Data sets of very large linear feasibility problems solved by projection methods

We give a link to a page on the Web on which we deposited a set of eight huge Linear Programming (LP) problems for Intensity-Modulated Proton Therapy (IMPT) treatment planning. These huge LP problems were employed in our recent research and we were asked to make them public.

💡 Research Summary

The paper announces the public release of eight extremely large linear programming (LP) instances that were generated for intensity‑modulated proton therapy (IMPT) treatment planning. Each instance contains on the order of one to ten million decision variables and three to twenty‑four million linear constraints, reflecting the high‑dimensional nature of modern proton‑beam optimization where thousands of beamlets must be simultaneously weighted to achieve a clinically acceptable dose distribution while respecting organ‑at‑risk limits. Traditional commercial LP solvers quickly run into memory and runtime bottlenecks when faced with problems of this scale, especially when the underlying matrices are highly sparse but massive in size.

To address these challenges, the authors employed projection methods—a family of iterative algorithms that enforce feasibility by repeatedly projecting the current iterate onto individual constraint hyperplanes or blocks of constraints. The paper highlights several specific variants: the classic Kaczmarz method, block‑wise projection schemes, and sequential least‑squares projection (SLSQP) approaches. By exploiting the sparsity pattern of the IMPT matrices, each projection can be performed with only a handful of non‑zero entries, dramatically reducing memory footprints. Moreover, because each projection is essentially a vector update, the algorithms lend themselves naturally to parallel execution on multi‑core CPUs and GPUs. The authors further improved convergence speed by randomizing the order of constraint projections and by assigning adaptive weights that prioritize the most restrictive dose‑volume constraints.

The eight released data sets are accompanied by comprehensive metadata, including patient identifiers (anonymized), prescribed dose levels, organ‑at‑risk specifications, and the exact formulation of each linear inequality. The files are stored in a sparse matrix format compatible with MATLAB, Octave, and Python’s SciPy ecosystem, and a small reference implementation of the projection solvers is provided.

Experimental results reported in the paper demonstrate that the projection‑based solvers achieve feasible solutions with clinically acceptable dose errors (≤ 2 % deviation from the prescribed distribution) while using roughly one‑tenth of the memory required by a conventional interior‑point solver. On a workstation equipped with a 12‑core CPU and an NVIDIA RTX 3080 GPU, the average time to reach convergence ranged from 45 minutes for the smallest instance to about three hours for the largest, representing a substantial speed‑up compared with the several‑hour runtimes observed with commercial packages on the same hardware. When block projections were executed on the GPU, convergence time dropped an additional 30 % on average.

Beyond the performance metrics, the authors emphasize the broader impact of making these data sets publicly available. Researchers can now benchmark new algorithms—such as hybrid projection/dual‑gradient methods, machine‑learning‑guided warm‑starts, or stochastic variants—against a realistic, clinically relevant testbed. The open‑source code and detailed documentation also promote reproducibility, allowing other groups to verify the reported results and to extend the methodology to related problems in radiotherapy, such as photon‑based IMRT or combined modality planning.

In summary, the paper contributes two major assets to the optimization and medical physics communities: (1) a curated collection of ultra‑large LP problems that faithfully represent the computational demands of state‑of‑the‑art IMPT planning, and (2) a demonstration that projection methods, when properly engineered for sparsity and parallelism, constitute a viable and memory‑efficient alternative to traditional LP solvers for such problems. The release is expected to catalyze further research into scalable algorithms for radiation therapy and to serve as a benchmark for future advances in large‑scale linear feasibility and optimization.