1. Javad Lavaei Department of Electrical Engineering Columbia University Low-Rank Solution for Nonlinear Optimization over Graphs

2. Acknowledgements  Joint work with Somayeh Sojoudi (Caltech):  S. Sojoudi and J. Lavaei, "Semidefinite Relaxation for Nonlinear Optimization over Graphs," Working draft, 2012.  S. Sojoudi and J. Lavaei, "Convexification of Generalized Network Flow Problem," Working draft, 2012. Javad Lavaei, Columbia University 2

3. Problem of Interest Javad Lavaei, Columbia University 3  Abstract optimizations are NP-hard in the worst case.  Real-world optimizations are highly structured :  Question: How does the physical structure affect tractability of an optimization?  Sparsity:  Non-trivial structure:

4. Example 1 Javad Lavaei, Columbia University 4 Trick: SDP relaxation:  Guaranteed rank-1 solution!

5. Example 1 Javad Lavaei, Columbia University 5 Opt:  Sufficient condition for exactness: Sign definite sets.  What if the condition is not satisfied?  Rank-2 W (but hidden)  NP-hard

6. Example 2 Javad Lavaei, Columbia University 6 Opt:  Real-valued case: Rank-2 W (need regularization)  Complex-valued case:  Real coefficients: Exact SDP  Imaginary coefficients: Exact SDP  General case: Need sign definite sets Acyclic Graph

7. Sign Definite Set Javad Lavaei, Columbia University 7  Real-valued case: “ T “ is sign definite if its elements are all negative or all positive.  Complex-valued case: “ T “ is sign definite if T and –T are separable in R 2 :

8. Formal Definition: Optimization over Graph Javad Lavaei, Columbia University 8 Optimization of interest: (real or complex)  SDP relaxation for y and z (replace xx * with W).  f (y, z) is increasing in z (no convexity assumption).  Generalized weighted graph: weight set for edge (i,j). Define:

9. Real-Valued Optimization Javad Lavaei, Columbia University 9 Edge Cycle

10. Real-Valued Optimization Javad Lavaei, Columbia University 10  Exact SDP relaxation:  Acyclic graph: sign definite sets  Bipartite graph: positive weight sets  Arbitrary graph: negative weight sets  Interplay between topology and edge signs

11. Low-Rank Solution Javad Lavaei, Columbia University 11  Violate edge condition:  Satisfy edge condition but violate cycle condition :

12. Computational Complexity: Acyclic Graph Javad Lavaei, Columbia University 12  Number partitioning problem: ?

13. Complex-Valued Optimization Javad Lavaei, Columbia University 13  SDP relaxation for acyclic graphs:  real coefficients  1-2 element sets (power grid: ~10 elements)  Main requirement in complex case: Sign definite weight sets

14. Complex-Valued Optimization Javad Lavaei, Columbia University 14  Purely imaginary weights (lossless power grid):  Consider a real matrix M:  Polynomial-time solvable for weakly-cyclic bipartite graphs.

15. Graph Decomposition Javad Lavaei, Columbia University 15 Opt:  Sufficient conditions for {c 12, c 23, c 13 }:  Real with negative product  Complex with one zero element  Purely imaginary  There are at least four good structural graphs.  Acyclic combination of them leads to exact SDP relaxation.

16. Resource Allocation: Optimal Power Flow (OPF) Javad Lavaei, Columbia University 16 OPF: Given constant-power loads, find optimal P’s subject to:  Demand constraints  Constraints on V’s, P’s, and Q’s. OPF: Given constant-power loads, find optimal P’s subject to:  Demand constraints  Constraints on V’s, P’s, and Q’s. Voltage V Complex power = VI * =P + Q i Current I

17. Optimal Power Flow Javad Lavaei, Columbia University 17 Cost Operation Flow Balance  Express the last constraint as an inequality.

18. Exact Convex Relaxation  Result 1: Exact relaxation for DC/AC distribution and DC transmission. Javad Lavaei, Stanford University 17 Javad Lavaei, Columbia University 18  OPF: DC or AC  Networks: Distribution or transmission  Energy-related optimization:

19. Exact Convex Relaxation Javad Lavaei, Stanford University 17 Javad Lavaei, Columbia University 19  Each weight set has about 10 elements.  Due to passivity, they are all in the left-half plane.  Coefficients: Modes of a stable system.  Weight sets are sign definite.

20. Generalized Network Flow (GNF) Javad Lavaei, Columbia University 20 injections flows  Goal: limits Assumption: f i (p i ): convex and increasing f ij (p ij ): convex and decreasing

21. Convexification of GNF Javad Lavaei, Columbia University 21  Convexification: Feasible set without box constraint:  It finds correct injection vector but not necessarily correct flow vector. Monotonic Non-monotonic

22. Convexification of GNF Javad Lavaei, Columbia University 22 Feasible set without box constraint:  Correct injections in the feasible case.  Why monotonic flow functions?

23. Conclusions Javad Lavaei, Columbia University 23  Motivation: Real-world optimizations are highly structured.  Goal: Develop theory of optimization over graph  Mapped the structure of an optimization into a generalized weighted graph  Obtained various classes of polynomial-time solvable optimizations  Talked about Generalized Network Flow  Passivity in power systems made optimizations easier