Why is the sum of smallest components of a vector a concave function?

Raiden Barr

Raiden Barr

Answered question

2022-10-23

Why is the sum of smallest components of a vector a concave function?

Answer & Explanation

Ostrakodec3

Ostrakodec3

Beginner2022-10-24Added 18 answers

The sum of a k specific components of a vector is a linear function. For example, the sum of the first two components would be:
i = 0 1 x i = a T x where a T = [ 1 1 0 0 0 ]
The set of all possible sums of k components is thus a finite set of linear functions. The sum of the largest components is the pointwise supremum of this set. The Pointwise supremum sup k f k ( x ) preserves convexity if f k is convex for all k , thus the sum of the k largest components of a vector is convex.
To extend this proof for the k smallest components, simply negate the vector.

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