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

2019
*
arXiv
*
pre-print

This establishes

arXiv:1811.01351v2
fatcat:w73ejvsl3fepdpgqebajs6cldu
*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*...##
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Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs

2019
*
Computational Complexity Conference
*

This establishes

doi:10.4230/lipics.ccc.2019.24
dblp:conf/coco/AtseriasH19
fatcat:ega7hny7tngcbbl342szz7f3ye
*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. ...##
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On the construction of converging hierarchies for polynomial optimization based on certificates of global positivity
[article]

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*...

##
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Convex relaxations for quadratic distance problems

2008
*
2008 47th IEEE Conference on Decision and Control
*

In this paper, several convex relaxations are presented

doi:10.1109/cdc.2008.4739051
dblp:conf/cdc/GarulliMV08
fatcat:dulogqhbnjgsrhy5fbllspwhki
*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*. ...##
###
Optimization over Nonnegative and Convex Polynomials With and Without Semidefinite Programming
[article]

2018
*
arXiv
*
pre-print

In the first part

arXiv:1806.06996v1
fatcat:ywvkxguvobh43jvdaedk2tf3ju
*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*. ...##
###
Polynomial invariant generation for non-deterministic recursive programs

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

doi:10.1145/3385412.3385969
dblp:conf/pldi/Chatterjee0GG20
fatcat:4bk64r4cxvexfnue4dbguwas4u
*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. ...##
###
The copositive way to obtain certificates of non-negativity over general semialgebraic sets
[article]

2021
*
arXiv
*
pre-print

Also, we derive a general

arXiv:1909.06689v3
fatcat:escd7z62uvcztajehivcaxke24
*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. ...##
###
Polynomial Invariant Generation for Non-deterministic Recursive Programs
[article]

2020
*
arXiv
*
pre-print

We consider the classical problem

arXiv:1902.04373v3
fatcat:bduo7cdmerb4rcs7v3vifn3voa
*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. ...##
###
Algorithmic Thresholds for Refuting Random Polynomial Systems
[article]

2021
*
arXiv
*
pre-print

We prove a lower bound in the restricted low-

arXiv:2110.08677v1
fatcat:snhosyteenbvrlrryh4o5tgjee
*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. ...##
###
Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions

2011
*
Fuzzy sets and systems (Print)
*

Fuzzy Lyapunov functions have been fruitfully used in literature

doi:10.1016/j.fss.2011.07.008
fatcat:zrj3dtyerze4tantjh2c2bae2u
*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*...##
###
Control and verification of high-dimensional systems with DSOS and SDSOS programming

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. ...

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

2021
*
arXiv
*
pre-print

This paper presents a method

arXiv:2103.03531v1
fatcat:kp4ys66lmzdcvmcknefdly33di
*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. ...##
###
DSOS and SDSOS Optimization: More Tractable Alternatives to Sum of Squares and Semidefinite Optimization
[article]

2018
*
arXiv
*
pre-print

In this paper, we introduce DSOS

arXiv:1706.02586v3
fatcat:a7kmmfs435ejbb5mtufd7jitli
*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. ...##
###
Sums of squares based approximation algorithms for MAX-SAT

2008
*
Discrete Applied Mathematics
*

We investigate the Semidefinite Programming based

doi:10.1016/j.dam.2007.08.036
fatcat:snh2mjyiofh2npmqkta3s2amoy
*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. ...##
###
The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs
[chapter]

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. ...

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