Exploiting Symmetries in SDP-Relaxations for Polynomial Optimization

In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems, show how the degree principle for the symmetric group can be computationally exploited, and also propose some methods to efficiently compute the geometric quotient.

[1]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[2]  Monique Laurent,et al.  Strengthened semidefinite programming bounds for codes , 2007, Math. Program..

[3]  Alexander Schrijver,et al.  New code upper bounds from the Terwilliger algebra and semidefinite programming , 2005, IEEE Transactions on Information Theory.

[4]  W. Specht,et al.  Zur Darstellungstheorie der symmetrischen Gruppe , 1937 .

[5]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[6]  Didier Henrion,et al.  Convergent relaxations of polynomial matrix inequalities and static output feedback , 2006, IEEE Transactions on Automatic Control.

[7]  Claus Scheiderer,et al.  Sums of squares and moment problems in equivariant situations , 2008, 0808.0034.

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

[9]  Fredrik Meyer,et al.  Representation theory , 2015 .

[10]  Alexander Schrijver,et al.  Reduction of symmetric semidefinite programs using the regular $$\ast$$-representation , 2007, Math. Program..

[11]  J. Lasserre,et al.  Detecting global optimality and extracting solutions in GloptiPoly , 2003 .

[12]  J. B. Lasserre.  Characterizing polynomials with roots in a semi-algebraic set , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[13]  Alexander Schrijver,et al.  Invariant Semidefinite Programs , 2010, 1007.2905.

[14]  Alexander Schrijver,et al.  A comparison of the Delsarte and Lovász bounds , 1979, IEEE Trans. Inf. Theory.

[15]  Dion Gijswijt,et al.  Matrix Algebras and Semidefinite Programming Techniques for Codes , 2005, 1007.0906.

[16]  G. James,et al.  The Representation Theory of the Symmetric Group , 2009 .

[17]  A. Barvinok,et al.  Combinatorial complexity of orbits in representations of the symmetric group , 1992 .

[18]  A. Ivic.  Sums of squares , 2020, An Introduction to 𝑞-analysis.

[19]  Markus Schweighofer,et al.  Optimization of Polynomials on Compact Semialgebraic Sets , 2005, SIAM J. Optim..

[20]  Anatoly M. Vershik,et al.  Methods of representations theory in combinatorial optimization problems , 1989 .

[21]  F. Vallentin.  Symmetry in semidefinite programs , 2007, 0706.4233.

[22]  John D. Dixon.  Constructing Representations of Finite Groups , 1991, Groups And Computation.

[23]  Gerald W. Schwarz,et al.  Inequalities defining orbit spaces , 1985 .

[24]  T. Inui,et al.  The Symmetric Group , 1990 .

[25]  David A. Cox,et al.  Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics) , 2007 .

[26]  N. Katoh,et al.  Group Symmetry in Interior-Point Methods for Semidefinite Program , 2001 .

[27]  Lajos Rónyai,et al.  Computing irreducible representations of finite groups , 1989, 30th Annual Symposium on Foundations of Computer Science.

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

[29]  Monique Laurent,et al.  Semidefinite representations for finite varieties , 2007, Math. Program..

[30]  Harm Derksen,et al.  Computational Invariant Theory , 2002 .

[31]  D. Kamenetsky.  Symmetry Groups , 2003 .

[32]  Vlad Timofte,et al.  On the positivity of symmetric polynomial functions. Part II: Lattice general results and positivity , 2005 .

[33]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[34]  Didier Henrion,et al.  GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi , 2003, TOMS.

[35]  J. Lasserre.  Moments, Positive Polynomials And Their Applications , 2009 .

[36]  Ludwig Bröcker.  On symmetric semialgebraic sets and orbit spaces , 1998 .

[37]  Thorsten Theobald,et al.  Radii minimal projections of polytopes and constrained optimization of symmetric polynomials , 2003 .

[38]  Randall R. Holmes.  Linear Representations of Finite Groups , 2008 .

[39]  M. Laurent.  Sums of Squares, Moment Matrices and Optimization Over Polynomials , 2009 .

[40]  Vahid Dabbaghian-Abdoly,et al.  An algorithm for constructing representations of finite groups , 2005, J. Symb. Comput..

[41]  Monique Laurent,et al.  Revisiting two theorems of Curto and Fialkow on moment matrices , 2005 .

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

[43]  Cordian Riener,et al.  On the degree and half-degree principle for symmetric polynomials , 2010, 1001.4464.

[44]  Vlad Timofte,et al.  On the positivity of symmetric polynomial functions.: Part I: General results , 2003 .