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

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?

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

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

 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 Positivstellensatz (Real Nullstellensatz) if and only if

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

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

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

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

AB 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

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

Entangled not-PPT Separable Entangled PPT New relaxations 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!

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?

Summary  Algorithmic construction of SDP relaxations  Generalization of many earlier schemes  Very powerful in practice  Done properly, can fully exploit structure  Lots of applications: bounds for combinatorial problems, control & dynamical systems, entanglement, …