Graph-Theoretic Algorithm for Arbitrary Polynomial
Optimization Problems with Applications to Distributed Control,
Power Systems, and Matrix Completion
Javad Lavaei
Department of Electrical Engineering
Columbia University
Joint work with:
Ramtin Madani, Ghazal Fazelnia, Abdulrahman Kalbat and Morteza Ashraphijuo
(Columbia University)
Somayeh Sojoudi (NYU Langone Medical Center)
Outline
 Theory: Convex relaxation
 Application 1: Optimization for power networks
 Application 2: Optimal decentralized control
 Implementation: High-performance solver handling 1B variables
 Theory: General polynomial optimization
Application: Matrix completion
Javad Lavaei, Columbia University
Penalized Semidefinite Programming (SDP) Relaxation
 Exactness of SDP relaxation:
 Existence of a rank-1 solution
 Implies finding a global solution
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Sparsity Graph
 Example:
1
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...
n
3
Treewidth
Tree decomposition
Vertices
Bags of vertices
 Treewidth of graph: The smallest width of all tree decompositions
-Rank of W at optimality ≤ Treewidth +1
- How to find it? (answered later)
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Outline
 Theory: Convex relaxation
 Application 1: Optimization for power networks
 Application 2: Optimal decentralized control
 Implementation: High-performance solver handling 1B variables
 Theory: General polynomial optimization
Application: Matrix completion
Javad Lavaei, Columbia University
Optimization for Power Systems
 Optimization:
 Optimal power flow (OPF)
 Security-constrained OPF
 State estimation
 Network reconfiguration
 Unit commitment
 Dynamic energy management
 Issue of non-convexity:
 Discrete parameters
 Nonlinearity in continuous variables
 Challenge: ~90% of decisions are made
in day ahead and ~10% are updated
iteratively during the day so a local
solution remains throughout the day.
Cost
local
global
Production
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Contingency Analysis
Contingency
G8#
37#
G10#
25#
26#
29#
28#
30#
 Contingency Analysis:
27#
2#
 Assume a line is disconnected.
18#
38#
17#
G9#
24#
1#
 Many generators cannot change productions quickly.
 The flows over other lines would increase.
G6#
3#
16#
15#
G1#
39#
22#
21#
14#
4#
 This triggers a cascading failure.
35#
5#
6#
12#
23#
19#
7#
13#
 Secure operation: Design an operating point such that the
20#
36#
11#
8#
31#
10#
34#
33#
G7#
9#
network survives under certain line or generator outages.
G2#
32#
G5#
G3#
G4#
Limited correction
by a generator
 Challenge 1: Number of constraints is prohibitive (our project with Ross Baldick proposes a new technique
to address this).
 Challenge 2: How to find the best operating point given the nonlinearity of the problem?
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Security-Constrained Optimal Power Flow (SCOPF)
Cost for pre-contingency case
Power flow equations for pre- and
post- contingencies
Physical and network limits for
pre- and post- contingencies
Preventive and corrective actions
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Power Networks
 What graph is important for SCOPF?
 Sparsity graph: Disjoint union of pre- and post contingency graphs:
Pre-contingency
Contingency 1
Contingency 2
 Observation: treewidth of sparsity graph of SCOPF = treewidth of power network
 Treewidth: IEEE 300 bus: 6, Polish 3120 ≤ 24, New York State ≤ 40.
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Decomposed SDP
0
0
0
Decomposed SDP
Full-scale SDP
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Power Networks
 Decomposed relaxed SCOPF: SDP problem with small-sized constraints
0,
0,
0
 Reduction of the number of variables for a Polish system from ~9,000,000 to ~90K.
 Result: Rank of W at optimality ≤ Treewidth +1
 Stronger Result: Rank of W at optimality ≤ maximum rank of bags
 By-product: Lines of network in not-rank-1 bags make SDP fail.
 Penalized SDP: Penalize the loss over problematic lines
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Example 1:
Performance of penalized relaxed OPF on IEEE and Polish systems:
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Example 2:
 Performance of penalized relaxed OPF on Bukhsh’s examples:
http://www.maths.ed.ac.uk/OptEnergy/LocalOpt/
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Example 3:
 New England 39 bus system under 10 contingencies:
 IEEE 300 bus system under 1 contingency corresponding to 3 outages:
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Outline
 Theory: Convex relaxation
 Application 1: Optimization for power networks
 Application 2: Optimal decentralized control
 Implementation: High-performance solver handling 1B variables
 Theory: General polynomial optimization
Application: Matrix completion
Javad Lavaei, Columbia University
Motivation
 Computational challenges arising in the control of real-world systems:
 Communication networks
 Electrical power systems
 Aerospace systems
 Large-space flexible structures
 Traffic systems
 Wireless sensor networks
 Various multi-agent systems
 Decentralized controller:
