Filters








1,787 Hits in 6.7 sec

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

Nebojša Gvozdenović, Monique Laurent
2006 Mathematical programming  
Two other hierarchies 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.  ... 
doi:10.1007/s10107-006-0062-8 fatcat:ltq2utolynbgbpvepydhsjnopi

Semidefinite Bounds for the Stability Number of a Graph via Sums of Squares of Polynomials [chapter]

Nebojša Gvozdenović, Monique Laurent
2005 Lecture Notes in Computer Science  
Two other hierarchies 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.  ... 
doi:10.1007/11496915_11 fatcat:rdw75rtmfvfntkx77hvgugil44

Optimization over structured subsets of positive semidefinite matrices via column generation

Amir Ali Ahmadi, Sanjeeb Dash, Georgina Hall
2017 Discrete Optimization  
We then apply these techniques to approximate 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.  ... 
doi:10.1016/j.disopt.2016.04.004 fatcat:fqlpa7ohdngm7opzgy3qe7nxtm

Towards scalable algorithms with formal guarantees for Lyapunov analysis of control systems via algebraic optimization

Amir Ali Ahmadi, Pablo A. Parrilo
2014 53rd IEEE Conference on Decision and Control  
In this paper, we give 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  ... 
doi:10.1109/cdc.2014.7039734 dblp:conf/cdc/AhmadiP14 fatcat:kaahigfyavfsta47vwldaiht64

Optimization over Structured Subsets of Positive Semidefinite Matrices via Column Generation [article]

Amir Ali Ahmadi, Sanjeeb Dash, Georgina Hall
2016 arXiv   pre-print
We then apply these techniques to approximate 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.  ... 
arXiv:1512.05402v2 fatcat:mynygd7w55hrvb6bq3evstmxq4

Certifying non-existence of undesired locally stable equilibria in formation shape control problems

Tyler H. Summers, Changbin Yu, Soura Dasgupta, Brian D.O. Anderson
2013 2013 IEEE International Symposium on Intelligent Control (ISIC)  
This paper shows how this question can be answered 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.  ... 
doi:10.1109/isic.2013.6658617 dblp:conf/IEEEisic/SummersYDA13 fatcat:hgsjnqflrracxl3wlsqvsmd4gm

Approximation of the Stability Number of a Graph via Copositive Programming

E. de Klerk, D. V. Pasechnik
2002 SIAM Journal on Optimization  
In this way we can compute 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.  ... 
doi:10.1137/s1052623401383248 fatcat:gtnjzetoc5e67hejifv7jrlbla

Algebraic Relaxations and Hardness Results in Polynomial Optimization and Lyapunov Analysis [article]

Amir Ali Ahmadi
2012 arXiv   pre-print
This thesis settles 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 forρ 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.  ... 
arXiv:1201.2892v1 fatcat:wazdhceidnfktmp4tkbkozha2m

Sum of Squares Basis Pursuit with Linear and Second Order Cone Programming [article]

Amir Ali Ahmadi, Georgina Hall
2016 arXiv   pre-print
We devise 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.  ... 
arXiv:1510.01597v2 fatcat:cuqmy2nping7thoji6i7knrkti

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

Amir Ali Ahmadi, Anirudha Majumdar
2018 arXiv   pre-print
These are optimization problems over certain subsets 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  ... 
arXiv:1706.02586v3 fatcat:a7kmmfs435ejbb5mtufd7jitli

Degree Bounds for Polynomial Verification of the Matrix Cube Problem [article]

Been-Der Chen, Sanjay Lall
2006 arXiv   pre-print
When 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.  ... 
arXiv:math/0604573v1 fatcat:642x77w76zdapbzfmgem66wt7a

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

Georgina Hall
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  ... 
arXiv:1806.06996v1 fatcat:ywvkxguvobh43jvdaedk2tf3ju

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

Amir Ali Ahmadi, Raphaël Jungers, Pablo A. Parrilo, Mardavij Roozbehani
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 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 forρ V SOS,2d for any finite degree 2d of the sum of squares polynomials.  ... 
doi:10.1145/1967701.1967706 dblp:conf/hybrid/AhmadiJPR11 fatcat:cjawo3ko55gvhjbfnqcj7oqm7a

Handelman's hierarchy for the maximum stable set problem

Monique Laurent, Zhao Sun
2013 Journal of Global Optimization  
Moreover we show two upper 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] .  ... 
doi:10.1007/s10898-013-0123-5 fatcat:ze3p7fa4pvhbdaoqdhskn5xspq

SOS-Convex Lyapunov Functions and Stability of Difference Inclusions [article]

Amir Ali Ahmadi, Raphael M. Jungers
2018 arXiv   pre-print
We then provide 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.  ... 
arXiv:1803.02070v1 fatcat:kmmqoqyvljf63phxggfdbcu73e
« Previous Showing results 1 — 15 out of 1,787 results