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Semidefinite bounds for the stability number of a graph via sums of squares of polynomials

2006
*
Mathematical programming
*

Two other hierarchies

doi:10.1007/s10107-006-0062-8
fatcat:ltq2utolynbgbpvepydhsjnopi
*of**semidefinite**bounds**for**the**stability**number*have been proposed by Lasserre (SIAM J. ... Optim. 1:166-190, 1991) have constructed*semidefinite*relaxations*for**the*stable set polytope*of**a**graph*G = (V, E) by*a*sequence*of*lift-and-project operations; their procedure finds*the*stable set polytope ...*The*authors thank Etienne de Klerk*for*several valuable discussions about*the*topic*of*this paper and two referees*for*their careful reading and useful suggestions. ...##
###
Semidefinite Bounds for the Stability Number of a Graph via Sums of Squares of Polynomials
[chapter]

2005
*
Lecture Notes in Computer Science
*

Two other hierarchies

doi:10.1007/11496915_11
fatcat:rdw75rtmfvfntkx77hvgugil44
*of**semidefinite**bounds**for**the**stability**number*have been proposed by Lasserre (SIAM J. ... Optim. 1:166-190, 1991) have constructed*semidefinite*relaxations*for**the*stable set polytope*of**a**graph*G = (V, E) by*a*sequence*of*lift-and-project operations; their procedure finds*the*stable set polytope ...*The*authors thank Etienne de Klerk*for*several valuable discussions about*the*topic*of*this paper and two referees*for*their careful reading and useful suggestions. ...##
###
Optimization over structured subsets of positive semidefinite matrices via column generation

2017
*
Discrete Optimization
*

We then apply these techniques to approximate

doi:10.1016/j.disopt.2016.04.004
fatcat:fqlpa7ohdngm7opzgy3qe7nxtm
*the**sum**of**squares*cone in*a*nonconvex*polynomial*optimization setting, and*the*copositive cone*for**a*discrete optimization problem. ... We develop algorithms*for*inner approximating*the*cone*of*positive*semidefinite*matrices*via*linear programming and second order cone programming. ... Acknowledgments We are grateful to Anirudha Majumdar*for*insightful discussions and*for*his help with some*of**the*numerical experiments in this paper. ...##
###
Towards scalable algorithms with formal guarantees for Lyapunov analysis of control systems via algebraic optimization

2014
*
53rd IEEE Conference on Decision and Control
*

In this paper, we give

doi:10.1109/cdc.2014.7039734
dblp:conf/cdc/AhmadiP14
fatcat:kaahigfyavfsta47vwldaiht64
*a*brief overview*of*our recent research efforts (with various coauthors) to (i) enhance*the*scalability*of**the*algorithms in this field, and (ii) understand their worst case performance ... Exciting recent developments at*the*interface*of*optimization and control have shown that several fundamental problems in dynamics and control, such as*stability*, collision avoidance, robust performance ...*For*example, consider*the*following sets: •*The*cone*of**polynomials*that are*sums**of*4-th powers*of**polynomials*: {p| p = q 4 i }, •*The*set*of**polynomials*that are*a**sum**of*three*squares**of**polynomials*...##
###
Optimization over Structured Subsets of Positive Semidefinite Matrices via Column Generation
[article]

2016
*
arXiv
*
pre-print

We then apply these techniques to approximate

arXiv:1512.05402v2
fatcat:mynygd7w55hrvb6bq3evstmxq4
*the**sum**of**squares*cone in*a*nonconvex*polynomial*optimization setting, and*the*copositive cone*for**a*discrete optimization problem. ... We develop algorithms*for*inner approximating*the*cone*of*positive*semidefinite*matrices*via*linear programming and second order cone programming. ... These*polynomials**for*us were always either*a*single*square*or*a**sum**of**squares**of**polynomials*. There are*polynomials*, however, that are nonnegative but not representable as*a**sum**of**squares*. ...##
###
Certifying non-existence of undesired locally stable equilibria in formation shape control problems

2013
*
2013 IEEE International Symposium on Intelligent Control (ISIC)
*

This paper shows how this question can be answered

doi:10.1109/isic.2013.6658617
dblp:conf/IEEEisic/SummersYDA13
fatcat:hgsjnqflrracxl3wlsqvsmd4gm
*for*any size formation in principle using*semidefinite*programming techniques*for*semialgebraic problems, involving solutions sets*of**polynomial*equations ...*A*fundamental control problem*for*autonomous vehicle formations is formation shape control, in which*the*agents must maintain*a*prescribed formation shape using only information measured or communicated ...*A*.*Sum**of**Squares**Polynomials*and*Semidefinite*Programming We begin with some basic definitions. ...##
###
Approximation of the Stability Number of a Graph via Copositive Programming

2002
*
SIAM Journal on Optimization
*

In this way we can compute

doi:10.1137/s1052623401383248
fatcat:gtnjzetoc5e67hejifv7jrlbla
*the**stability**number*α(G)*of*any*graph*G(V, E) after at most α(G) 2 successive liftings*for**the*LP-based approximations. ... In this paper we present*a*similar idea. We show how*the**stability**number*can be computed as*the*solution*of**a*conic linear program (LP) over*the*cone*of*copositive matrices. ...*The*authors would like to thank Immanuel Bomze, Pablo Parrilo, and Kees Roos*for*their comments on*a*draft version*of*this paper. ...##
###
Algebraic Relaxations and Hardness Results in Polynomial Optimization and Lyapunov Analysis
[article]

