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Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs [article]

Albert Atserias, Tuomas Hakoniemi
2019 arXiv   pre-print
This establishes size-degree trade-offs for SOS and PS that match their analogues for weaker proof systems such as Resolution, Polynomial Calculus, and the proof systems for the LP and SDP hierarchies  ...  We show that if a system of degree-k polynomial constraints on n Boolean variables has a Sums-of-Squares (SOS) proof of unsatisfiability with at most s many monomials, then it also has one whose degree  ...  We are grateful to Michal Garlik, Moritz Müller and Aaron Potechin for comments on an earlier version of this paper. We are also grateful to Jakob Nordström for  ... 
arXiv:1811.01351v2 fatcat:w73ejvsl3fepdpgqebajs6cldu

Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs

Albert Atserias, Tuomas Hakoniemi, Michael Wagner
2019 Computational Complexity Conference  
This establishes size-degree trade-offs for SOS and PS that match their analogues for weaker proof systems such as Resolution, Polynomial Calculus, and the proof systems for the LP and SDP hierarchies  ...  We show that if a system of degree-k polynomial constraints on n Boolean variables has a Sums-of-Squares (SOS) proof of unsatisfiability with at most s many monomials, then it also has one whose degree  ...  We are also grateful to Jakob Nordström for initiating a discussion on the several variants of the definition of monomial size as discussed in Section 2.  ... 
doi:10.4230/lipics.ccc.2019.24 dblp:conf/coco/AtseriasH19 fatcat:ega7hny7tngcbbl342szz7f3ye

On the construction of converging hierarchies for polynomial optimization based on certificates of global positivity [article]

Amir Ali Ahmadi, Georgina Hall
2018 arXiv   pre-print
of bisection and increase the number of variables (resp. degree) of the problem by the number of inequality constraints plus three (resp. by a factor of two).  ...  We remark that the scope of this paper is theoretical at this stage as our hierarchies-though they involve at most two sum of squares constraints or only basic arithmetic at each level-require the use  ...  We are grateful to Pablo Parrilo for very insightful comments, particularly as regards Section 4 and the observation that any form can be made even by only doubling the number of variables and the degree  ... 
arXiv:1709.09307v2 fatcat:bnks6dbefjb7hmatquefb2dy2q

Convex relaxations for quadratic distance problems

Andrea Garulli, Alfio Masi, Antonio Vicino
2008 2008 47th IEEE Conference on Decision and Control  
In this paper, several convex relaxations are presented for quadratic distance problems which are based on the sum-of squares representation of positive polynomials.  ...  Convex relaxations of nonconvex problems are a powerful tool for the analysis and design of control systems.  ...  Moreover, the only source of conservatism for such relaxations is due to the gap between positive semidefinite forms and sums-of-squares.  ... 
doi:10.1109/cdc.2008.4739051 dblp:conf/cdc/GarulliMV08 fatcat:dulogqhbnjgsrhy5fbllspwhki

Optimization over Nonnegative and Convex Polynomials With and Without Semidefinite Programming [article]

Georgina Hall
2018 arXiv   pre-print
In the first part of this thesis, we present two methods for approximately solving large-scale sum of squares programs that dispense altogether with semidefinite programming and only involve solving a  ...  squares hierarchies (e.g., due to Lasserre and Parrilo).  ...  Note that these three Positivstellensätze involve in their expressions sum of squares polynomials of unspecified degree.  ... 
arXiv:1806.06996v1 fatcat:ywvkxguvobh43jvdaedk2tf3ju

Polynomial invariant generation for non-deterministic recursive programs

Krishnendu Chatterjee, Hongfei Fu, Amir Kafshdar Goharshady, Ehsan Kafshdar Goharshady
2020 Proceedings of the 41st ACM SIGPLAN Conference on Programming Language Design and Implementation  
More specifically, in Step 3, we are applying a special case of Putinar's positivstellensatz, in which the sum-of-square polynomials can have a degree of at most ϒ. Example 8.  ...  Then д(x) > 0 for all x ∈ Π if and only if: д = ϵ + h 0 + m i=1 h i · д i (2) where ϵ > 0 is a real number and each polynomial h i is the sum of squares of some polynomials in R[X ]. Proof.  ...  [Farzan and Kincaid 2015] . Hence, there is currently a trade-off between exactness (completeness guarantees) and efficiency.  ... 
doi:10.1145/3385412.3385969 dblp:conf/pldi/Chatterjee0GG20 fatcat:4bk64r4cxvexfnue4dbguwas4u

The copositive way to obtain certificates of non-negativity over general semialgebraic sets [article]

Olga Kuryatnikova, Juan C. Vera, Luis F. Zuluaga
2021 arXiv   pre-print
Also, we derive a general Positivstellensatz which allows us to prove the existence of certificates of non-negativity for any semialgebraic compact set, based on any class of non-negative polynomials such  ...  Certificates of non-negativity are fundamental tools in optimization, and they underlie powerful algorithmic techniques for various types of optimization problems.  ...  Acknowledgements The work of the last author was supported by NSF CMMI grant 1300193.  ... 
arXiv:1909.06689v3 fatcat:escd7z62uvcztajehivcaxke24

Polynomial Invariant Generation for Non-deterministic Recursive Programs [article]

Krishnendu Chatterjee and Hongfei Fu and Amir Kafshdar Goharshady and Ehsan Kafshdar Goharshady
2020 arXiv   pre-print
We consider the classical problem of invariant generation for programs with polynomial assignments and focus on synthesizing invariants that are a conjunction of strict polynomial inequalities.  ...  To the best of our knowledge, this is the first invariant generation method to provide completeness guarantees for invariants consisting of polynomial inequalities.  ...  Hence, there is currently a trade-off between accuracy (completeness guarantees) and efficiency.  ... 
arXiv:1902.04373v3 fatcat:bduo7cdmerb4rcs7v3vifn3voa

