Interior point methods are not worse than Simplex

TL;DR

The paper presents a novel interior point method for linear programming that achieves near-optimal iteration behavior by navigating the central path using subspace layered least squares (SLLS) steps. It introduces the max central path as a combinatorial surrogate for the central path, and connects the number of iterations to the straight-line complexity (SLC) of the LP, showing an exponential upper bound on iterations via SLC while also providing strongly polynomial time per-iteration complexity. A key innovation is the use of cheap lift subspaces and SLLS directions to efficiently traverse polarized segments of the central path, with an amortized analysis showing the total iterations scale with sums of coordinate-wise straight-line complexities. The approach links path-following IPMs and shadow-vertex/polynomial LP structures, offering a framework that matches the iteration complexity of any path-following method up to a polynomial factor and enabling strong polynomial-time computation of the required linear-algebra subroutines, including a deterministic strongly polynomial approximate SVD solver. Overall, the work provides a principled, worst-case bound on IPM iteration counts that are exponential in dimension in the worst case (via SLC) while achieving practical, structure-exploiting steps that can be computed in strongly polynomial time, and it sharpens the understanding of the relationship between interior-point trajectories and simplex-like paths.

Abstract

We develop a new `subspace layered least squares' interior point method (IPM) for solving linear programs. Applied to an $n$-variable linear program in standard form, the iteration complexity of our IPM is up to an $O(n^{1.5} \log n)$ factor upper bounded by the \emph{straight line complexity} (SLC) of the linear program. This term refers to the minimum number of segments of any piecewise linear curve that traverses the \emph{wide neighborhood} of the central path, a lower bound on the iteration complexity of any IPM that follows a piecewise linear trajectory along a path induced by a self-concordant barrier. In particular, our algorithm matches the number of iterations of any such IPM up to the same factor $O(n^{1.5}\log n)$. As our second contribution, we show that the SLC of any linear program is upper bounded by $2^{n + o(1)}$, which implies that our IPM's iteration complexity is at most exponential. This in contrast to existing iteration complexity bounds that depend on either bit-complexity or condition measures; these can be unbounded in the problem dimension. We achieve our upper bound by showing that the central path is well-approximated by a combinatorial proxy we call the \emph{max central path}, which consists of $2n$ shadow vertex simplex paths. Our upper bound complements the lower bounds of Allamigeon, Benchimol, Gaubert, and Joswig (SIAGA 2018), and Allamigeon, Gaubert, and Vandame (STOC 2022), who constructed linear programs with exponential SLC. Finally, we show that each iteration of our IPM can be implemented in strongly polynomial time. Along the way, we develop a deterministic algorithm that approximates the singular value decomposition of a matrix in strongly polynomial time to high accuracy, which may be of independent interest.

Figures (1)

• Figure 1: The projection of the max central path on the primal coordinates $(x_1,x_2)$ for the cost function $x_1 + x_2$. Dashed lines correspond to level sets at breakpoints. Note that the max central path does not lie in the feasible region $\mathcal{P}$.

Theorems & Definitions (178)

• Theorem 1.1
• Lemma 1.1
• Definition 1.2: Straight-line complexity
• Theorem 1.3
• Lemma 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1: Linear Operator
• Remark 2.2