**Javad Lavaei Department of Electrical EngineeringColumbia** University Low-Rank Solution for Nonlinear Optimization over Graphs

**Acknowledgements** • Joint work with SomayehSojoudi (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

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

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

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

**Example 2** Opt: Acyclic Graph • 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 Javad Lavaei, Columbia University 6

**Sign Definite Set** • 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 R2: Javad Lavaei, Columbia University 7

**Formal Definition: Optimization over Graph** Optimization of interest: (real or complex) Define: • 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). Javad Lavaei, Columbia University 8

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

**Real-Valued Optimization** • ExactSDP relaxation: • Acyclic graph: sign definite sets • Bipartite graph: positive weight sets • Arbitrary graph: negative weight sets • Interplay between topology and edge signs Javad Lavaei, Columbia University 10

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

**Computational Complexity: Acyclic Graph** ? • Number partitioning problem: Javad Lavaei, Columbia University 12

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

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

**Graph Decomposition** • There are at least four good structural graphs. • Acyclic combination of them leads to exact SDP relaxation. Opt: • Sufficient conditions for {c12 , c23 ,c13 }: • Real with negative product • Complex with one zero element • Purely imaginary Javad Lavaei, Columbia University 15

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

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

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

**Exact Convex Relaxation** • 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. Javad Lavaei, Stanford University Javad Lavaei, Columbia University 17 19

**Generalized Network Flow (GNF)** injections flows limits • Goal: • Assumption: • fi(pi): convex and increasing • fij(pij): convex and decreasing Javad Lavaei, Columbia University 20

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

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

**Conclusions** • 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 Javad Lavaei, Columbia University 23