Symmetry groups, semidefinite programs, and sums of squares

  title={Symmetry groups, semidefinite programs, and sums of squares},
  author={Karin Gatermann and Pablo A. Parrilo},
  journal={Journal of Pure and Applied Algebra},
  • K. GatermannP. Parrilo
  • Published 28 November 2002
  • Mathematics, Computer Science
  • Journal of Pure and Applied Algebra

Sums of Squares, Moment Matrices and Optimization Over Polynomials

This work considers the problem of minimizing a polynomial over a semialgebraic set defined byPolynomial equations and inequalities, which is NP-hard in general and reviews the mathematical tools underlying these properties.

On Symmetry Groups of Some Quadratic Programming Problems

A larger group of invertible linear transformations is considered, where the objective function and constraints are given by quadratic forms, and the sum of all matrices of quadratics forms, involved in the constraints, is a positive definite matrix.

Semidefinite Relaxations of Products of Nonnegative Forms on the Sphere

A series of intermediate relaxations of increasing complexity that interpolate to the full Sum-of-Squares relaxation, as well as a rounding algorithm that finds an approximate solution from the solution of any intermediate relaxation, which can be generalized for Relaxations of products of non-negative forms of any degree.

On the equivalence of algebraic conditions for convexity and quasiconvexity of polynomials

It is shown that contrary to a claim made in the same related work, if an even degree polynomial is homogeneous, then it is quasiconvex if and only if it is convex.

On matrix algebras associated to sum-of-squares semidefinite programs

Abstract To each semidefinite program (SDP) in primal form, we associate the matrix algebra generated by its constraint matrices. In this note, we show that this algebra is always a full matrix

Symmetric Non-Negative Forms and Sums of Squares

It is shown that in degree 4 the cones of non-negative symmetric forms and symmetric sums of squares approach the same limit, thus these two cones asymptotically become closer as the number of variables grows.

Block Factor-width-two Matrices and Their Applications to Semidefinite and Sum-of-squares Optimization

This paper introduces a new notion of block factor-width-two matrices and builds a new hierarchy of inner and outer approximations of the cone of positive semidefinite (PSD) matrices, which leads to a block extension of the scaled diagonally dominant sum-of-squares (SDSOS) polynomials.



Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization

In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where the

Semidefinite programming relaxations for semialgebraic problems

  • P. Parrilo
  • Mathematics, Computer Science
    Math. Program.
  • 2003
It is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility and provide a constructive approach for finding bounded degree solutions to the Positivstellensatz.

Invariants of Finite Groups and Involutive Division

The invariant ring of a finite matrix group is known to be well behaved for reflection groups and messy in general. Involutive division is a newly discovered tool in commutative algebra and in this

Minimizing Polynomial Functions

  • P. ParriloB. Sturmfels
  • Computer Science, Mathematics
    Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science
  • 2001
It is demonstrated that existing algebraic methods are dramatically outperformed by a relaxation technique, due to N.Z. Shor and the first author, which involves sums of squares and semidefinite programming.

Squared Functional Systems and Optimization Problems

It is proved that such cones can be always seen as a linear image of the cone of positive semidefinite matrices, and a description of the cones of univariate and non-negative trigonometric polynomials is given.

Zeros of Equivariant Vector Fields: Algorithms for an Invariant Approach

  • P. Worfolk
  • Mathematics, Computer Science
    J. Symb. Comput.
  • 1994
A computationally effective algorithm to solve for the zeros of a polynomial vector field equivariant with respect to a finite subgroup of O (n) is presented and it is proved that the module of equivariants is Cohen-Macaulay.

Group Symmetry in Interior-Point Methods for Semidefinite Program

A class of group symmetric Semi-Definite Program (SDP) is introduced by using the framework of group representation theory. It is proved that the central path and several search directions of

An algorithm for sums of squares of real polynomials

Global Optimization with Polynomials and the Problem of Moments

It is shown that the problem of finding the unconstrained global minimum of a real-valued polynomial p(x): R n to R, in a compact set K defined byPolynomial inequalities reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems.

Invariants of finite groups and their applications to combinatorics

1 CONTENTS 1. Introduction 2. Molien's theorem 3. Cohen-Macaulay rings 4. Groups generated by pseudo-reflections 5. Three applications 6. Syzygies 7. The canonical module 8. Gorenstein rings 9.