By which criteria ???

Trivially you buy feta in the cheapest supermarket F and olives in the cheapest O ...

Or do you say going to multiple supermarkets isn't practical and you want to minimize the costs for time and fuel?

Or do you say you need a minimum amount f of feta and o of olives and you need the supermarket where you can buy the whole package to the best condition?

I guess the last one. Mainly from the point of view of discovering some kind of supermarket price policy (someone claimed that one day olives are up and fetta down, then the other day the opposite happen. you can replace fetta+olives with salt+pepper, tomato+cucumber, chicken+noodles and on to bigger groups, etc.etc.

Anyway, the interesting question is what you asked: can the intersection of M hyperplanes in N dimensions be solved as a linear algebra problem, perhaps by the simplex algorithm. Will it be faster?

I'm still confused about what you want, and I have trouble linking to the right pages. Likely a problem of terminology...

So from memory, and without guarantee that terminology corresponds to English mathematics

• Every hyperplane can be expressed as a linear equation derived from the normal vector
• hence multiple hyperplanes form a System of linear equations
• solutions to those equations are the intersections of the corresponding hyperplanes
• if you have less equations than variables you get a solution set in the form of a linear equation with free variables.
• the later is just another hyperplane /line of dimension of mentioned free variables, (a point is just a dimension 0 solution, i.e no free variables)
Matrix calculations:
• there are various numeric algorithms to solve equation systems represented as a matrix, like Gauss elimination. (Others might be better in terms of limiting rounding errors)
• they also produce a solution set for under defined systems (less rows than columns)

There are some generalizations applied when dealing with linear optimization methods:

• the equations are just the boundaries of a < "less than" inequalities, i.e. like all points "left" all boundaries are valid solutions,
• the intersections of those boundary equations are the corners of a convex polygon/polytope then, whose body represents solutions
• a further target/weight function might be given to choose the "optimal" solution in this solution set
• finally: sometimes only integer solutions are sought after, while the exact solution is non-integer. (You don't wanna buy a third of an olive)

I hope this helps you finding the words to express your questions

Again I'm rusted, and might have confused some (English) terminology.

Cheers Rolf
_{(addicted to the Perl Programming Language :)
see Wikisyntax for the Monastery}

> supermarket price policy

Don't expect those to be "rational", it's long known that people tend to drive many kilometers because they are baited by a 1€ discount on some beer or such.

Marketing is more about psychology.

Cheers Rolf
_{(addicted to the Perl Programming Language :)
see Wikisyntax for the Monastery}