Semialgebraic Relaxations and Semidefinite Programs Pablo A. Parrilo ETH Zürich control.ee.ethz.ch/~parrilo

1. Semialgebraic Relaxationsand Semidefinite ProgramsPablo A. ParriloETH Zürichcontrol.ee.ethz.ch/~parrilo

2. Semialgebraic modeling • Many problems in different domains can be modeled by polynomial inequalities • Continuous, discrete, hybrid • NP-hard in general • Tons of examples: spin glasses, max-cut, nonlinear stability, robustness analysis, entanglement, etc. How to prove things about them, in an algorithmic, certified and efficient way?

3. Proving vs. disproving • Really, it’s automatic theorem proving • Big difference: finding counterexamples vs. producing proofs (NP vs. co-NP) • A good decision theory exists (Tarski-Seidenberg, etc), but practical performance is generally poor • Want unconditionally valid proofs, but may fail to get them sometimes • We use a particular proof system from real algebra: the Positivstellensatz

4. An example Is empty, since with Reason: consider signs on candidate feasible points

5. Positivstellensatz (Real Nullstellensatz) if and only if • Generalizes Hilbert’s Nullstellensatz, LP duality. • Infeasibility certificates for polynomial systems over the reals. • Sums of squares (SOS) are essential • Conditions are convex in f,g • Bounded degree solutions can be computed! • A convex optimization problem. • Furthermore, it’s a semidefinite program

6. P-satz proofs(P., Caltech thesis 2000, Math Prog 2003) • Proofs are given by algebraic identities • Extremely easy to verify • Use convex optimization to search for them • Convexity, hence a duality structure: • On the primal, simple proofs. • On the dual, weaker models (liftings, etc) • General algorithmic construction • Techniques for exploiting problem structure

7. Polynomial descriptions P-satz relaxations Symmetry reduction Exploit structure Sparsity Ideal structure Semidefinite programs Graph structure

8. Exploiting structure Isolate algebraic properties! • Symmetry reduction: invariance under a group • Sparsity: Few nonzeros, Newton polytopes • Ideal structure: Equalities, quotient ring • Graph structure: use the dependency graph to simplify the SDPs Methods (mostly) commute, can mix and match

9. A few applications • Continuous and combinatorial optimization • Optimization of polynomials • Graphs: stability numbers, cuts, … • Dynamical systems: Lyapunov and Bendixson-Dulac functions • Robustness analysis • Reachability analysis: set mappings, … • Geometric theorem proving • Today: deciding quantum entanglement

10. A B QM state described by PSD Hermitian matrices ρ(density matrix, mixed states) • States of multipartite systems are described by operators on the tensor product of vector spaces

11. Z Separablestates ρ Separable states: convex combination of product states. Interpretation: statistical ensemble of locally prepared states. Entangled states:all the rest Q: How to determine whether or not a given quantum state is entangled ? Decision problem is NP-hard (Gurvits)

12. Entangled not-PPT Entangled PPT New relaxations Separable Use the techniques to find certified “entanglement witnesses,”generalizations of Bell’s inequalities. The witnesses are self-certified, e.g. Obtain a hierarchy of SDP-testable conditions. For all entangled states tried, the second level of the hierarchy is enough!

13. Future challenges • Structure: we know a lot, can we do more? • A good algorithmic use of abstractions, and randomization. • Infinite # of variables? Possible, but not too nice computationally. PSD integral operators, discretizations, etc. • Other kinds of structure to exploit? • Algorithmics: alternatives to interior point?