Algebraic relaxations and hardness results in polynomial optimization and Lyapunov analysis

This thesis settles a number of questions related to computational complexity and algebraic, semidefinite programming based relaxations in optimization and control.

[1]  Gianna Stefani,et al.  Global stabilizability of homogeneous vector fields of odd degree , 1988 .

[2]  V. Kharitonov.  Asympotic stability of an equilibrium position of a family of systems of linear differntial equations , 1978 .

[3]  Amir Ali Ahmadi,et al.  Non-monotonic Lyapunov functions for stability of discrete time nonlinear and switched systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[4]  N. C. A. da Costa,et al.  Undecidable hopf bifurcation with undecidable fixed point , 1994 .

[5]  N. Z. Shor.  Class of global minimum bounds of polynomial functions , 1987 .

[6]  ASYMPTOTIC STABILITY OF AN EQUILIBRIUM P . OSITION OF A FAMILY OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS , 2022 .

[7]  Antonis Papachristodoulou,et al.  A converse sum-of-squares Lyapunov result: An existence proof based on the Picard iteration , 2010, 49th IEEE Conference on Decision and Control (CDC).

[8]  Y. Nesterov,et al.  On the accuracy of the ellipsoid norm approximation of the joint spectral radius , 2005 .

[9]  Pablo A. Parrilo,et al.  Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming , 2006, at - Automatisierungstechnik.

[10]  Tingshu Hu,et al.  On Several Composite Quadratic Lyapunov Functions for Switched Systems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[11]  Amir Ali Ahmadi,et al.  On the equivalence of algebraic conditions for convexity and quasiconvexity of polynomials , 2010, 49th IEEE Conference on Decision and Control (CDC).

[12]  Amir Ali Ahmadi.  Non-monotonic Lyapunov functions for stability of nonlinear and switched systems : theory and computation , 2008 .

[13]  A. Papachristodoulou,et al.  Analysis of switched and hybrid systems - beyond piecewise quadratic methods , 2003, Proceedings of the 2003 American Control Conference, 2003..

[14]  W. Rudin.  Real and complex analysis , 1968 .

[15]  T. Motzkin,et al.  Maxima for Graphs and a New Proof of a Theorem of Turán , 1965, Canadian Journal of Mathematics.

[16]  J. Tsitsiklis,et al.  The boundedness of all products of a pair of matrices is undecidable , 2000 .

[17]  Pablo A. Parrilo,et al.  Minimizing Polynomial Functions , 2001, Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science.

[18]  P.A. Parrilo,et al.  Polynomial games and sum of squares optimization , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[19]  P. Parrilo,et al.  Symmetry groups, semidefinite programs, and sums of squares , 2002, math/0211450.

[20]  D. Hilbert.  Über die Darstellung definiter Formen als Summe von Formenquadraten , 1888 .

[21]  Man-Duen Choi.  Positive semidefinite biquadratic forms , 1975 .

[22]  M. Kojima.  Sums of Squares Relaxations of Polynomial Semidefinite Programs , 2003 .

[23]  Jiawang Nie,et al.  Polynomial Matrix Inequality and Semidefinite Representation , 2009, Math. Oper. Res..

[24]  Ian R. Manchester,et al.  LQR-trees: Feedback Motion Planning via Sums-of-Squares Verification , 2010, Int. J. Robotics Res..

[25]  Dirk Aeyels,et al.  Homogeneous systems: stability, boundedness and duality , 2000 .

[26]  P. Pardalos.  Complexity in numerical optimization , 1993 .

[27]  Mardavij Roozbehani,et al.  Optimization of Lyapunov invariants in analysis and implementation of safety-critical software systems , 2008 .

[28]  John Baillieul,et al.  The geometry of homogeneous polynomial dynamical systems , 1980 .

[29]  J. William Helton,et al.  Semidefinite representation of convex sets , 2007, Math. Program..

[30]  Grigoriy Blekherman,et al.  Convex Forms that are not Sums of Squares , 2009, 0910.0656.

[31]  Matthew M. Peet,et al.  Exponentially Stable Nonlinear Systems Have Polynomial Lyapunov Functions on Bounded Regions , 2007, IEEE Transactions on Automatic Control.

[32]  Jacques A. Ferland,et al.  On pseudo-convex functions of nonnegative variables , 1971, Math. Program..

[33]  J. Gross,et al.  Graph Theory and Its Applications , 1998 .

[34]  J. Paquet,et al.  Second Order Conditions for Pseudo-Convex Functions , 1974 .

[35]  Newton C. A. da Costa,et al.  On Arnold's Hilbert Symposium Problems , 1993, Kurt Gödel Colloquium.

