Global Optimization of continuous MinMax problem Dominique Monnet, - PowerPoint PPT Presentation

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Global Optimization of continuous MinMax problem Dominique Monnet, Jordan Ninin, Benoˆ ıt Cl´ ement LAB-STICC, UMR 6285 / ENSTA-Bretagne 1 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Min Max problems Min Max problems appear in: Robust control Game theory Risk management Every problem involving uncertainty 2 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Plan 1 Min Max problem in control 2 Global optimization for Min max problems 3 Benchmark 4 Conclusion 3 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion What is control? Dynamic system (Robot, Missile, Dam, Washing machine...). G 4 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion What is control? Dynamic system (Robot, Missile, Dam, Washing machine...). Actuators (Motor, Steering wheel, Flap, ...). u G 4 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion What is control? Dynamic system (Robot, Missile, Dam, Washing machine...). Actuators (Motor, Steering wheel, Flap, ...). Sensors (INS, Sonar, Temperature/Pressure sensor, ...). u y G 4 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion What is control? Dynamic system (Robot, Missile, Dam, Washing machine...). Actuators (Motor, Steering wheel, Flap, ...). Sensors (INS, Sonar, Temperature/Pressure sensor, ...). Reference to follow. u y r G 4 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion What is control? Dynamic system (Robot, Missile, Dam, Washing machine...). Actuators (Motor, Steering wheel, Flap, ...). Sensors (INS, Sonar, Temperature/Pressure sensor, ...). Reference to follow. + e u y r G − 4 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion What is control? Dynamic system (Robot, Missile, Dam, Washing machine...). Actuators (Motor, Steering wheel, Flap, ...). Sensors (INS, Sonar, Temperature/Pressure sensor, ...). Reference to follow. Controller to close the loop. + e u y r K G − 4 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Frequency constraint Frequency constraint on e ( iω ): we want | e ( iω ) r ( iω ) | = | T r → e ( K, iω ) | to be small ω ( | T r → e ( K, iω ) W − 1 ( iω ) | ) ≤ 1 ∀ ω ≥ 0 , | T r → e ( K, iω ) | ≤ | W ( iω ) | ⇐ ⇒ sup 5 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Min max problem formulation Stability constraint: The closed loop system is stable ⇐ ⇒ R ( K ) ≤ 0 (Routh criterion). R ( K ) ≤ 0 is a non-convex rational system. Problem formulation ω | T r → e ( K, iω ) W − 1 ( iω ) | ,  min K sup    s.t. R ( K ) ≤ 0  We want: an enclosure of the minimum. reliable computation. → Interval Based Branch and Bound Algorithm 6 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Plan 1 Min Max problem in control 2 Global optimization for Min max problems 3 Benchmark 4 Conclusion 7 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Min max problem formulation We search x ∗ ∈ X such that sup f ( x ∗ , y ) is minimal. y ∈Y Constrained Min max problem  min x ∈X sup f ( x, y ) ,   y ∈Y    s.t. C x ( x ) ≤ 0     C xy ( x, y ) ≤ 0  X and Y are bounded. f , C x and C xy can be evaluated with interval computation. 8 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Main Branch and bound algorithm: minimization Interval Based Branch and Bound Algorithm Init: push X in L 1 Choose a box x from L . 2 Contract x w.r.t C x ( x ) ≤ 0 using CSP techniques. 3 Compute [ lb x , ub x ] an enclosure of sup f ( x , y ). y ∈Y 4 Try to find a good feasible solution in x . 5 Update best current solution. 6 Bisect x into x 1 and x 2 , push x 1 and x 2 in L . Stop criterion: width ([min x ∈L lb x , x ∈L , C ( x ) ≤ 0 ub x ]) ≤ ǫ . min 9 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Main Branch and bound x 10 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Main Branch and bound C x = 0 x 10 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Main Branch and bound C x = 0 x 10 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Main Branch and bound C x = 0 x 10 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization f ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization f ( x , . ) y C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization f ( x , . ) y C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization ub x f ( x , . ) y lb x C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization ub x f ( x , . ) y lb x C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization ub x f ( x , . ) y lb x C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization ub x f ( x , . ) y lb x C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization ub x f ( x , . ) y lb x C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization ub x f ( x , . ) y lb x C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization ub x f ( x , . ) lb x y C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization ub x f ( x , . ) lb x y C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization ub x f ( x , . ) lb x y C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization ub x f ( x , . ) lb x y C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization ub x f ( x , . ) lb x y C xy ( x , . ) y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Secondary Branch and Bound algorithm: maximization ub x f ( x , . ) lb x y ∪ C xy ( x , . ) [ y x ,max ] y 11 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Inclusion properties Let be x ⊆ X and y ⊆ Y , we denote f max ( x ) = { sup f ( x, y ) , x ∈ x } y y x ,max = { y ∈ Y|∃ x ∈ x , y maximizes f ( x, y ) } Let be x 1 ⊆ x . Proposition f max ( x 1 ) ⊆ f max ( x ) y x 1 ,max ⊆ y x ,max C x ( x ) ≤ 0 = ⇒ C x ( x 1 ) ≤ 0 C xy ( x , y ) ≤ 0 = ⇒ C xy ( x 1 , y ) ≤ 0 12 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Main Branch and bound C x = 0 C x = 0 x [ f max ( x )] , [ y x ,max ] 13 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Main Branch and bound C x = 0 C x = 0 x [ f max ( x )] , [ y x ,max ] x 1 x 2 13 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Main Branch and bound C x = 0 C x = 0 x [ f max ( x )] , [ y x ,max ] x 1 x 2 [ f max ( x )] , [ y x ,max ] [ f max ( x )] , [ y x ,max ] 13 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Main Branch and bound C x = 0 C x = 0 x [ f max ( x )] , [ y x ,max ] x 1 x 2 [ f max ( x )] , [ y x ,max ] [ f max ( x )] , [ y x ,max ] 13 / 23

Min Max problem in control Global optimization for Min max problems Benchmark Conclusion Main Branch and bound C x = 0 C x = 0 x [ f max ( x )] , [ y x ,max ] x 1 x 2 [ f max ( x )] , [ y x ,max ] [ f max ( x )] , [ y x ,max ] 13 / 23