Numerical Stability and Scalability of Secure Private Linear Programming

Astract:

Linear programming (LP) has numerous applications in different fields. In some scenarios, e.g. supply chain master planning (SCMP), the goal is solving linear programs involving multiple parties reluctant to sharing their private information. In this case, methods from the area of secure multi-party computation (SMC) can be used. Secure multi-party versions of LP solvers have been known to be impractical due to high communication complexity. To overcome this, solutions based on problem transformation have been put forward. In this thesis, one such algorithm, proposed by Dreier and Kerschbaum, is discussed, implemented, and evaluated with respect to numerical stability and scalability. Results obtained with different parameter sets and different test cases are presented and some problems are exposed. It was found that the algorithm has some unforeseen limitations, particularly when implemented within the bounds of normal primitive data types. Random numbers generated during the protocol have to be extremely small so as to not cause problems with overflows after a series of multiplications. The number of peers participating additionally limits the size of numbers. A positive finding was that results produced when none of the aforementioned problems occur are generally quite accurate. We discuss a few possibilities to overcome some of the problems with an implementation using arbitrary precision numbers.