optimal solution by a single-objective optimization algorithm. Although not often practiced, the procedure can be used to find multiple trade-off solutions by using a different preference vector and repeating the above procedure."

1. Multi-objective optimization problems

- minimize f1

- minimize f2

- .

- .

- .

- minimize fk

subject to constraints

2. Higher-level information

3. Estimate a relative importance vector (w1, w2, ..., wm)

4. Single-objective optimization problem

F = w1.f1 + w2.f2 + ... + wm.fm

or

a composite function

5. Single-objective optimizer

6. One optimum solution

It is important to realize that the trade-off solution obtained by using the preference-based strategy is largely sensitive to the relative preference vector used in forming the composite function. A change in this preference vector will result in a (hopefully) different trade-off solution.

Beside this difficulty, it is intuitive to realize that finding a relative preference vector itself is highly subjective and not straightforward. This requires an analysis of the non-technical, qualitative and experience-driven information to find a quantitative relative preference vector. Without any knowledge of the likely trade-off solutions, this is an even more difficult task.

The ideal multi-objective optimization procedure is less subjective.