2012
*
arXiv
*
pre-print

This thesis settles

arXiv:1201.2892v1
fatcat:wazdhceidnfktmp4tkbkozha2m
*a**number**of*questions related to computational complexity and algebraic,*semidefinite*programming based relaxations in optimization and control. ... Moreover, we remark that this*bound*is tighter, in terms*of*its dependence on n, than*the*known*bounds**for*ρ V SOS,2d*for*any finite degree 2d*of**the**sum**of**squares**polynomials*. ... This, together with Theorem 4.8, would imply that asymptotic*stability**of*homogeneous*polynomial*systems can always be decided*via**sum**of**squares*programming. ...##
###
Sum of Squares Basis Pursuit with Linear and Second Order Cone Programming
[article]

2016
*
arXiv
*
pre-print

We devise

arXiv:1510.01597v2
fatcat:cuqmy2nping7thoji6i7knrkti
*a*scheme*for*solving an iterative sequence*of*linear programs (LPs) or second order cone programs (SOCPs) to approximate*the*optimal value*of*any*semidefinite*program (SDP) or*sum**of**squares*...*The*first LP and SOCP-based*bounds*in*the*sequence come from*the*recent work*of*Ahmadi and Majumdar on diagonally dominant*sum**of**squares*(DSOS) and scaled diagonally dominant*sum**of**squares*(SDSOS)*polynomials*...*bounding**the**stability**number**of**the*complement*of**the*Petersen*graph*Finally, in 7 Theorem 5.3. ...##
###
DSOS and SDSOS Optimization: More Tractable Alternatives to Sum of Squares and Semidefinite Optimization
[article]

2018
*
arXiv
*
pre-print

These are optimization problems over certain subsets

arXiv:1706.02586v3
fatcat:a7kmmfs435ejbb5mtufd7jitli
*of**sum**of**squares**polynomials*(or equivalently subsets*of*positive*semidefinite*matrices), which can be*of*interest in general applications*of**semidefinite*... In recent years, optimization theory has been greatly impacted by*the*advent*of**sum**of**squares*(SOS) optimization. ... Our gratitude extends to Russ Tedrake and*the*members*of**the*Robot Locomotion Group at MIT*for*several insightful discussions, particularly around control applications. We thank ...##
###
Degree Bounds for Polynomial Verification of the Matrix Cube Problem
[article]

2006
*
arXiv
*
pre-print

When

arXiv:math/0604573v1
fatcat:642x77w76zdapbzfmgem66wt7a
*the*semialgebraic set is*a*hypercube, we give*bounds*on*the*degree*of**the*required certificate*polynomials*. ... In this paper we consider*the*problem*of*how to computationally test whether*a*matrix inequality is positive*semidefinite*on*a*semialgebraic set. ... We define*the*notion*of**sum*-*of*-*squares**for*matrix*polynomials*as follows Definition 3. ...##
###
Optimization over Nonnegative and Convex Polynomials With and Without Semidefinite Programming
[article]

2018
*
arXiv
*
pre-print

*A*

*number*

*of*breakthrough papers in

*the*early 2000s showed that this problem, long thought to be out

*of*reach, could be tackled by using

*sum*

*of*

*squares*programming. ... 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*... (

*a*) Complement

*of*Petersen

*graph*(b)

*The*Lovász theta

*number*and iterative

*bounds*

*bounds*obtained by LP and SOCP Upper

*bounding*

*the*

*stability*

*number*

*of*

*the*complement

*of*

*the*Petersen

*graph*Finally, in ...

##
###
Analysis of the joint spectral radius via lyapunov functions on path-complete graphs

2011
*
Proceedings of the 14th international conference on Hybrid systems: computation and control - HSCC '11
*

Inspired by concepts in automata theory and symbolic dynamics, we define

doi:10.1145/1967701.1967706
dblp:conf/hybrid/AhmadiJPR11
fatcat:cjawo3ko55gvhjbfnqcj7oqm7a
*a*class*of**graphs*called path-complete*graphs*, and show that any such*graph*gives rise to*a*method*for*proving*stability**of**the*...*For**the*general case*of*any set*of*n × n matrices we propose*semidefinite*programs*of*modest size that approximate*the*JSR within*a*multiplicative factor*of*1/ 4 √ n*of**the*true value. ... Moreover, we remark that these*bounds*are tighter, in terms*of*their dependence on n, than*the*known*bounds**for*ρ V SOS,2d*for*any finite degree 2d*of**the**sum**of**squares**polynomials*. ...##
###
Handelman's hierarchy for the maximum stable set problem

2013
*
Journal of Global Optimization
*

Moreover we show two upper

doi:10.1007/s10898-013-0123-5
fatcat:ze3p7fa4pvhbdaoqdhskn5xspq
*bounds*on*the*Handelman rank in terms*of**the*(fractional)*stability**number**of**the**graph*and compute*the*Handelman rank*for*several classes*of**graphs*including odd cycles and ...*The*maximum stable set problem is*a*well-known NP-hard problem in combinatorial optimization, which can be formulated as*the*maximization*of**a*quadratic*square*-free*polynomial*over*the*(Boolean) hypercube ... Vera*for*useful discussions. We also thank two anonymous referees*for*their comments which helped improve*the*clarity*of**the*paper and*for*drawing our attention to*the*paper by Krivine [13] . ...##
###
SOS-Convex Lyapunov Functions and Stability of Difference Inclusions
[article]

2018
*
arXiv
*
pre-print

We then provide

arXiv:1803.02070v1
fatcat:kmmqoqyvljf63phxggfdbcu73e
*a**semidefinite*programming-based procedure*for*computing*a*full-dimensional subset*of**the*region*of*attraction*of*equilibrium points*of*switched*polynomial*systems, under*the*condition ... We show*via*an explicit example however that*the*minimum degree*of**a*convex*polynomial*Lyapunov function can be arbitrarily higher than*a*non-convex*polynomial*Lyapunov function. ...*The*authors are thankful to Alexandre Megretski*for*insightful discussions around convex Lyapunov functions. ...
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