# Constrained optimization with multiple variables

I'm bringing a word problem/conceptual problem to the math stack exchange. Feel free to edit the title of this question if it does not reflect the following word problem/conceptual problem.

What I want to do is maximize the value called "health" subject to the constraint of budget. "Health" is determined by a number of "micronutrients". (Say twenty different kinds of micronutrients.) For each micronutrient, you can have a certain percentage of your daily value up to a 100%, and there is no penalty for exceeding the daily value but there is no "health" advantage for exceeding this value either.

In the grocery store there are "products" with attributes, aka "micronutrients". One product may have (for daily values) (90% vitamin A, 10% vitamin C, 0% calcium... and so on) another product may have different values. Each product has a price, and we can simplify by assuming this price is "per serving".

1) What I am interested in is finding how to describe the "healthiest" possible solution for a given average daily budget, algorithmically and as a general set of equations.

2) I am certain that there must be a description missing in my account for how "health" relates to the (twenty) micronutrients. I am interested in some general formulas that have been used elsewhere in mathematics that might be applicable to this situation too.

$$H = w_1 + w_2 + \ldots + w_{20}$$ where $w_i$ is the fraction of the RDA, but limited to a maximum of $1$, then this is a constrained optimization problem, and pretty well adapted to standard techniques like the simplex method.