Polynomial Optimization over the Unit Sphere Vijay Bhattiprolu

Polynomial Optimization over the Unit Sphere Vijay Bhattiprolu (CMU) Mrinal Ghosh (TTIC) Venkat Guruswami (CMU) Euiwoong Lee (CMU / Simons) Madhur Tulsiani (TTIC)

Problem •

Complexity •

What we will prove •

What we will prove •

First Step •

First Step •

Tuples and Monomials •

Multi-Indices and Monomials •

Matrix Representations •

1, 1 1, 2 1, 3 2, 1 2, 2 2, 3 3, 1 3, 2 3, 3 1, 1 1, 2 1, 3 2, 1 2, 2 2, 3 3, 1 3, 2 3, 3

1, 1 1, 2 1, 3 2, 1 1, 2 1, 3 2, 1 2, 2 2, 3 3, 1 3, 2 3, 3 1 1 2, 2 1 2, 3 1 3, 1 1 3, 2 1 3, 3 1

1, 1 1, 2 1, 3 2, 1 2, 2 2, 3 3, 1 3, 2 3, 3 1 1 1 1, 2 1, 3 2, 1 2, 2 2, 3 3, 1 3, 2 3, 3 1 1 1

Our “Relaxation” • •

Strategy •

Multilinear Decomposition Inequality •

Multilinear Decomposition Inequality •

Finding a good Matrix Representation •

• 1, 1 1, 2 1, 3 2, 1 2, 2 2, 3 3, 1 3, 2 3, 3

• 1, 1 1, 2 1, 3 2, 1 2, 2 2, 3 3, 1 3, 2 3, 3

• 1, 1 1, 2 1, 3 2, 1 2, 2 2, 3 3, 1 3, 2 3, 3

Saving 1 in Exponent •

Conclusion •

Thank you!