The Computational Complexity of Duality

TL;DR

This paper explores the computational complexity relationships between a convex set or function and its dual, showing polynomial-time reductions and equivalences in their membership and computation problems, with implications for convex optimization.

Contribution

It establishes polynomial-time reducibility between weak membership problems and dual computations for convex sets, functions, and norms, revealing their complexity equivalences.

Findings

Weak membership in a convex set is polynomial-time reducible to its dual.

Computing a norm or convex function's dual is polynomial-time reducible to the original.

NP-hardness of computing a convex function or norm is equivalent to that of its dual.

Abstract

We show that for any given norm ball or proper cone, weak membership in its dual ball or dual cone is polynomial-time reducible to weak membership in the given ball or cone. A consequence is that the weak membership or membership problem for a ball or cone is NP-hard if and only if the corresponding problem for the dual ball or cone is NP-hard. In a similar vein, we show that computation of the dual norm of a given norm is polynomial-time reducible to computation of the given norm. This extends to convex functions satisfying a polynomial growth condition: for such a given function, computation of its Fenchel dual/conjugate is polynomial-time reducible to computation of the given function. Hence the computation of a norm or a convex function of polynomial-growth is NP-hard if and only if the computation of its dual norm or Fenchel dual is NP-hard. We discuss implications of these results on the weak membership problem for a symmetric convex body and its polar dual, the polynomial approximability of Mahler volume, and the weak membership problem for the epigraph of a convex function with polynomial growth and that of its Fenchel dual.