Is this function of a sum of indicator functions convex

convex optimizationconvex-analysisnon-convex-optimizationnonlinear optimizationoptimization

let $x=x_1,x_2,…,x_n$ for $x \ge 0$,

let $I(x) =max(x_i, 0) $ for $i \in [1,n]$

$f(x) = I(c-x)a^T+I(x-c)b^T$ where $a,b,c$ are constant vectors

Is this a convex function? How is this proven?

Best Answer

Yes, $f$ is a convex function. You just need to know some some basic properties that imply convexity:

  • Linear and convex functions are convex.
  • The maximum of convex functions is convex.
  • The composition of a convex function with an affine map is convex.
  • Nonnegative linear combinations of convex functions are convex.

From that we have:

  • The functions $x_i$ and $0$ are convex.
  • The components of $I(x)$, namely $\max(x_i,0)$ are convex.
  • The components of $I(c-x)$ and $I(x-c)$ are convex.
  • The nonnegative combinations $I(c-x)a^T$ and $I(x-c)b^T$ are convex, provided $a,b\ge0$. Likewise $f$ itself is convex.
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