In optimization, is there a distinction between "implicit/natural" and "explicit/designed" constraints?

Question

For example, I wish to optimization a function which has a log term $\log(x)$

Now the very presence of the log term induces a constraint which says $x > 0$. The case $x = 0 $ might be a bit ambiguous but certainly $x \not< 0$

Now, this to me is some sort of an implicit constraint associated with this problem.

But suppose for whatever reason, I impose $x < M$ where $M$ is a large number, i.e., the maximum current that can be flow through my circuit. Then this is a designer imposed constraint, and to me this is an explicit constraint.

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As another example, you could have a $\log\det$ term in your optimization problem. Just by having that term alone, you are working with a constrained problem: your solution must be square and positive definite. https://angms.science/doc/LA/logdet.pdf Hence this to me is a "natural/implicit" constraint.

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Is this distinguished in the optimization literature? I don't think the term active/inactive constraint helps or is precise enough. I want to find a precise term that distinguished a natural/implicit constraint due to geometry or the objective function versus a designed/explicit constraint imposed by some external problem factors.

Thanks! References would be great!

Answer 1

First, active/inactive is definitely not the answer. An active constraint is one that is affecting the result (perhaps only the interim result at a particular step in optimisation).

I think the distinction you're looking for needs clarification. If you have a constraint that rules out negative values for a parameter because they would be non-physical (eg, negative masses or concentrations or probabilities) but the objective function extends to negative values with no singularities or discontinuities or other mathematical unpleasantness, does that count as a 'natural' constraint in your sense?

TO be more concrete, consider the positive matrix factorisation (PMF) problem in air pollution measurement. We have measurements $X_{ts}$ of mass concentrations of chemical species $s$ in the air at time $t$. We want to factorise it as a combination of sources $$X_{ts}=\sum_k F_{tk}G_{ks}+\epsilon_{ts}$$ where $\epsilon$ are errors with known (but not equal) variances $\sigma^2_{ts}$, $F$ is a matrix of % concentrations of species $s$ in source $k$, and $G$ is matrix of mass concentrations of source $k$ in the air at time $t$. We do this by minimising $Q=(X-FG)^2)$.

Now, $Q$ is defined for all real $F$ and $G$, but:

• On elementary physical grounds, all the entries of $F$ and $G$ must be non-negative
• Based on knowledge of the chemistry of specific sources, we may have additional constraints on $F$ (eg that particular entries are zero, or that they fall in a specific range or that they should be close to particular values).
• Based on knowledge of the emission of air pollution, we may have additional constraints on $G$ (eg, fireworks on New Year's Eve but not on other data; more trucks on weekdays than weekends)

Do any of these constraints count as 'natural' in your sense, or do you just mean constraints where there is some sort of barrier in the objective function, so that imposing constrained optimisation is not logically necessary (even if it might be practically important)?

Answer 2

Ideally, all optimisation problems should be well-defined by being clear about the domain over which the optimisation occurs. So really, all optimisation problems should be constrained to a clear domain for the objective function. One would hope that this is either done explicitly, or if it is implicit, the domain of optimisation should be clear from the context. Certainly there is a distinction between explicit constraints stipulated in the optimisation problem and implicit constraints that are imposed by recognising context.

There are certainly cases where a person stipulates an optimisation problem without being clear about the domain of the function they are working with. Sometimes users stipulate functions and take the domain to be obvious by consideration of the context. In the case you give, you are right that the use of the $\log x$ part implicitly suggests the intention to restrict attention to $x \geqslant 0$ (since negative values give complex outputs, which would then make it ambiguous as to what constitutes "optimisation" of the output).

The distinction you are referring to would usually just be called "explicit" versus "implicit" --- not "natural" versus "designed". For the most part, literature on optimisation is pretty careful in specifying the domain of optimisation (i.e., likely to give explicit statements of the domain), so the distinction between explicit and implicit constraints is more likely to arise outside of this literature.