Sparsity in Sum of Squares and Semi Definite

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Sparsity in Sum of Squares and Semi. Definite Programmming Relaxations of Polynomial Optimization Problems

Sparsity in Sum of Squares and Semi. Definite Programmming Relaxations of Polynomial Optimization Problems Workshop on Polyhedral Computation Montreal, October 17 -20, 2006 Masakazu Kojima Tokyo Institute of Technology

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares)

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares) polynomials 3. Sparse SOS relaxation of unconstrained POPs 4. Sparse SOS relaxation of constrained POPs --- briefly 5. Numerical results 6. Concluding remarks

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares)

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares) polynomials 3. Sparse SOS relaxation of unconstrained POPs 4. Sparse SOS relaxation of constrained POPs --- briefly 5. Numerical results 6. Concluding remarks

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares)

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares) polynomials 3. Sparse SOS relaxation of unconstrained POPs 4. Sparse SOS relaxation of constrained POPs --- briefly 5. Numerical results 6. Concluding remarks

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares)

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares) polynomials 3. Sparse SOS relaxation of unconstrained POPs 4. Sparse SOS relaxation of constrained POPs --- briefly 5. Numerical results 6. Concluding remarks

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares)

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares) polynomials 3. Sparse SOS relaxation of unconstrained POPs 4. Sparse SOS relaxation of constrained POPs --- briefly 5. Numerical results 6. Concluding remarks

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares)

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares) polynomials 3. Sparse SOS relaxation of unconstrained POPs 4. Sparse SOS relaxation of constrained POPs --- briefly 5. Numerical results 6. Concluding remarks

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares)

Contents 1. POPs (Polynomial Optimization Problems) 2. Nonnegative polynomials and SOS (Sum of Squares) polynomials 3. Sparse SOS relaxation of unconstrained POPs 4. Sparse SOS relaxation of constrained POPs --- briefly 5. Numerical results 6. Concluding remarks