Performance, Information Pattern Trade-offs and Computational - PowerPoint PPT Presentation

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Alireza Farhadi in collaboration with M. Cantoni and P. M. Dower Department of Electrical and Electronic Engineering The University of Melbourne October 30, 2012

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Motivation Distributed Optimization Method Computational Complexity Analysis Future Work References

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Motivation � Figure: An irrigation network. � � � � − � ��������� ��������� �������� ���������� � � − � � � � � � − � � ���������� ����������� � � � ����� �� + � � � − � � � � � − � � Figure: An automated irrigation network via distributed distant downstream feedback control. Ki Ti s + Ki z i ( s ) = C i ( s ) e i ( s ) , C i ( s ) = s ( Ti Fi s + Ti ) , e i = u i − y i .

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Motivation e37:blue,e38:green,e39:red,e40:cyan,e41:magenta,e42:black 0.2 0.1 error(m) 0 -0.1 -0.2 0 1000 2000 3000 4000 5000 6000 time(minutes) Figure: Downstream errors. e1:blue,e2:green,e3:red,e4:cyan,e5:magenta,e6:black 150 100 50 error(m) 0 -50 -100 -150 0 1000 2000 3000 4000 5000 6000 time(minutes) Figure: Upstream errors.

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Motivation w1:blue,w2:green,w3:red,w4:cyan,w5:magenta,w6:black 2000 1500 input flow 1000 500 0 0 1000 2000 3000 4000 5000 6000 time(minutes) Figure: Upstream input flows. e1:blue,e2:green,e3:red,e4:cyan,e5:magenta,e6:black 1.5 1 error(m) 0.5 0 -0.5 0 1000 2000 3000 4000 5000 6000 time(minutes) Figure: Upstream errors.

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Motivation � � − � � � ��������� ��������� �������� ���������� � � � � � − � � � − � � ���������� ���������� �� � � � ����� + � � − � � � � � − � � � Figure: An automated irrigation network via distributed distant downstream feedback and Ki Ti s + Ki feedforward control. z i ( s ) = C i ( s ) e i ( s ) + f i v i +1 , C i ( s ) = s ( Ti Fi s + Ti ) , e i = u i − y i .

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Motivation � ���������� � � � − � � ����������������������� � � ��������� ��������� ���������� �������� � � − � � � � � − � � � ���������� ����������� � � + � � ����� �� � � � � � − � − � � � Figure: An automated irrigation network equipped with a supervisory controller.

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Motivation 2000 1500 C cen (seconds) 1000 500 0 0 10 20 30 40 50 number of subsystems Figure: Computational complexity of the centralized optimization method versus the number of subsystems.

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Distributed Optimization Method Distributed supervisory control Scheduler d d Distributed Distributed supervisory supervisory control+ control+ Decision Decision maker i-1 y maker i y 1 i − i z i − 1 z i Subsystem Subsystem v v v i-1 i 1 i + 1 i − i d d i − 1 i Figure: An automated irrigation network equipped with distributed supervisory controller.

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Distributed Optimization Method Distributed optimization method (problem formulation) u =( u 1 ,..., u n ) { J ( u 1 , ..., u n ) , min u i ⊂ U i } U i ⊂ R m i , argmin u i J ( u 1 , ..., u n ) ∈ R Nm i . � �eighborhood � � Figure: Two-level architecture for exchanging information between distributed decision makers.

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Distributed Optimization Method Distributed optimization method (steps 1 ) N 1 = { S 1 , S 2 } , N 2 = { S 3 , S 4 } ◮ Initialization: The information exchange between neighborhoods at outer iterate t makes it possible for subsystem S i to initialize its local decision variables as h 0 i = u t i , where u 0 i ∈ U i are chosen arbitrarily at time t = 0. ◮ Inner Iterate: Then, subsystem S i performs ¯ p inner iterates as follows: For inner iterate p ∈ { 0 , 1 , ..., ¯ p − 1 } , it first updates its decision variable via h p +1 = π i h ∗ i + (1 − π i ) h p i , i where π 1 + π 2 = 1 , π 3 + π 4 = 1 and h ∗ 1 = argmin h 1 ∈U 1 J ( h 1 , h p 2 , h 0 3 , h 0 h ∗ 2 = argmin h 2 ∈U 2 J ( h p 1 , h 2 , h 0 3 , h 0 4 ) , 4 ) , 2 , h 3 , h p 2 , h p h ∗ 3 = argmin h 3 ∈U 3 J ( h 0 1 , h 0 h ∗ 4 = argmin h 4 ∈U 4 J ( h 0 1 , h 0 4 ) , 3 , h 4 ) . 1 [ACC2010] B. T. Stewart, J. B. Rawlings, and S. J. Wright.

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Distributed Optimization Method Distributed optimization method (steps) ◮ Inner Iterate (continued): Then, subsystem S i trades its updated decision variable h p +1 with all other subsystems within its neighborhood. i ◮ Outer Iterate: After ¯ p inner iterates there is an outer iterate update as follows u t +1 = λ i h ¯ i + (1 − λ i ) u t p i , i where λ 1 = λ 2 , λ 3 = λ 4 , λ 1 + λ 3 = 1 . Then, there is an outer iterate communication, in which the updated decision variables u t +1 are shared between all neighborhoods and subsequently between all i subsystems.

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Distributed Optimization Method Feasibility, convergence and optimality results 2 Feasibility: Given any collection of disjoint neighborhoods, above strictly convex finite horizon cost functional J , convex control constraint sets U i and a feasible initialization i ∈ U i ), the inner and outer iterates are feasible (i.e., h p +1 (i.e., u 0 , u t +1 ∈ U i ). i i Convergence: Given any collection of disjoint neighborhoods and a feasible initialization, the strictly convex finite horizon cost functional J ( u t 1 , ..., u t n ) is non-increasing at each outer iterate t and converges as t → ∞ . Optimality: Given any collection of disjoint neighborhoods, a feasible initialization, strictly convex and quadratic cost J , and closed convex control constraint sets U i , the cost J ( u t 1 , ..., u t n ) converges to the optimal cost J ( u ∗ 1 , ..., u ∗ n ), and the iterates ( u t 1 , ..., u t n ) converge to the unique optimal solution ( u ∗ 1 , ..., u ∗ n ), as t → ∞ . 2 [AUCC2012]A. Farhadi, M. Cantoni, and P. M. Dower.

Performance, Information Pattern Trade-offs and Computational Complexity Analysis of a Consensus Based Distributed Optimization Method Distributed Optimization Method Interaction strength decomposition method � � � � ��� � � � � � � �� �� �� � � � � � �� �� � � � � Figure: Left: Communication graph. Right: Interaction strength graph summarizing the effects of decision variables on subsystems. No hopping is allowed for intra-neighborhood communication ⇒ Following the communication graph, the size of each neighborhood must be at most 2: Option1: { S 2 , S 3 } , { S 4 , S 5 } , { S 6 , S 1 } Option2: { S 1 , S 2 } , { S 3 , S 4 } , { S 5 , S 6 } Following interaction strength graph, option 2 is selected.