Automatic Reformulations for Convex Mixed-Integer Nonlinear Optimization: Perspective and Separability
Tight reformulations of combinatorial optimization problems like Convex Mixed-Integer Nonlinear Programs (MINLPs) enable one to solve these problems faster by obtaining tight bounds on optimal value. We consider two techniques for reformulation: perspective reformulation and separability detection. We develop routines for automatic detection of problem structures suitable for these reformulations, and implement new extensions. Since detecting all "on-off" sets for perspective reformulation in a problem can be as hard as solving the original problem, we develop heuristic methods to automatically identify them. The LP/NLP branch-and-bound method is strengthened via "perspective cuts" derived from these automatic routines. We also provide methods to generate tight perspective cuts at different nodes in the branch-and-bound tree. The second structure, i.e., separability of nonlinear functions, is detected by means of the computational graph of the function. Our routines have been implemented in the open-source Minotaur solver for general convex MINLPs. Computational results show an improvement of up to 45% in the solution time and the size of the branch-and-bound tree for convex instances from benchmark library MINLPLib. On instances where reformulation using function separability induces structures that are amenable to perspective reformulation, we observe an improvement of up to 88% in the solution time.
Convex MINLP
perspective reformulation
branch-and-bound
outer approximation
function separability
Mathematics of computing~Combinatorial optimization
Mathematics of computing~Solvers
Applied computing~Operations research
23:1-23:20
Regular Paper
http://www.optimization-online.org/DB_HTML/2022/04/8888.html
Meenarli
Sharma
Meenarli Sharma
Institute of Mathematics, University of Augsburg, Germany
https://orcid.org/0000-0002-5677-2822
Ashutosh
Mahajan
Ashutosh Mahajan
Industrial Engineering and Operations Research, Indian Institute of Technology Bombay, India
https://orcid.org/0000-0002-4602-7431
10.4230/LIPIcs.SEA.2022.23
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Meenarli Sharma and Ashutosh Mahajan
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