A problem is an objective(s), condition, or situation that demands change, correction, or solution. And a complex problem is a problem that has obvious patterns and relationships, that are conflicting, competing, or have dynamic objectives (or equilibrium states). States which may or may not be immediately identifiable; some of which might be recursive. The problem itself is also an equilibrium that satisfies the objectives defined to be the problem. Otherwise, it is only transient.
With such problems, the solution is not immediately obvious to the parties, and typically seeks to derive one equilibrium state that is designed or acceptable. It may be straightforwardly arrived at or straightforward in itself, with hindsight.
Making an otherwise ‘fixed’ problem state transient, such that it may evolve into a solution (or desirable) state marks a process of solving the problem. It is also possible to make the problem worse by shifting its equilibrium point along the wrong path. We typically find out that this is what we think we have done, with hindsight.
A complicated problem is marked by apparently convoluted and patternless relationships. It becomes less complicated when the understanding of the associated situations increases. (Understanding is defined to be the recognition of patterns and principles; the realisation of truth.)
Whether or not a problem is complex or complicated is a function of perception. The two are the same because there is nothing that exists, no situation, without pattern or principle. A solution to a complex problem, hence, whether or not optimal, is also a function of pattern and principle. It should implicitly require simplicity.
We may thus say that there are no complex or complicated problems for the one who can be God. Because his perception must be perfect and full, and his ability to effect optimal solutions, illimitable.
Still, we can conjecture that there are fundamentally no complex problems as it relates to an individual in his personal life, and that every solution required by mankind for himself is simple to him, however challenging it might be to discover or implement.
But complex problems are everywhere in our human society, and in the sciences and engineering. All demanding change, correction, or some sort of a solution.
If a solution to a complex or complicated problem isn’t simple, then the solution itself is a problem to be solved. And if this complex solution cannot be solved in simple(r) ‘terms,’ then, the chosen solution is sub-optimal, and may represent a problem for the future.
Practically, the best solutions are those whose relationships to simple effected elements can be simply or directly traced. A simple effected element is the most basic building block of the solution to a complex problem, whose effect can be traced to helping solve the problem with no (or little) side effects, and without contravening natural law.
When the full solution is solved, it can then be said to be beautiful. This is the reason why love, the natural world, and natural law are beautiful; beautiful solutions.