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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Trick:

SDP relaxation:

Guaranteed rank-1 solution!

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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Edge

Cycle

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