[36]  A. M. Lyapunov.  The general problem of the stability of motion , 1992 .

[37]  J. Lasserre.  Representation of nonnegative convex polynomials , 2008, 0801.3754.

[38]  Tingshu Hu,et al.  Conjugate convex Lyapunov functions for dual linear differential inclusions , 2006, IEEE Transactions on Automatic Control.

[39]  Claus Scheiderer,et al.  A Positivstellensatz for projective real varieties , 2011, 1104.1772.

[40]  John N. Tsitsiklis,et al.  The Stability of Saturated Linear Dynamical Systems Is Undecidable , 2001, J. Comput. Syst. Sci..

[41]  John N. Tsitsiklis,et al.  NP-hardness of deciding convexity of quartic polynomials and related problems , 2010, Math. Program..

[42]  Hendrik W. Lenstra,et al.  Integer Programming with a Fixed Number of Variables , 1983, Math. Oper. Res..

[43]  L. Grüne.  Homogeneous state feedback stabilization of homogeneous systems , 2000, CDC.

[44]  Wolfgang Hahn,et al.  Stability of Motion , 1967 .

[45]  J. Ferland.  Matrix-theoretic criteria for the quasiconvexity of twice continuously differentiable functions , 1981 .

[46]  Bruce Reznick.  On the absence of uniform denominators in Hilbert's 17th problem , 2003 .

[47]  Christos H. Papadimitriou,et al.  On the complexity of integer programming , 1981, JACM.

[48]  Chen Ling,et al.  Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations , 2009, SIAM J. Optim..

[49]  Vincent D. Blondel,et al.  Computationally Efficient Approximations of the Joint Spectral Radius , 2005, SIAM J. Matrix Anal. Appl..

[50]  Emilio Frazzoli,et al.  Distributed Lyapunov Functions in Analysis of Graph Models of Software , 2008, HSCC.

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

[52]  Ryan Feeley,et al.  Some controls applications of sum of squares programming , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[53]  Carsten W. Scherer,et al.  Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs , 2006, Math. Program..

[54]  A. Garulli,et al.  Positive Polynomials in Control , 2005 .

[55]  Claus Scheiderer,et al.  An elementary proof of Hilbertʼs theorem on ternary quartics , 2010, 1009.3144.

[56]  Yeung Sam Hung,et al.  Establishing Convexity of Polynomial Lyapunov Functions and Their Sublevel Sets , 2008, IEEE Transactions on Automatic Control.

[57]  Tingshu Hu,et al.  Absolute stability analysis of discrete-time systems with composite quadratic Lyapunov functions , 2005, IEEE Transactions on Automatic Control.

[58]  John N. Tsitsiklis,et al.  Complexity of stability and controllability of elementary hybrid systems , 1999, Autom..

[59]  P. Olver.  Nonlinear Systems , 2013 .

[60]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[61]  V. I. Arnol'd.  Algebraic unsolvability of the problem of lyapunov stability and the problem of topological classification of singular points of an analytic system of differential equations , 1970 .

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

[63]  Pablo A. Parrilo,et al.  Nonlinear control synthesis by convex optimization , 2004, IEEE Transactions on Automatic Control.

[64]  O. Mangasarian.  PSEUDO-CONVEX FUNCTIONS , 1965 .

[65]  G. Rota,et al.  A note on the joint spectral radius , 1960 .

[66]  Hai Lin,et al.  Stability and Stabilizability of Switched Linear Systems: A Short Survey of Recent Results , 2005, Proceedings of the 2005 IEEE International Symposium on, Mediterrean Conference on Control and Automation Intelligent Control, 2005..

[67]  Nikola Samardzija,et al.  Stability properties of autonomous homogeneous polynomial differential systems , 1983 .

[68]  A. Papachristodoulou,et al.  On the construction of Lyapunov functions using the sum of squares decomposition , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[69]  John N. Tsitsiklis,et al.  Deciding stability and mortality of piecewise affine dynamical systems , 2001, Theor. Comput. Sci..

[70]  Pablo A. Parrilo,et al.  Explicit SOS decompositions of univariate polynomial matrices and the Kalman-Yakubovich-Popov lemma , 2007, 2007 46th IEEE Conference on Decision and Control.

[71]  John N. Tsitsiklis,et al.  The Lyapunov exponent and joint spectral radius of pairs of matrices are hard—when not impossible—to compute and to approximate , 1997, Math. Control. Signals Syst..

[72]  Tingshu Hu,et al.  Stabilization of Switched Systems via Composite Quadratic Functions , 2008, IEEE Transactions on Automatic Control.

