Capturing some thoughts while commuting to the office.
I’ve been asked countless times what exactly is Optimization and why is it different.
Mathematically, optimization is the term used for the minimization or maximization of a function.
This function is called the objective function. We will denote the objective function as F(x), where x is, in general, an n-vector of variables that we are trying to determine.
The variables denoted by x may have no limitations on their value, in which case the problem is called unconstrained.
A simple bound constrained problem merely imposes bounds on the range of values that the variables may take. For example the variables might be constrained to lie within the interval [-1,1], or may be constrained so that the values of x are all positive.
More often the values of x are constrained in some other way, typically constraint function G(x) are such that A<= G(x)<=B If the functions G(x) are linear (i.e. a weighted sum of the variables x) then the problem is linearly constrained; otherwise the problem is nonlinearly constrained.
It is also possible to meet constraints that require the solution to be integral i.e. only integer values are required. Such problems are termed integer programming problems.
A point which satisfies the constraints is called a feasible point.
High school chemistry illustrates Optimization rather elegantly; Na + Cl = NaCl.
Adding more parts of Na isn't going to yield more salt; you meet saturation!
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