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

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

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

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

“convexify” nonconvex problems 12/2/2018

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

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