In many real-world scenarios, we need to optimize more than one objective function simultaneously. For instance, in engineering, we might want to minimize cost while maximizing performance. The challenge lies in finding a solution that balances these competing objectives.
My questions is how can we efficiently solve multi-objective optimization problems with a large number of objectives?
The simple way is to take a weighted sum of objective functions and select gains in accordance with your idea of the significance of a particular objective function. Also, take a look at the Pareto optimisation. The task is to select a parameter region such that each optimized function is within the specified limits.
The previous comment is spot on. An easy way to do this in practice is to first "normalize" every output to be on the same scale, so that the units and scales of the corresponding objective functions do not interfere. Then multiply the resulting normalized values by the relative weights (based on your design intentions, requirements, etc.).
For example if the range of one objective function is between 1-10000, while the other is .001 to 0.1, the first objective function will nearly dominate everything with no normalizing or weighting. As such, normalizing both to be in the range between 0 and 1 (or something similar), will make it easier for any weights to be applied. After normalizing, if we want the second objective to be 10 times more important than the first, we simply multiply it by 10, and so on.
One thing to keep in mind, though is that the addition of additional objective functions will often result in more complicated objective function "landscapes" whereby they may no longer be differentiable, or even continuous. As such, more complicated (and potentially slower) optimization functions may be required to get good results. For example, simple gradient-based solvers may no longer work, and more meta-heuristic solvers such as "particle swarm" or "simulated annealing" may be required.
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