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Mathematical Programming (towards programming with math)

Explore distributed systems with global objectives, local computation, and communication constraints from an optimization perspective. Learn about convexity's role, hard problems and their relaxations, and the promises of decentralization. Discover how to engineer local interactions for desired global goals and tackle nonconvex problems using convexification methods. Delve into relaxations, NP-hard issues, interpretations, and technical tools like SDP and SOS. Discover challenges, applications in optimization, dynamic systems, entanglement, and more.

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Mathematical Programming (towards programming with math)

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

  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

  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?

  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?

  5. “convexify” nonconvex problems

  6. Relaxations NP-hard Convexity is relative • 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.

  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

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