# Calculable Programs (afterall)

## Simple dual

There is in simple arithmetics (could we call complex ones "arhythmetics" ?) something very essential in the concept of a program. It appears in optimization, calculus of an extrema, solution of a program submitted to constraints. Be this linear or non linear. Whichever forms, we seek its solution in a program. Our purpose is not to explain you all the mathematics of such programs, but to mind enough on how to pose your quantitative problems so. Intuitively it is not difficult. Such program seeks the maximum possible solution or minimum possible one, considering an equation of value, constraints and criteria. This is modelled by a function (mathematical) of you gain, like a benefit. The program will be defined as the way to make it optimum considering your constraints. Also since you do, with what you have, and what you have is not unlimited, you understand that there are constraints like the maximum of quantities you can use or, minima you should use.

Now, program-goal function can often be reverted taking other perspective: minimization when maximisation at first and at the reverse. Constraints can also reverse they approach turning less and equal when more or more and equal when less. Relation-combination is called primal-dual. Typically, what is interesting in such program is that you can switch formula and take the easiest formula expression of the function to find your solution. Example: very often to maximize the benefit of a program-situation is quite similar to the minimization of the costs. This is very helpful: switch constraints superior to inferior (the direction of inequalities) and the primal (max or min) to its dual (min or max) and you solve your program taking the easiest pathway (all that in optimization calculus).

Now if not the mathematician for that, what you still can do is:

- Learn how to set your common problems so, at best with a good computed interface and if your program enough well specified the machine will do it for you and you may not need a specialist. Mind too that if mathematician has the knowledge of formats that you need to input, you often know better the practical and realist flexibility on components and constraints of the problems, while the reasonable criterias are yours.
- Mind also that just able to picture the program on a plane graph, constraints and criteria are often lines or planes topping or underlying delimited space of production, by constraints and/ or possibilities. Visually some known point of a combination of extrema is easy to observe, it may be enough to join them, show of profit curve and how rights lines or plane match maximum combinations. Since complexity is there and what you managed is the graphical expression of your understanding. Your picture shows you the solution of the program, it helps better than formula, to discuss of management and of constraints.Visual solution can well have enough precision: considering the complexity of many practical stituations and empirical knowledge could make not very wise to be as rigorous as formulas look.
- This requires logical mind and knowledge that you are you used to, rather than many mathematical skills.

We have suggested a simple arithmetic sense of program, able you to design your own in a similar way. Graphic and geometric manipulations can do it for more complex purpose.

## Complex Primal Dual

Now, one step further is with complex program. This is more with our (intuitive) concept, you will find in literature some complex pictures but no "wholistic overview": confusion is not what mathematicians and logicians like to practice within their so rigorously purified world. Programming in mathematics can be much more sophisticated and we do not dicuss of that. Lesser naive profanes could like more detailed introduction. Complications can be managed in an intuitive and qualitative way. Imagine we want to set some complex notions about such sort of program. How complications would look like in simplified forms? - There the complex primal-dual.With our strange (among nature) human will, it could be not a complex-complex thing, eventually related by complex properties of complementarity, but how to care a complex program and issues (without being abused by labels) with programs of reductions up to an acceptable social model and that could not produce too unacceptable humane disaster.

Another important point has to do with specifications and measures, including mistakes prevention, but these will be treated in other pages.

May be there in a simple-complex program, the gordian knot of our policies. Not enough cared formally by our policies makers, especially those wanting one special result with simplest means; often uncaring that methods should be used under constraints they want to ignore.There are similar hard or complex concepts of program, already intuitive but still unexplained in practices of methodologists. For example robust statistics care a lot the effect of one special (value, outlier, etc.) on all other possible issues. This is the sort of thought we could mind in "complex programming". Like, for example what, as an invidual or a team work, could program or manage, considering as far as possible other citizens. Good for them and me adverse effect be minimum in a repairable way may turn more important than Max-maximum.

Indeed both picture of complex programs can receive some help from added humane criteria: democracy, precautionnary principle and the difficulty of intermediation: how it is ? - between individual (I or you) and all others (both, same - democratically and different - personnally). Some will have easy answers. We prefer to mind that there are points, criteria, in any complicated programs, to discuss democratically, technically while formally caring also efficacy with efficiency.

Nevertheless, if we cannot mind all the situations and the solutions (which feasability could vary, in one place or the other) and, do not want to forbid any potential solutions; there is something that we can nevertheless care. For example learn together to set program according democratic and humane frames. It will nevertheless help us to discuss points and too, attract the attention of people that could help us better in calculating what we could make together.