eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-08-27
82:1
82:17
10.4230/LIPIcs.MFCS.2018.82
article
Optimization over the Boolean Hypercube via Sums of Nonnegative Circuit Polynomials
Dressler, Mareike
1
Kurpisz, Adam
2
de Wolff, Timo
3
Goethe-Universität, FB 12 - Institut für Mathematik, Robert-Mayer-Str. 6-10, 60054 Frankfurt am Main, Germany
ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland
Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany
Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems are based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS based certificates remain valid: First, there exists a SONC certificate of degree at most n+d for polynomials which are nonnegative over the n-variate boolean hypercube with constraints of degree d. Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate, that includes at most n^O(d) nonnegative circuit polynomials. Finally, we show certain differences between SOS and SONC cones: we prove that, in contrast to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar's Positivestellensatz. We discuss these results both from algebraic and optimization perspective.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol117-mfcs2018/LIPIcs.MFCS.2018.82/LIPIcs.MFCS.2018.82.pdf
nonnegativity certificate
hypercube optimization
sums of nonnegative circuit polynomials
relative entropy programming
sums of squares