Convex separation from convex optimization for large-scale problems

TL;DR

This work introduces a practical, memory-efficient scheme to certify separation from any convex set S using an oracle that solves strong optimization problems. By reducing weak separation (WSEP) to weak minimum distance (WDIST) via Gilbert's algorithm and its memory/heuristic variants, the authors obtain convergence guarantees with favorable dependence on geometry (including curvature) and derive concrete complexity bounds. They validate the theory with extensive numerical experiments and showcase significant improvements in large-scale quantum-information tasks, including nonlocality and steering analyses for Werner and GHZ states, as well as bounds related to Grothendieck's constant. The approach enables solving high-dimensional separation problems that were previously intractable with traditional linear-programming methods, offering practical benefits for quantum information science and beyond.

Abstract

We present a scheme, based on Gilbert's algorithm for quadratic minimization [SIAM J. Contrl., vol. 4, pp. 61-80, 1966], to prove separation between a point and an arbitrary convex set $S\subset\mathbb{R}^{n}$ via calls to an oracle able to perform linear optimizations over $S$. Compared to other methods, our scheme has almost negligible memory requirements and the number of calls to the optimization oracle does not depend on the dimensionality $n$ of the underlying space. We study the speed of convergence of the scheme under different promises on the shape of the set $S$ and/or the location of the point, validating the accuracy of our theoretical bounds with numerical examples. Finally, we present some applications of the scheme in quantum information theory. There we find that our algorithm out-performs existing linear programming methods for certain large scale problems, allowing us to certify nonlocality in bipartite scenarios with upto $42$ measurement settings. We apply the algorithm to upper bound the visibility of two-qubit Werner states, hence improving known lower bounds on Grothendieck's constant $K_G(3)$. Similarly, we compute new upper bounds on the visibility of GHZ states and on the steerability limit of Werner states for a fixed number of measurement settings.

Figures (8)

• Figure 1: Geometrical description of Gilbert's algorithm.
• Figure 2: Geometrical meaning of curvature. The plot depicts a convex set $S$ with a distinguished boundary point $\bar{s}\in S$ with normal vector $\bar{h}$. $\bar{h}$ defines a linear approximation to the set via the inequality $\bar{h}\cdot(\bar{s}'-\bar{s})\leq0$ for all $\bar{s}'\in S$. If the set has a certified curvature $R$, this approximation can be improved via the quadratic witness defined by Eq. (\ref{['quadratic']}).
• Figure 3: Lower bounds on the speed of convergence. In the picture, the convex set $S$ is a rectangle. It can be seen that, for all $k$, $s_{2k}'$, $s_{2k+1}'$ are the right and left upper vertices of the rectangle, respectively.
• Figure 4: Numerical convergence for the rectangle. Blue: plot of $\log(d_{k}-d^{*})$ versus $\log(k)$ for the rectangle for $10^{5}$ iterations. Red: best linear fit. Except for the first iterations, both curves are virtually indistinguishable.
• Figure 5: Numerical convergence for the rectangle. Blue: plot of $\log(d_{k}-d^{*})$ versus $\log(k)$ for the rectangle for $10^{5}$ iterations. Red: best linear fit.

Theorems & Definitions (3)

• Definition
• Definition
• Definition