1. Mathematical Programming (towards programming with math)
Pablo A. Parrilo
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
12/2/2018

2. Outline
Distributed systems with global goals, local computation, and communication constraints.
The optimization perspective
Role of convexity
Hard problems and their relaxations
The promises of decentralization
12/2/2018

3. Distributed systems: global to local
Variational principles (least action, Dirichlet, etc.)
From these, local actions (e.g., via Euler-Lagrange)
Want to engineer local interactions towards desired global goals (no “emergence”)
How? And why should this work?
12/2/2018

4. Convexity is crucial
"…in fact, the great watershed in optimization isn't between linearity and nonlinearity, but convexity and nonconvexity"
R. Tyrrell Rockafellar, in SIAM Review, 1993
Global geometry from local information
Gradient methods will converge
Systematically exploit structure
However, if not convex, what to do?
12/2/2018

5. “convexify” nonconvex problems
12/2/2018

6. Convexity is relative Relaxations NP-hard
Interpretations: probabilistic, algebraic, geometric, proof theoretic, etc.
Technical tools:
Semidefinite programming (SDP) and sum of squares (SOS)
Results from real algebraic geometry
Hierarchies of relaxations
Many apps: global optimization, dynamical systems, entanglement, geometric theorem proving, etc.
12/2/2018

7. Many challenges and connections
Extreme parallelization and decentralization
Interaction: convexity and decentralization
Much recent work on consensus-type schemes, currently being extended
Economics: shadow prices, mechanism design
Relations with belief-propagation
12/2/2018