[73]  Jamal Daafouz,et al.  Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties , 2001, Syst. Control. Lett..

[74]  Etienne de Klerk,et al.  On the Lasserre Hierarchy of Semidefinite Programming Relaxations of Convex Polynomial Optimization Problems , 2011, SIAM J. Optim..

[75]  A. Magnani,et al.  Tractable fitting with convex polynomials via sum-of-squares , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[76]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[77]  John F. Canny,et al.  Some algebraic and geometric computations in PSPACE , 1988, STOC '88.

[78]  Grigoriy Blekherman,et al.  Dimensional Differences between Nonnegative Polynomials and Sums of Squares , 2009, 0907.1339.

[79]  M. Branicky.  Multiple Lyapunov functions and other analysis tools for switched and hybrid systems , 1998, IEEE Trans. Autom. Control..

[80]  A. Jadbabaie,et al.  Approximation of the joint spectral radius using sum of squares , 2007, 0712.2887.

[81]  Amir Ali Ahmadi,et al.  Analysis of the joint spectral radius via lyapunov functions on path-complete graphs , 2011, HSCC '11.

[82]  R. Jungers.  The Joint Spectral Radius: Theory and Applications , 2009 .

[83]  Amir Ali Ahmadi,et al.  A convex polynomial that is not sos-convex , 2009, Mathematical Programming.

[84]  A. Seidenberg.  A NEW DECISION METHOD FOR ELEMENTARY ALGEBRA , 1954 .

[85]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .

[86]  P. Parrilo,et al.  Distinguishing separable and entangled states. , 2001, Physical review letters.

[87]  D. J. Newman,et al.  Arithmetic, Geometric Inequality , 1960 .

[88]  Ji-Woong Lee,et al.  Uniform stabilization of discrete-time switched and Markovian jump linear systems , 2006, Autom..

[89]  Halil Mete Soner.  Lecture Notes on Stochastic Optimal Control DO NOT CIRCULATE: Preliminary Version , 2009 .

[90]  K. Arrow,et al.  QUASI-CONCAVE PROGRAMMING , 1961 .

[91]  Katta G. Murty,et al.  Some NP-complete problems in quadratic and nonlinear programming , 1987, Math. Program..

[92]  R. Tyrrell Rockafellar,et al.  Lagrange Multipliers and Optimality , 1993, SIAM Rev..

[93]  Eduardo Sontag.  From linear to nonlinear: some complexity comparisons , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[94]  Y. Pyatnitskiy,et al.  Criteria of asymptotic stability of differential and difference inclusions encountered in control theory , 1989 .

[95]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[96]  Man-Duen Choi,et al.  Extremal positive semidefinite forms , 1977 .

[97]  Amir Ali Ahmadi,et al.  A positive definite polynomial Hessian that does not factor , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[98]  J. Tsitsiklis,et al.  Overview of complexity and decidability results for three classes of elementary nonlinear systems , 1999 .

[99]  Douglas Lind,et al.  An Introduction to Symbolic Dynamics and Coding , 1995 .

[100]  Pablo A. Parrilo,et al.  An Inequality for Circle Packings Proved by Semidefinite Programming , 2004, Discret. Comput. Geom..

[101]  D. Henrion,et al.  Guest Editorial: Special Issue on Positive Polynomials in Control , 2009 .

[102]  Robert G. Jeroslow,et al.  There Cannot be any Algorithm for Integer Programming with Quadratic Constraints , 1973, Oper. Res..

[103]  John N. Tsitsiklis,et al.  A survey of computational complexity results in systems and control , 2000, Autom..

[104]  Vincent D. Blondel,et al.  Joint Spectral Characteristics of Matrices: A Conic Programming Approach , 2010, SIAM J. Matrix Anal. Appl..

[105]  Etienne de Klerk,et al.  The complexity of optimizing over a simplex, hypercube or sphere: a short survey , 2008, Central Eur. J. Oper. Res..

[106]  P. H. Diananda.  On non-negative forms in real variables some or all of which are non-negative , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[107]  Grigoriy Blekherman,et al.  Nonnegative Polynomials and Sums of Squares , 2010, 1010.3465.

[108]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[109]  T. Andô,et al.  Simultaneous Contractibility , 1998 .

[110]  Emmanuel Hainry.  Decidability and Undecidability in Dynamical Systems , 2009 .

[111]  Etienne de Klerk,et al.  Approximation of the Stability Number of a Graph via Copositive Programming , 2002, SIAM J. Optim..

[112]  B. Guo.  On the difficulty of deciding the convexity of polynomials over simplexes , 1996, Int. J. Comput. Geom. Appl..

