An iterative process for the feasibility-seeking problem with sets that are unions of convex sets
Yair Censor, Alexander J. Zaslavski
TL;DR
The paper tackles feasibility-seeking for unions of convex sets ($C_i=\cup_{j=1}^{m_i}C_{i,j}$) with a nonempty intersection $C=\cap_i C_i$, and introduces the Projections onto Unions of Convex Sets (PUCS) algorithm. To ensure unique projections onto UCS, the authors augment the problem with an extra UCS identical to $C_1$, and provide a convergence analysis under Condition 1 (e.g., singleton projections $P_{C_{i+1}}(x)$ for $x\in C_i$). They prove that, for each orbit not returning to its initial inner set, $\lim_{k\to\infty}\|y_r^k-y_r^{k+1}\|=0$ and $\lim_{k\to\infty}\mathrm{dist}(y_r^k,C_i)=0$, effectively solving a sequence of pruned convex feasibility problems in parallel. The work highlights a mathematically rigorous pathway to handle non-convex UCS feasibility with potential floorplanning applications and points to future work on identifying problem classes satisfying Condition 1 and practical performance evaluations.
Abstract
In this paper we deal with the feasibility-seeking problem for unions of convex sets (UCS) sets and propose an iterative process for its solution. Renewed interest in this problem stems from the fact that it was recently discovered to serve as a modeling approach in fields of applications and from the ongoing recent research efforts to handle non-convexity in feasibility-seeking.