Semialgebraic Relaxationsand Semidefinite Programs

Pablo A. ParriloPablo A. Parrilo ETH ZürichETH 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?

0)(,0)( xgxf ii

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

}02:,03:|),{( 22 xygyxfyx

Is empty, since

with

1121 gtfss

6,2,)(6)(2 122

612

23

31

1 tsxys

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

1:)(ideal),(cone gfggff ii

Positivstellensatz (Real Nullstellensatz)

empty is}0)(,0)(:{ xgxfRx iin

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-satzrelaxations

Exploitstructure

Symmetry reduction

Ideal structure

Sparsity

Graph structureSemidefiniteprograms

Polynomialdescriptions

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

11 1 1A 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

Separable states:

convex combination of product states.

1,0, iiiii ppp

Entangled states: all the rest

Interpretation: statistical ensemble of locally prepared states.

Separablestates

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?