[113]  Antonis Papachristodoulou,et al.  A Converse Sum of Squares Lyapunov Result With a Degree Bound , 2012, IEEE Transactions on Automatic Control.

[114]  Frank Sottile,et al.  A new approach to Hilbert's theorem on ternary quartics , 2004 .

[115]  Jacques A. Ferland,et al.  Criteria for quasi-convexity and pseudo-convexity: Relationships and comparisons , 1982, Math. Program..

[116]  Amir Ali Ahmadi.  On the difficulty of deciding asymptotic stability of cubic homogeneous vector fields , 2011, 2012 American Control Conference (ACC).

[117]  Jean B. Lasserre,et al.  Convexity in SemiAlgebraic Geometry and Polynomial Optimization , 2008, SIAM J. Optim..

[118]  Panos M. Pardalos,et al.  Open questions in complexity theory for numerical optimization , 1992, Mathematical programming.

[119]  Amir Ali Ahmadi,et al.  Converse results on existence of sum of squares Lyapunov functions , 2011, IEEE Conference on Decision and Control and European Control Conference.

[120]  Santosh S. Vempala,et al.  Testing Geometric Convexity , 2004, FSTTCS.

[121]  Bruce Reznick,et al.  On Hilbert's construction of positive polynomials , 2007, 0707.2156.

[122]  Bruce Reznick,et al.  Real zeros of positive semidefinite forms. I , 1980 .

[123]  B. Reznick,et al.  Sums of squares of real polynomials , 1995 .

[124]  Olga Taussky-Todd.  SOME CONCRETE ASPECTS OF HILBERT'S 17TH PROBLEM , 1996 .

[125]  Arkadi Nemirovski,et al.  Several NP-hard problems arising in robust stability analysis , 1993, Math. Control. Signals Syst..

[126]  B. Reznick.  Uniform denominators in Hilbert's seventeenth problem , 1995 .

[127]  Olivier Bournez,et al.  A Survey on Continuous Time Computations , 2009, ArXiv.

[128]  L. Rosier.  Homogeneous Lyapunov function for homogeneous continuous vector field , 1992 .

[129]  A. Bacciotti,et al.  Liapunov functions and stability in control theory , 2001 .

[130]  Miroslav Krstic,et al.  A globally asymptotically stable polynomial vector field with no polynomial Lyapunov function , 2011, IEEE Conference on Decision and Control and European Control Conference.

[131]  Amir Ali Ahmadi,et al.  On higher order derivatives of Lyapunov functions , 2011, Proceedings of the 2011 American Control Conference.

[132]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[133]  Yacine Chitour,et al.  Common Polynomial Lyapunov Functions for Linear Switched Systems , 2006, SIAM J. Control. Optim..

[134]  P. Parrilo.  On a decomposition of multivariable forms via LMI methods , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[135]  Mohamed Ali Hammami,et al.  The stabilization of homogeneous cubic vector fields in the plane , 1994 .

[136]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[137]  Anders Rantzer,et al.  Computation of piecewise quadratic Lyapunov functions for hybrid systems , 1997, 1997 European Control Conference (ECC).

[138]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[139]  S. Basu,et al.  Algorithms in real algebraic geometry , 2003 .

[140]  Tingshu Hu,et al.  Dual Matrix Inequalities in Stability and Performance Analysis of Linear Differential/Difference Inclusions , 2006 .

[141]  Leonid Gurvits.  Classical deterministic complexity of Edmonds' Problem and quantum entanglement , 2003, STOC '03.

[142]  G. Chesi.  Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems , 2009 .

[143]  Jean B. Lasserre.  Certificates of convexity for basic semi-algebraic sets , 2010, Appl. Math. Lett..

[144]  G. Sacks.  A DECISION METHOD FOR ELEMENTARY ALGEBRA AND GEOMETRY , 2003 .

[145]  James Demmel,et al.  Minimizing Polynomials via Sum of Squares over the Gradient Ideal , 2004, Math. Program..

[146]  E. Artin.  Über die Zerlegung definiter Funktionen in Quadrate , 1927 .

[147]  Amir Ali Ahmadi,et al.  A Complete Characterization of the Gap between Convexity and SOS-Convexity , 2011, SIAM J. Optim..

[148]  Daniel S. Graça,et al.  Boundedness of the Domain of Definition is Undecidable for Polynomial ODEs , 2008, CCA.

[149]  B. Poonen.  Undecidability in number theory , 2007 .

[150]  Claus Scheiderer,et al.  Sums of squares on real algebraic surfaces , 2006 .

[151]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[152]  Graziano Chesi,et al.  Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach , 2005, IEEE Transactions on Automatic Control.