Solving Pooling Problems by LP and SOCP Relaxations and Rescheduling Methods

TL;DR

This paper introduces LP and SOCP relaxations for the pooling problem, demonstrating their equivalence to SDP relaxations and proposing a rescheduling method to improve solutions, with computational validation on various test instances.

Contribution

It establishes the equivalence of LP, SOCP, and SDP relaxations for the pooling problem and introduces an efficient rescheduling method to refine solutions.

Findings

LP and SOCP relaxations match SDP relaxation optimal values

Rescheduling method improves solution quality

Numerical tests confirm efficiency on diverse instances

Abstract

The pooling problem is an important industrial problem in the class of network flow problems for allocating gas flow in pipeline transportation networks. For P-formulation of the pooling problem with time discretization, we propose second order cone programming (SOCP) and linear programming (LP) relaxations and prove that they obtain the same optimal value as the semidefinite programming relaxation. The equivalence among the optimal values of the three relaxations is also computationally shown. Moreover, a rescheduling method is proposed to efficiently refine the solution obtained by the SOCP or LP relaxation. The efficiency of the SOCP and the LP relaxation and the proposed rescheduling method is illustrated with numerical results on the test instances from the work of Nishi in 2010, some large instances, and Foulds 3, 4, 5 test problems.