[Paper Review] An Efficient Polynomial Interior-Point Algorithm for Linear Programming

Yaguang Yang|arXiv (Cornell University)|Apr 12, 2013

Advanced Optimization Algorithms Research15 references3 citations

TL;DR

This paper proposes a novel interior-point algorithm for linear programming that simultaneously optimizes the centering parameter and line-search step size to minimize the duality gap at each iteration. By integrating these selections systematically rather than heuristically, the algorithm achieves the best-known polynomial complexity bound and demonstrates superior computational efficiency on Netlib test problems, outperforming even the widely used Mehrotra predictor-corrector method in iteration count without requiring a corrector step.

ABSTRACT

For interior-point algorithms in linear programming, it is well-known that the selection of the centering parameter is crucial for proving polynomility in theory and for efficiency in practice. However, the selection of the centering parameter is usually by heuristics and separate from the selection of the line-search step size. The heuristics are quite different while developing practically efficient algorithms, such as MPC, and theoretically efficient algorithms, such as short-step path-following algorithm. This introduces a dilemma that some algorithms with the best-known polynomial bound are least efficient in practice, and some most efficient algorithms may not be polynomial. In this paper, we propose a systematic way to optimally select the centering parameter and line-search step size at the same time, and we show that the algorithm based on this strategy has the best-known polynomial bound and may be very efficient in computation for real problems.

Motivation & Objective

• To resolve the theoretical-practical gap in interior-point methods where algorithms with strong polynomial bounds are inefficient, and efficient algorithms lack polynomial complexity proofs.
• To eliminate the heuristic separation between centering parameter and line-search step size selection, which undermines both theoretical guarantees and practical efficiency.
• To develop a systematic, optimization-based strategy for selecting both parameters simultaneously to minimize duality gap reduction per iteration.
• To demonstrate that this unified selection strategy yields both the best-known polynomial complexity bound and high computational efficiency in practice.

Proposed method

• The algorithm formulates the joint selection of centering parameter α and line-search step size σ as an optimization subproblem to minimize the duality gap in each iteration.
• It derives closed-form expressions for the search directions using sparse QR decomposition of the KKT system, enabling accurate and efficient computation of primal and dual predictor steps.
• The method uses a primal-dual predictor-corrector framework but omits the corrector step, relying on optimal α and σ to achieve fast duality gap reduction.
• It employs a preprocessing step to compute a feasible initial point with x = s = e, ensuring the problem has a strict interior point as required by interior-point methods.
• The algorithm is implemented in MATLAB using a function called optimalAlphaSigma, which solves the subproblem at each iteration to determine α and σ.
• It leverages the structure of the KKT system to compute search directions via orthogonal projections, ensuring numerical stability and accuracy, especially near optimality.

Experimental results

Research questions

• RQ1 Can a systematic, joint optimization of the centering parameter and line-search step size lead to both improved theoretical complexity and practical efficiency in interior-point methods?
• RQ2 Does eliminating the heuristic separation between centering and line search selection result in faster convergence and lower iteration counts?
• RQ3 Can an algorithm without a corrector step still achieve high efficiency if the centering and step size are optimally selected?
• RQ4 Is the proposed method capable of achieving the best-known polynomial complexity bound while remaining competitive in real-world performance?

Key findings

• The proposed algorithm achieves the best-known polynomial complexity bound for linear programming, with a theoretical iteration count of O(√n log(1/ε)) for ε-optimal solutions.
• On Netlib test problems, the algorithm consistently requires fewer iterations than MATLAB's linprog (which uses Mehrotra's predictor-corrector method), including 4 iterations on AFIRO versus 7, and 5 on SCAGR25 versus 16.
• The algorithm outperforms linprog on all tested problems except one (SCTAP1), where both require 17 iterations, despite not using a corrector step.
• The method demonstrates that optimal selection of α and σ leads to significantly faster duality gap reduction per iteration, even without additional computational cost from corrector steps.
• Numerical results show that the algorithm is highly efficient in practice, with iteration counts competitive with or better than state-of-the-art implementations, despite its theoretical focus.