Linearization Functors on Real Convex Sets

We prove that linearizing certain families of polynomial optimization problems leads to new functorial operations in real convex sets. We show that under some conditions these operations can be computed or approximated in ways amenable to efficient computation. These operations are convex analogues of Hom functors, tensor products, symmetric powers, exterior powers and general Schur functors on vector spaces and lead to novel constructions even for polyhedra.

[1]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[2]  A. Calderón,et al.  A note on biquadratic forms , 1973 .

[3]  S. Lane.  Categories for the Working Mathematician , 1971 .

[4]  Aharon Ben-Tal,et al.  Lectures on modern convex optimization , 1987 .

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

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

[7]  João Gouveia,et al.  Positive Polynomials and Projections of Spectrahedra , 2009, SIAM J. Optim..

[8]  Jean B. Lasserre,et al.  A New Look at Nonnegativity on Closed Sets and Polynomial Optimization , 2010, SIAM J. Optim..

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

[10]  Ellen Veomett.  A Positive Semidefinite Approximation of the Symmetric Traveling Salesman Polytope , 2007, Discret. Comput. Geom..

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

[12]  Alexander Barvinok,et al.  A course in convexity , 2002, Graduate studies in mathematics.

[13]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[14]  Rekha R. Thomas,et al.  Semidefinite Optimization and Convex Algebraic Geometry , 2012 .

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

[16]  G. P. Barker.  Faces and duality in convex cones , 1978 .

[17]  Rekha R. Thomas,et al.  Theta Bodies for Polynomial Ideals , 2008, SIAM J. Optim..

[18]  Julian Romero,et al.  Semidefinite Approximations of Conical Hulls of Measured Sets , 2014, Discret. Comput. Geom..

[19]  Joe Harris,et al.  Representation Theory: A First Course , 1991 .

[20]  A. J. Goldman,et al.  Some geometric results in semidefinite programming , 1995, J. Glob. Optim..

[21]  S. Maclane,et al.  Categories for the Working Mathematician , 1971 .

[22]  Rekha R. Thomas,et al.  Convex Hulls of Algebraic Sets , 2010, 1007.1191.

[23]  Jiawang Nie,et al.  Discriminants and nonnegative polynomials , 2010, J. Symb. Comput..

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

[25]  Moritz Beckmann,et al.  Young tableaux , 2007 .

[26]  G. Ziegler.  Lectures on Polytopes , 1994 .

[27]  P. K. Suetin,et al.  Linear Algebra and Geometry , 1989 .