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

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

Jan 18, 2023 311 likes •971 views

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

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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

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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

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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

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

What is control?

Dynamic system (Robot, Missile, Dam, Washing machine...). G

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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, ...). G u

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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, ...). G u y

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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. G u y r

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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. G u y r e + −

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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. K G u y r e + −

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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)| = |Tr→e(K, iω)|

to be small ∀ω ≥ 0, |Tr→e(K, iω)| ≤ |W(iω)| ⇐ ⇒ sup

ω (|Tr→e(K, iω)W −1(iω)|) ≤ 1

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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      min

K sup ω |Tr→e(K, iω)W −1(iω)|,

s.t. R(K) ≤ 0 We want: an enclosure of the minimum. reliable computation. → Interval Based Branch and Bound Algorithm

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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

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Min max problem formulation

We search x∗ ∈ X such that sup

y∈Y

f(x∗, y) is minimal. Constrained Min max problem            min

x∈X sup y∈Y

f(x, y), s.t. Cx(x) ≤ 0 Cxy(x, y) ≤ 0 X and Y are bounded. f, Cx and Cxy can be evaluated with interval computation.

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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 Cx(x) ≤ 0 using CSP techniques. 3 Compute [lbx, ubx] an enclosure of sup

y∈Y

f(x, y).

4 Try to find a good feasible solution in x. 5 Update best current solution. 6 Bisect x into x1 and x2, push x1 and x2 in L.

Stop criterion: width([min

x∈L lbx,

min

x∈L,C(x)≤0 ubx]) ≤ ǫ.

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Main Branch and bound

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound

x Cx = 0 10 / 23

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound

x Cx = 0 10 / 23

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound

x Cx = 0 10 / 23

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Secondary Branch and Bound algorithm: maximization

f(x, .) y Cxy(x, .) y ubx lbx

∪

[yx,max]

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Inclusion properties

Let be x ⊆ X and y ⊆ Y, we denote fmax(x) = {sup

y

f(x, y), x ∈ x} yx,max = {y ∈ Y|∃x ∈ x, y maximizes f(x, y)} Let be x1 ⊆ x. Proposition fmax(x1) ⊆ fmax(x) yx1,max ⊆ yx,max Cx(x) ≤ 0 = ⇒ Cx(x1) ≤ 0 Cxy(x, y) ≤ 0 = ⇒ Cxy(x1, y) ≤ 0

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound

x Cx = 0 Cx = 0 [fmax(x)], [yx,max] 13 / 23

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound

x Cx = 0 Cx = 0 [fmax(x)], [yx,max] x1 x2 13 / 23

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound

x Cx = 0 Cx = 0 [fmax(x)], [yx,max] x1 x2 [fmax(x)], [yx,max] [fmax(x)], [yx,max] 13 / 23

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound

x Cx = 0 Cx = 0 [fmax(x)], [yx,max] x1 x2 [fmax(x)], [yx,max] [fmax(x)], [yx,max] 13 / 23

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound

x Cx = 0 Cx = 0 [fmax(x)], [yx,max] x1 x2 [fmax(x)], [yx,max] [fmax(x)], [yx,max] 13 / 23

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound

x Cx = 0 Cx = 0 [fmax(x)], [yx,max] x1 x2 [fmax(x)], [yx,max] [fmax(x)], [yx,max] 13 / 23

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x, .) y Cxy(x, .) y

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x1, .) y Cxy(x1, .) y

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x1, .) y Cxy(x1, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x1, .) y Cxy(x1, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x1, .) y Cxy(x1, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x1, .) y Cxy(x1, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x1, .) y Cxy(x1, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x1, .) y Cxy(x1, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x1, .) y Cxy(x1, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x1, .) y Cxy(x1, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x1, .) y Cxy(x1, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x1, .) y Cxy(x1, .) y ubx lbx

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound with Inclusion properties

f(x1, .) y Cxy(x1, .) y ubx lbx

∪

[yx1,max]

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound

x Cx = 0 Cx = 0 x1 [fmax(x)], [yx,max] x2 [fmax(x)], [yx,max] [fmax(x)], [yx,max] [fmax(x1)], [yx1,max] 15 / 23

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound

x Cx = 0 Cx = 0 x1 [fmax(x)], [yx,max] x2 [fmax(x)], [yx,max] [fmax(x)], [yx,max] [fmax(x1)], [yx1,max] x11 x12 15 / 23

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Main Branch and bound

x Cx = 0 Cx = 0 x1 [fmax(x)], [yx,max] x2 [fmax(x)], [yx,max] [fmax(x)], [yx,max] [fmax(x1)], [yx1,max] x11 x12 [fmax(x1)], [yx1,max] [fmax(x1)], [yx1,max] 15 / 23

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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

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Examples

Problems

  • Obj. func.

x dim y dim Cx Cxy Article example[1]

  • ther

2 2 no no Article example[3] polynomial 1 1 no yes Article example[3] trigonometric 1 1 no yes Control rational 3 1 yes no Control rational 4 1 yes no Control rational 2 1 yes no Control rational 4 1 yes no Control rational 4 1 yes no Risk Management[2] polynomial 2 2 no no Risk Management[2] polynomial 2 2 no no Risk Management[2] polynomial 2 2 no no Risk Management[2] polynomial 2 3 no no Risk Management[2] polynomial 3 3 no no

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Algorithm features

Algorithm is tested with four features: 10 bisections are performed in the maximization problem, inclusion properties used → B 10. 100 bisections are performed in the maximization problem, inclusion properties used → B 100. 1000 bisections are performed in the maximization problem, inclusion properties used → B 1000. 10 bisections are performed in the maximization problem, inheritance properties not used → NH 10.

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Performance profile: cpu time

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Performance profile: number of function evaluation

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Plan

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

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Min Max problem in control Global optimization for Min max problems Benchmark Conclusion

Conclusion

Solver for non-convex problems (non-convex objective function and non-convex constraints). Taking advantage of Inclusion properties save computation time. Finding the best number of bisection is difficult. Next steps: Test the algorithm on more examples. Improve convergence time (monotonicity tests, affine arithmetic, ...). How to find the number of bisection?

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  • E. Carrizosa and F. Messine. A branch and bound method

for global robust optimization. Proc. 12th global

  • ptimization workshop (M´

alaga, Spain, September 2014), 2014.

  • B. Rustem and M. Howe. Algorithms for Worst-Case

Design and Applications to Risk Management. Princeton University Press, 2002.

  • M. Sainz, P. Herrero, J. Armengol, and J. Veh´

ı. Continuous minimax optimization using modal intervals. Journal of Mathematical Analysis and Applications, 339(1):18–30, 2008.

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