Like in this picture:

I understand the first part, but what does the “subject to” part mean?

Help? Couldn’t find anything on google searching “sigma subject to”.

In: 1

You are trying to minimize a value “subject to a constraint” / “while being constrained” / “only while this thing is true” . In this example, you are not necessarily trying to find the lowest x in a set; you are trying to find the lowest x that *also* has the property that the x times its corresponding c = S.

If the x’s were [1, 2, 3, 4], the c’s were [1, 2, 4, 3], and S = 12, then the minimum value of x subject to the constraint would be x = 3. x = 1 and x = 2 are smaller x’s, but they don’t satisfy the constraint. x = 4 satisfies the constraint, but it’s not less than x = 3.

“Subject to” had nothing to do with Sigma. That Sigma is just the regular, standard summation symbol.

This problem is using the phrase “subject to” to express constraints. If you Google “constrained optimization,” you’ll find a bunch of relevant stuff, although it may not be approachable.

Say you want to find the largest marble in a bag of marbles. That’s an optimization problem: If B is your bag of marbles, find the x in B that maximizes size(x).

Say you want to find the largest **red** marble in a bag of marbles. That’s a *constrained* optimization problem: find the x in B that maximizes size(x), **subject to** color(x) = red. There might be values of x that are higher (and there might not), but you are *constrained* to which solutions you are willing to consider.

Constrained Optimization is important is real-world optimization problems. For example if you’re building a bridge, you’ll be trying to find the bridge design that minimizes cost, **subject to** minimum weight that it can hold. And other constraints like build time, geological constraints, maybe additional stuff like ecological impact… basically *everything* you might want to achieve can be modeled as a constraint.

In the mathematical formalization, if you have multiple constraints you might squish them together into a single equation. That’s probably what the Sigma (addition) is doing: combining a multi-dimensional constraint into a single equation.

In this case, it means a constraint on the “min” in the previous problem. You want to find the *x*’s that minimize the sum x0 + x1 + x2 + … + x_(n-1), but you’re only considering candidate options that satisfy the requirement that sum x_i * c_i = S. Other choices for x aren’t allowed.