Javad Lavaei Department of Electrical Engineering Columbia University Graph-Theoretic Algorithm for Arbitrary Polynomial Optimization Problems with Applications to Distributed Control, Power Systems, and Matrix Completion Joint work with: Ramtin Madani, Ghazal Fazelnia, Abdulrahman Kalbat and Morteza Ashraphijuo (Columbia University) Somayeh Sojoudi (NYU Langone Medical Center)

Outline Javad Lavaei, Columbia University  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

Penalized Semidefinite Programming (SDP) Relaxation Javad Lavaei, Columbia University 2  Exactness of SDP relaxation:  Existence of a rank-1 solution  Implies finding a global solution

Sparsity Graph Javad Lavaei, Columbia University 3  Example: n n...

Treewidth Javad Lavaei, Columbia University 4 Tree decomposition  Treewidth of graph: The smallest width of all tree decompositions Vertices Bags of vertices -Rank of W at optimality ≤ Treewidth +1 - How to find it? (answered later)

Outline Javad Lavaei, Columbia University  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

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. Javad Lavaei, Columbia University 6 Production Cost local global

Javad Lavaei, Columbia University 8 Contingency Analysis  Contingency Analysis:  Assume a line is disconnected.  Many generators cannot change productions quickly.  The flows over other lines would increase.  This triggers a cascading failure. Contingency Limited correction by a generator  Secure operation: Design an operating point such that the network survives under certain line or generator outages.  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?

Security-Constrained Optimal Power Flow (SCOPF) Cost for pre-contingency case Javad Lavaei, Columbia University 9 Power flow equations for pre- and post- contingencies Physical and network limits for pre- and post- contingencies Preventive and corrective actions

Power Networks Javad Lavaei, Columbia University 10  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.

Decomposed SDP Javad Lavaei, Columbia University Full-scale SDP Decomposed SDP

Power Networks Javad Lavaei, Columbia University 11  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  Decomposed relaxed SCOPF: SDP problem with small-sized constraints 0, 0

Example 1: Javad Lavaei, Columbia University 12  Performance of penalized relaxed OPF on IEEE and Polish systems:

Example 2: Javad Lavaei, Columbia University 13  Performance of penalized relaxed OPF on Bukhsh’s examples:

Example 3: Javad Lavaei, Columbia University 14  New England 39 bus system under 10 contingencies:  IEEE 300 bus system under 1 contingency corresponding to 3 outages:

Outline Javad Lavaei, Columbia University  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

Motivation Javad Lavaei, Columbia University 15  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

Optimal Distributed control (ODC) Javad Lavaei, Columbia University 16  Stochastic ODC: Find a structured control for the system: to minimize the cost functional: disturbance noise  Finite-horizon ODC: Deterministic system with the objective:  Infinite-horizon ODC: Terminal time equal to infinity.

Motivation: Maximum Penetration of Renewable Energy Javad Lavaei, Columbia University Bus New England System Four Communication Topologies

Motivation: Maximum Penetration of Renewable Energy Javad Lavaei, Columbia University 18  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).

Two Formulations of SODC Javad Lavaei, Columbia University 19 SODC: Convex Non-convex (NP-hard) Reformulated SODC: Convex Non-convex but quadratic

Quadratic Formulation in Static Case Javad Lavaei, Columbia University 20  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 domainTime domain

Computationally-Cheap SDP Relaxation Javad Lavaei, Columbia University 21  Dimension of SDP variable = O(n 2 )  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.

Javad Lavaei, Columbia University 22 First Stage of SDP Relaxation Lyapunov Direct and two-hop pattern Inversion of variables Exactness: Rank n Automatic penalty of the trace of W

Two-Stage SDP Relaxation Javad Lavaei, Columbia University 23  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.

Mass-Spring System Javad Lavaei, Columbia University 24  Mass-spring system:  Goal: Design two constrained controllers for 10 masses.  Solution: (under various level of measurement noise)

Outline Javad Lavaei, Columbia University  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

Low-Complex Algorithm for Sparse SDP Javad Lavaei, Columbia University 25 Slides for this section are removed.

Outline Javad Lavaei, Columbia University  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

Polynomial Optimization Javad Lavaei, Columbia University 28  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.

Outline Javad Lavaei, Columbia University  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

Matrix Completion Javad Lavaei, Columbia University 29 ? ? ? ? x1x1 x2x2 x5x5 x3x3 x4x4 x 10 x9x9 x8x8 x7x7 x6x6 x 11 Matrix completion: Fill a partially know matrix to a low-rank matrix (PSD or not PSD) 0  There are rank-1 solutions.  Minimize an arbitrary nonzero sum of X entries to get a rank-1 solution.

Matrix Completion Javad Lavaei, Columbia University 30 0 ?? ? ? ? ? x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 x9x9 x 10  Minimize a nonzero sum of X entries if all blocks are nonsingular and square.

Matrix Completion Javad Lavaei, Columbia University 31 0 ? ? 0 x3x3 x2x2 x4x4 x5x5 x6x6 x7x7 x8x8 x1x1 u1u1 u2u2 u3u3 u4u4  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.

Conclusions Javad Lavaei, Columbia University  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. 32