Exact Optimization via Sums of Nonnegative Circuits and Sums of AM/GM Exponentials

We provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision algorithm for polynomials lying in the interior of the SAGE cone. Each framework, inspired by previous contributions of Parrilo and Peyrl, is a rounding-projection procedure. For a polynomial lying in the interior of the SAGE cone, we prove that the decision algorithm terminates within a number of arithmetic operations, which is polynomial in the degree and number of terms of the input, and singly exponential in the number of variables. We also provide experimental comparisons regarding the implementation of the two optimization algorithms.

[1]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[2]  Masakazu Muramatsu,et al.  Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity , 2004 .

[3]  Tobias Nipkow,et al.  A FORMAL PROOF OF THE KEPLER CONJECTURE , 2015, Forum of Mathematics, Pi.

[4]  Mohab Safey El Din,et al.  On Exact Polya, Hilbert-Artin and Putinar's Representations , 2018, ArXiv.

[5]  Timo de Wolff,et al.  A Positivstellensatz for Sums of Nonnegative Circuit Polynomials , 2016, SIAM J. Appl. Algebra Geom..

[6]  V. Chandrasekaran,et al.  Newton Polytopes and Relative Entropy Optimization , 2018, Foundations of Computational Mathematics.

[7]  Clarence Zener,et al.  Geometric Programming : Theory and Application , 1967 .

[8]  Pablo A. Parrilo,et al.  Computing sum of squares decompositions with rational coefficients , 2008 .

[9]  Stephen P. Boyd,et al.  CVXPY: A Python-Embedded Modeling Language for Convex Optimization , 2016, J. Mach. Learn. Res..

[10]  Timo de Wolff,et al.  An Experimental Comparison of SONC and SOS Certificates for Unconstrained Optimization , 2018, ArXiv.

[11]  Jean B. Lasserre,et al.  A bounded degree SOS hierarchy for polynomial optimization , 2015, EURO J. Comput. Optim..

[12]  Dima Grigoriev,et al.  Solving Systems of Polynomial Inequalities in Subexponential Time , 1988, J. Symb. Comput..

[13]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[14]  James Renegar,et al.  A faster PSPACE algorithm for deciding the existential theory of the reals , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[15]  S. Basu,et al.  A New Algorithm to Find a Point in Every Cell Defined by a Family of Polynomials , 1998 .

[16]  Mehdi Ghasemi,et al.  Lower Bounds for a Polynomial on a basic closed semialgebraic set using geometric programming , 2013 .

[17]  George A. Constantinides,et al.  Certified Roundoff Error Bounds Using Semidefinite Programming , 2015, ACM Trans. Math. Softw..

[18]  Thorsten Theobald,et al.  Exploiting Symmetries in SDP-Relaxations for Polynomial Optimization , 2011, Math. Oper. Res..

[19]  James Renegar,et al.  A mathematical view of interior-point methods in convex optimization , 2001, MPS-SIAM series on optimization.

[20]  Jean B. Lasserre,et al.  Sparse-BSOS: a bounded degree SOS hierarchy for large scale polynomial optimization with sparsity , 2016, Mathematical Programming Computation.

[21]  Stephen P. Boyd,et al.  A tutorial on geometric programming , 2007, Optimization and Engineering.

[22]  A. Ivic.  Sums of squares , 2020, An Introduction to 𝑞-analysis.

[23]  S. Pearson.  Moments , 2020, Narrative inquiry in bioethics.

[24]  Timo de Wolff,et al.  Amoebas, nonnegative polynomials and sums of squares supported on circuits , 2014, 1402.0462.

[25]  Victor Magron,et al.  Formal Proofs for Nonlinear Optimization , 2014, J. Formaliz. Reason..

[26]  W. Böge,et al.  Quantifier Elimination for Real Closed Fields , 1985, AAECC.

[27]  O. Taussky.  Sums of Squares , 1970 .

[28]  Stephen P. Boyd,et al.  ECOS: An SOCP solver for embedded systems , 2013, 2013 European Control Conference (ECC).

[29]  J. Harrison,et al.  Efficient and accurate computation of upper bounds of approximation errors , 2011, Theor. Comput. Sci..

[30]  Mohab Safey El Din,et al.  On Exact Polya and Putinar's Representations , 2018, ISSAC.

[31]  Maurice Mignotte,et al.  Mathematics for computer algebra , 1991 .

[32]  Mohab Safey El Din,et al.  Algorithms for Weighted Sums of Squares Decomposition of Non-negative Univariate Polynomials , 2017, ArXiv.

[33]  James G. Oxley,et al.  Matroid theory , 1992 .

[34]  David Joyner,et al.  Open source computer algebra systems: SymPy , 2012, ACCA.

[35]  Marc Giusti,et al.  Generalized polar varieties: geometry and algorithms , 2005, J. Complex..

[36]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[37]  Mehdi Ghasemi,et al.  Lower Bounds for Polynomials Using Geometric Programming , 2012, SIAM J. Optim..

[38]  Jean B. Lasserre,et al.  Convergent SDP-Relaxations in Polynomial Optimization with Sparsity , 2006, SIAM J. Optim..

[39]  Mohab Safey El Din,et al.  Realcertify: a maple package for certifying non-negativity , 2018, ACCA.

[40]  Parikshit Shah,et al.  Relative Entropy Relaxations for Signomial Optimization , 2014, SIAM J. Optim..

[41]  Timo de Wolff,et al.  An approach to constrained polynomial optimization via nonnegative circuit polynomials and geometric programming , 2016, J. Symb. Comput..

[42]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.