do constraints simplify or further complicate non-linear optimization problems?
i think i can think of cases of both:
cases where certain constraints simplify the non-linear problem, possibly by decreasing the solution space, etc
and cases where certain constraints further complicate the non-linear problem, because they are complex themselves.
hence, are all constraints "good" or a "good thing"?
are there types of constraints one wishes to avoid?
i am particularly referring to non-convex optimization problems
Restricting solution space with valid equations may be a good idea however I had many cases where new constraints increased the computation costs. On the other hand, I had cases where relaxations increased the solution space but helped in terms of computation costs.
If you can find similar problems to yours that are solved in literature, you may try to make your formulation similar to their formulation with valid equations or relaxations. This may be an initial strategy to start with.
It depends very much on the types of constraints that are used and the optimisation problem itself.
Are the constraints an integral part of the problem, or are they ones that the user can choose as a means of reducing the time and effort to find desirable solutions?
Reducing the hyper-volume to be searched can be very helpful.
Convex boundaries, e.g. circular areas in 2D usually are easier than complicated star-shaped boundaries.
For some multi-modal problems, constraints might not hinder or be of any use in the search for an optimum. If the objective function has many equal sized peaks, and you are searching for any one peak, then boundary constraints might not be any help. If, on the other hand, you want to find as many peaks as possible, then restricting the search domain would not be prudent.
Now that I have a few more minutes free, I should add that the objective functions used in the Black Box Optimisation Benchmarks I recommended are mathematically defined. The difference between the complexity of those functions and what is required in real world applications is unimaginably great.
For example, there is a lot of research into the siting of wind farms and the distribution of turbines within those farms. But to get the components of the turbines to the site, to build them and to maintain them efficiently is a far more difficult task than the optimisation of those three tasks separately.
The requirements of industry, the military and commerce are different and complicated because optimisation must be performed within legislative, economic, ethical, political, social, and sometimes even religious frameworks.
Even in the initial planning stages of long construction projects, some account might need to be taken of the effect of a possible change of government during the project.
Economic factors, such as interest rates on loans is something that cannot be treated exactly. There are other financial risks that are even more difficult to model.
Construction of a wind farm in a stable 3rd world country is vastly different to that in a narco-state, or one in which bribes are part of the social and economic fabric.
Once a site has been found to be optimal in some sense, the transportation of materials to the site might be on roads on which trucks cannot be driven. Finding out that fact might mean the original "optimum" found is not practical.
Employment of labour and management is also something that cannot be modelled in an ad hoc simple fashion.
The amount of work that can actually be performed can depend on many factors. You might want to use a simple model where work can be done on any day of the week, but you'd soon find that's not accurate because of public holidays, or in countries with strong labour laws that require workers to be paid more on weekends, or that truck drivers must not drive for longer than 8 hours without rest.
In short, constraints in mathematical toy worlds are nothing like what real constraints are like in the real world.
Before going on a possibly lengthy and fruitless search for available software, I would advise you to think very carefully about the many factors that are required for your actual problem.
Of course, it is also very important to consider the computational hardware resources you have available and how long you have to come up with a reasonable answer. A near optimal value for a large travelling salesman problem might be good enough after a day or two if the true optimum would take millions of years to find!
All the best with your research!
Leo.
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