javad lavaei department of electrical engineering columbia university low-rank solution for...
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Javad Lavaei
Department of Electrical EngineeringColumbia University
Low-Rank Solution for Nonlinear Optimization
over Graphs
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
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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:
Example 1
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Opt:
Sufficient condition for exactness: Sign definite sets.
What if the condition is not satisfied?
Rank-2 W (but hidden)
NP-hard
Example 2
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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
Sign Definite Set
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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:
Formal Definition: Optimization over Graph
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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:
Real-Valued Optimization
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Exact SDP relaxation:
Acyclic graph: sign definite sets
Bipartite graph: positive weight sets
Arbitrary graph: negative weight sets
Interplay between topology and edge signs
Low-Rank Solution
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Violate edge condition:
Satisfy edge condition but violate cycle condition :
Computational Complexity: Acyclic Graph
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Number partitioning problem: ?
Complex-Valued Optimization
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SDP relaxation for acyclic graphs:
real coefficients
1-2 element sets (power grid: ~10 elements)
Main requirement in complex case: Sign definite weight sets
Complex-Valued Optimization
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Purely imaginary weights (lossless power grid):
Consider a real matrix M:
Polynomial-time solvable for weakly-cyclic bipartite graphs.
Graph Decomposition
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Opt:
Sufficient conditions for {c12 , c23 , c13 }:
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.
Resource Allocation: Optimal Power Flow (OPF)
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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
Optimal Power Flow
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Cost
Operation
Flow
Balance
Express the last constraint as an inequality.
Exact Convex Relaxation
Result 1: Exact relaxation for DC/AC distribution and DC transmission.
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OPF: DC or AC
Networks: Distribution or transmission
Energy-related optimization:
Exact Convex Relaxation
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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.
Generalized Network Flow (GNF)
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injections
flows
Goal:
limits
Assumption: • fi(pi): convex and increasing• fij(pij): convex and decreasing
Convexification of GNF
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Convexification:
Feasible set without box constraint:
It finds correct injection vector but not necessarily correct flow vector.
Monotonic Non-monotonic
Convexification of GNF
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Feasible set without box constraint:
Correct injections in the feasible case.
Why monotonic flow functions?
Conclusions
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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