scholarly journals Convexity in SemiAlgebraic Geometry and Polynomial Optimization

2009 ◽  
Vol 19 (4) ◽  
pp. 1995-2014 ◽  
Author(s):  
Jean B. Lasserre
Keyword(s):  
Polynomial Optimization ◽  
Semialgebraic Geometry

scholarly journals Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

2021 ◽  
Author(s):  
Mareike Dressler ◽  
Adam Kurpisz ◽  
Timo de Wolff
Keyword(s):  
Computer Science ◽  
Affine Transformation ◽  
Optimization Problems ◽  
Polynomial Optimization ◽  
Theoretical Computer Science ◽  
Sums Of Squares ◽  
Theoretical Computer ◽  
Complexity Bounds ◽  
Polynomial Optimization Problems ◽  
Transformation Of Variables

AbstractVarious 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 is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems 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, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most $$n+d$$ n + 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 $$n^{O(d)}$$ n O ( d ) nonnegative circuit polynomials. Moreover, we prove that, in opposite 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 Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

scholarly journals Efficiency improvement in an nD systems approach to polynomial optimization

2007 ◽  
Vol 42 (1-2) ◽  
pp. 30-53 ◽  
Author(s):  
Ivo Bleylevens ◽  
Ralf Peeters ◽  
Bernard Hanzon
Keyword(s):  
Systems Approach ◽  
Polynomial Optimization ◽  
Efficiency Improvement

scholarly journals Finding unstable periodic orbits: A hybrid approach with polynomial optimization

2021 ◽  
Vol 427 ◽  
pp. 133009
Author(s):  
Mayur V. Lakshmi ◽  
Giovanni Fantuzzi ◽  
Sergei I. Chernyshenko ◽  
Davide Lasagna
Keyword(s):  
Periodic Orbits ◽  
Polynomial Optimization ◽  
Hybrid Approach ◽  
Unstable Periodic Orbits

scholarly journals Integer Polynomial Optimization in Fixed Dimension

2006 ◽  
Vol 31 (1) ◽  
pp. 147-153 ◽  
Author(s):  
Jesús A. De Loera ◽  
Raymond Hemmecke ◽  
Matthias Köppe ◽  
Robert Weismantel
Keyword(s):  
Polynomial Optimization ◽  
Fixed Dimension ◽  
Integer Polynomial

scholarly journals Solving Fractional Polynomial Problems by Polynomial Optimization Theory

2018 ◽  
Vol 25 (10) ◽  
pp. 1540-1544 ◽  
Author(s):  
Andrea Pizzo ◽  
Alessio Zappone ◽  
Luca Sanguinetti
Keyword(s):  
Polynomial Optimization ◽  
Optimization Theory ◽  
Fractional Polynomial
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