Hi Wander.
Lineal means linear relations between variables. Also implies that you have all the lineal algebra tools for solve it.
Nonlineal means complications. There are some few theory about nonlinear systems, but in the practice what can be done is linearize the system near points of interest. This is taking tangent planes which linearize the system. Newton-Raphson method do that: substitute the curve or the surface by the lineal equivalent :a rect for one variable, planes for two.
Programming mean nothing. It's just the cooknie from economists and similar professionals for call the function or system of functions for minimize.
Optimization means find minimal's (or max's: if min(f) = ym, then max(-f) = -ym, so, it is the same problem). Minimals appear in critical points, which are the set of borders, non-domain (isolated or not) discontinuities, and where diff (or grad) is zero.
Borders are very important no not forgot: the min for f(x) = 3 x -1 subject to x>5 is 14. Also, remember to distinguish between where the min occur (x=5) and which value the min it is (14). There are also two kinds of min's: extremes (the point isn't in the domain set, as this example) and true mins (point is in the set). Is not casual that the simbol for the border of a set B is dB (well, delta, in Greek). For some few figures there are different words for each one: circle and circumference, for example.
Restrictions means restrictions to the "normal" domain of the independent variables (the x's). Suppose that you want to study the better place for the sound in a cinema. So your restrictions are the walls: if there are a better solution outside de cinema, isn't very well solution.
In other situations, domains are not rectangles, but any domain in the (x,y) plane, like circles, paraboles, etc. That's a topology question, and actually it is very complicated (for example, what happen with holes). This is more interesting if you go to study complex variable calculus.
Usual situations for "programming" problems are, like your example, one function of multiple variables, system of functions of several variables, and differential equations system.
In your first example, I guess that you mixes two notations: f(x,y) = f(X), where X[1]=x, X[2]=y. So, X[3] in your example is actually a mistake. In matchad can use both for minimizing.
Werner present the well construct for your second example.
About mixing differential equations searching minimals, it is more delicate questions, for a lot of reasons. For instance, numerical methods are very sensible (inexact's) in the borders, and we see that they are very important. Other important thing about minimizing diff systems of equations, is that diff is a punctual property, so can vary a lot from points very closed. But, in the other hand, integrals are much more stables, are a interval property of the function, and, at least, is always continue "by the left". So, for systems more or less complicated, (and in your example seems very complicated one) the better way is transform the diff equations into integral equations. Benefits of that, is stability in point variations, and in the borders. (remember that the integral for a<x<b is the same of a<=x<=b, this is, can take out the borders and not change the value of the integral, that's stability in the borders).
Best regards.
Alvaro.
