Low-Rank Solution for Nonlinear Optimization
over Graphs
Javad Lavaei
Department of Electrical Engineering
Columbia University
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.
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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?
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Example 1
Trick:
SDP relaxation:
 Guaranteed rank-1 solution!
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Example 1
Opt:
 Sufficient condition for exactness: Sign definite sets.
 What if the condition is not satisfied?
 Rank-2 W (but hidden)
 NP-hard
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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
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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:
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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
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for edge (i,j).
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Real-Valued Optimization
Edge
Cycle
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Real-Valued Optimization
 Exact SDP relaxation:
 Acyclic graph: sign definite sets
 Bipartite graph: positive weight sets
 Arbitrary graph: negative weight sets
Interplay between topology and edge signs
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Low-Rank Solution
 Violate edge condition:
 Satisfy edge condition but violate cycle condition :
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Computational Complexity: Acyclic Graph
 Number partitioning problem:
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?
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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)
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Complex-Valued Optimization
 Purely imaginary weights (lossless power grid):
 Consider a real matrix M:
 Polynomial-time solvable for weakly-cyclic bipartite graphs.
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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
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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.
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Optimal Power Flow
Cost
Operation
Flow
Balance
 Express the last constraint as an inequality.
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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:
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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.
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Generalized Network Flow (GNF)
injections
flows
limits
 Goal:
Assumption:
• fi(pi): convex and increasing
• fij(pij): convex and decreasing
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Convexification of GNF
Feasible set without box constraint:
Monotonic
Non-monotonic
 Convexification:
 It finds correct injection vector but not necessarily correct flow vector.
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Convexification of GNF
Feasible set without box constraint:
 Correct injections in the feasible case.
 Why monotonic flow functions?
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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
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