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Algorithm 996: BBCPOP: A Sparse Doubly Nonnegative Relaxation of Polynomial Optimization Problems With Binary, Box, and Complementarity Constraints
- Algorithm 996: BBCPOP: A Sparse Doubly Nonnegative Relaxation of Polynomial Optimization Problems With Binary, Box, and Complementarity Constraints
- Ito, Naoki; Kim, Sunyoung; Kojima, Masakazu; Takeda, Akiko; Toh, Kim-Chuan
- Ewha Authors
- SCOPUS Author ID
- Issue Date
- Journal Title
- ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
- ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE vol. 45, no. 3
- MATLAB software package; high-degree polynomial optimization problems with binary; box and complementarity constraints; hierarchy of doubly nonnegative relaxations; sparsity; bisection and projection methods; tight lower bounds; efficiency
- ASSOC COMPUTING MACHINERY
- SCI; SCIE; SCOPUS
- Document Type
- The software package BBCPOP is a MATLAB implementation of a hierarchy of sparse doubly nonnegative relaxations of a class of polynomial optimization (minimization) problems (POPs) with binary, box, and complementarity (BBC) constraints. Given a POP in the class and a relaxation order, BBCPOP constructs a simple conic optimization problem (COP), which serves as a doubly nonnegative relaxation of the POP, and then solves the COP by applying the bisection and projection method. The COP is expressed with a linear objective function and constraints described as a single hyperplane and two cones, which are the Cartesian product of positive semidefinite cones and a polyhedral cone induced from the BBC constraints. BBCPOP aims to compute a tight lower bound for the optimal value of a large-scale POP in the class that is beyond the comfort zone of existing software packages. The robustness, reliability, and efficiency of BBCPOP are demonstrated in comparison to the state-of-the-art software SDP package SDPNAL+ on randomly generated sparse POPs of degree 2 and 3 with up to a few thousands variables, and ones of degree from 5 to 8 with up to a few hundred variables. Numerical results on BBC-constrained POPs that arise from quadratic assignment problems are also reported. The software package BBCPOP is available at https://sites.google.com/sit/bbcpop1/.
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