Algorithmic Thresholds for Refuting Random Polynomial Systems [article]

Jun-Ting Hsieh, Pravesh K. Kothari
2021 arXiv   pre-print
We prove a lower bound in the restricted low-degree polynomial model of computation which suggests that this trade-off between SoS degree and the number of equations is nearly tight for all d.  ...  We also confirm the predictions of this lower bound in a limited setting by showing a lower bound on the canonical degree-4 sum-of-squares relaxation for refuting random quadratic polynomials.  ...  Finally, we would like to thank Sidhanth Mohanty and Jeff Xu for discussions on low-degree hardness and SoS lower bounds in general.  ... 
arXiv:2110.08677v1 fatcat:snhosyteenbvrlrryh4o5tgjee

Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions

Miguel Bernal, Antonio Sala, Abdelhafidh Jaadari, Thierry-Marie Guerra
2011 Fuzzy sets and systems (Print)  
Fuzzy Lyapunov functions have been fruitfully used in literature for local analysis of Takagi-Sugeno models, a particular class of the polynomial fuzzy ones.  ...  In this paper, stability of continuous-time polynomial fuzzy models by means of a polynomial generalization of fuzzy Lyapunov functions is studied.  ...  Simulation shows that x  4.15 is the maximum admissible value for x Table 1 : 1 Comparing polynomial Lyapunov functions versus polynomial fuzzy Lyapunov functions in Example 3: maximum size of a square  ... 
doi:10.1016/j.fss.2011.07.008 fatcat:zrj3dtyerze4tantjh2c2bae2u

Control and verification of high-dimensional systems with DSOS and SDSOS programming

Anirudha Majumdar, Amir Ali Ahmadi, Russ Tedrake
2014 53rd IEEE Conference on Decision and Control  
of a 50 node network of oscillators, searching for degree 3 controllers and degree 8 Lyapunov functions for an Acrobot system (with the resulting controller validated on a hardware platform), and a balancing  ...  In this paper, we consider linear programming (LP) and second order cone programming (SOCP) based alternatives to sum of squares (SOS) programming and apply this framework to high-dimensional problems  ...  of the code that sets up DSOS and SDSOS programs.  ... 
doi:10.1109/cdc.2014.7039413 dblp:conf/cdc/MajumdarAT14 fatcat:jxh2f3l3mza7ponnwjjprxrk4m

Spatio-Temporal Decomposition of Sum-of-Squares Programs for the Region of Attraction and Reachability [article]

Vít Cibulka, Milan Korda, Tomáš Haniš
2021 arXiv   pre-print
This paper presents a method for calculating Region of Attraction of a target set (not necessarily an equilibrium) for controlled polynomial dynamical systems, using a hierarchy of semidefinite programming  ...  Our approach builds on previous work and addresses its main issue, the fast-growing memory demands for solving large-scale SDPs.  ...  Practical implications The ROA with splits is expected to improve accuracy of the original formulation by allowing one to trade off the degree of the polynomials for number of splits.  ... 
arXiv:2103.03531v1 fatcat:kp4ys66lmzdcvmcknefdly33di

DSOS and SDSOS Optimization: More Tractable Alternatives to Sum of Squares and Semidefinite Optimization [article]

Amir Ali Ahmadi, Anirudha Majumdar
2018 arXiv   pre-print
In this paper, we introduce DSOS and SDSOS optimization as linear programming and second-order cone programming-based alternatives to sum of squares optimization that allow one to trade off computation  ...  In recent years, optimization theory has been greatly impacted by the advent of sum of squares (SOS) optimization.  ...  We would like to thank Pablo Parrilo for acquainting us with scaled diagonally dominant matrices, and Georgina Hall for many corrections and simplifications on the first draft of this work.  ... 
arXiv:1706.02586v3 fatcat:a7kmmfs435ejbb5mtufd7jitli

Sums of squares based approximation algorithms for MAX-SAT

H. van Maaren, L. van Norden, M.J.H. Heule
2008 Discrete Applied Mathematics  
We investigate the Semidefinite Programming based sums of squares (SOS) decomposition method, designed for global optimization of polynomials, in the context of the (Maximum) Satisfiability problem.  ...  Mach. 42(6) (1995) 1115-1145] and Feige and Goemans [Approximating the value of two prover proof systems, with applications to MAX2SAT and MAXDICUT, in: which are based on Semidefinite Programming as well  ...  Introduction Hilbert's Positivstellensatz states that a non-negative polynomial in R[x 1 , . . . , x n ] is a sums of squares (SOS) in case n = 1, or has degree 2, or n = 2 and the degree is 4.  ... 
doi:10.1016/j.dam.2007.08.036 fatcat:snh2mjyiofh2npmqkta3s2amoy

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs [chapter]

Albert Atserias
2013 Lecture Notes in Computer Science  
For example, resolution admits polynomial-size proofs of the least-number principle (every finite linear order has a least element) [40] , which underlies many inductive proofs.  ...  And I will close with bounded-degree semialgebraic proof systems, whose proof-search problem turned out to hide the complexity of systems of linear equations over finite fields, among other problems.  ...  Acknowledgments We thank the comments of Allen Van Gelder and an anonymous referee on the preliminary draft of this paper.  ... 
doi:10.1007/978-3-642-39071-5_1 fatcat:4vpknc2xqrbe3fu7hw3nzi7t5m
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