 A group of isolated local controllers
 Distributed controller:
 Partially interacting local controllers
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Optimal Distributed control (ODC)
 Stochastic ODC: Find a structured control
for the system:
disturbance
noise
to minimize the cost functional:
 Finite-horizon ODC: Deterministic system with the objective:
 Infinite-horizon ODC: Terminal time equal to infinity.
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Motivation: Maximum Penetration of Renewable Energy
39-Bus New England System
Four Communication Topologies
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Motivation: Maximum Penetration of Renewable Energy
 Assumption: Every generator has access to the rotor angle and frequency of its
neighbors (if any).
 Problem: Design a near-global distributed controller for adjusting the mechanical
power
 We solve three ODC problems for various values of alpha (gain) and sigma (noise).
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Two Formulations of SODC
SODC:
Convex
Reformulated SODC:
Non-convex (NP-hard)
Convex
Non-convex but
quadratic
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Quadratic Formulation in Static Case
 SDP relaxation for SODC:
W
 Theorem: W has rank 1, 2, or 3 at optimality for finite-horizon ODC, infinite-horizon ODC, and SODC.
Lyapunov domain
Time domain
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Computationally-Cheap SDP Relaxation
 Dimension of SDP variable = O(n2)
 Is lower bound tight enough?
 Can penalize the trace of W in the objective to make it penalized SDP.
 Goal: Design a new SDP relaxation such that:
 Dimension of SDP variable = O(n)
 The entries of W are automatically penalized in the problem.
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First Stage of SDP Relaxation
Automatic penalty
of the trace of W
Lyapunov
Inversion of variables
Exactness: Rank n
Direct and two-hop pattern
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Two-Stage SDP Relaxation
 Stage 1: Solve SDP relaxation.
 Stage 2: Recover a near-global controller.
 Direct Method: Read off the controller from the SDP solution.
 Indirect Method: Read off G from the SDP solution and solve a convex program.
 ϒ helps to make the problem feasible, but it’s penalized to keep its value low.
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Mass-Spring System
 Mass-spring system:
 Goal: Design two constrained controllers for 10 masses.
 Solution: (under various level of measurement noise)
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Outline
 Theory: Convex relaxation
 Application 1: Optimization for power networks
 Application 2: Optimal decentralized control
 Implementation: High-performance solver handling 1B variables
 Theory: General polynomial optimization
Application: Matrix completion
Javad Lavaei, Columbia University
Low-Complex Algorithm for Sparse SDP
Slides for this section are removed.
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Outline
 Theory: Convex relaxation
 Application 1: Optimization for power networks
 Application 2: Optimal decentralized control
 Implementation: High-performance solver handling 1B variables
 Theory: General polynomial optimization
Application: Matrix completion
Javad Lavaei, Columbia University
Polynomial Optimization
 Vertex Duplication Procedure:
 Edge Elimination Procedure:
 This gives rise to a sparse QCQP with a sparse graph.
 The treewidth can be reduced to 1.
Theorem: Every polynomial optimization has a QCQP formulation whose
SDP relaxation has a solution with rank 1 or 2.
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Outline
 Theory: Convex relaxation
 Application 1: Optimization for power networks
 Application 2: Optimal decentralized control
 Implementation: High-performance solver handling 1B variables
 Theory: General polynomial optimization
Application: Matrix completion
Javad Lavaei, Columbia University
Matrix Completion
Matrix completion: Fill a partially know matrix to a low-rank matrix (PSD or not PSD)
x1
? ?
?
?
x2
x3
x4
x5
x6
x7
x8
0
x9
x10
x11

There are

Minimize an arbitrary nonzero sum of X entries to get a rank-1 solution.
rank-1 solutions.
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Matrix Completion
x1
x2
? ?
?
?
? ?
x3
x4
x5
x6

0
x7
x8
x9
x10
Minimize a nonzero sum of X entries if all blocks are nonsingular and square.
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Matrix Completion
?
?
0
0
x1
x2
x3
x4
x5
x6
x7
x8
u1
u2
u3
u4

Minimize a nonzero sum of X entries by choosing U’s to be 1.

General theory based on msr, OS, and treewidth for a general non-block case.

Form a penalized SDP or nuclear-norm minimization to find a low-rank solution.
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Conclusions
 Theory: Low-rank optimization
 Applications: Power, Control, nonlinear optimization,…
 Implementation: High-performance solver handling 1B variables
 Two of our solvers posted online and another one is forthcoming.
 Collaboration with industry for demonstration on real data.
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