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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 1 / 23
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Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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Page 1: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Global Optimization of continuous MinMaxproblem

Dominique Monnet, Jordan Ninin, Benoıt ClementLAB-STICC, UMR 6285 / ENSTA-Bretagne

1 / 23

Page 2: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Page 3: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Page 4: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

What is control?

Dynamic system (Robot, Missile, Dam, Washingmachine...).

Actuators (Motor, Steering wheel, Flap, ...).

Sensors (INS, Sonar, Temperature/Pressure sensor, ...).

Reference to follow.

Controller to close the loop.

K

G

u yre+

4 / 23

Page 5: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

What is control?

Dynamic system (Robot, Missile, Dam, Washingmachine...).

Actuators (Motor, Steering wheel, Flap, ...).

Sensors (INS, Sonar, Temperature/Pressure sensor, ...).

Reference to follow.

Controller to close the loop.

K

Gu

yre+

4 / 23

Page 6: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

What is control?

Dynamic system (Robot, Missile, Dam, Washingmachine...).

Actuators (Motor, Steering wheel, Flap, ...).

Sensors (INS, Sonar, Temperature/Pressure sensor, ...).

Reference to follow.

Controller to close the loop.

K

Gu y

re+

4 / 23

Page 7: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

What is control?

Dynamic system (Robot, Missile, Dam, Washingmachine...).

Actuators (Motor, Steering wheel, Flap, ...).

Sensors (INS, Sonar, Temperature/Pressure sensor, ...).

Reference to follow.

Controller to close the loop.

K

Gu yr

e+

4 / 23

Page 8: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

What is control?

Dynamic system (Robot, Missile, Dam, Washingmachine...).

Actuators (Motor, Steering wheel, Flap, ...).

Sensors (INS, Sonar, Temperature/Pressure sensor, ...).

Reference to follow.

Controller to close the loop.

K

Gu yr

e+

4 / 23

Page 9: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

What is control?

Dynamic system (Robot, Missile, Dam, Washingmachine...).

Actuators (Motor, Steering wheel, Flap, ...).

Sensors (INS, Sonar, Temperature/Pressure sensor, ...).

Reference to follow.

Controller to close the loop.

K Gu yr

e+

4 / 23

Page 10: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Frequency constraint

10 -3 10 -2 10 -1 10 0 10 1

Pulsation (rad/s)

-100

-80

-60

-40

-20

0

20

40

Magnitude

(dB)

Bode Diagrams

WTr!e

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)| ⇐⇒ supω

(|Tr→e(K, iω)W−1(iω)|) ≤ 1

5 / 23

Page 11: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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 (Routhcriterion).

R(K) ≤ 0 is a non-convex rational system.

Problem formulationminK

supω|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 Algorithm6 / 23

Page 12: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Page 13: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Min max problem formulation

We search x∗ ∈ X such that supy∈Y

f(x∗, y) is minimal.

Constrained Min max problem

minx∈X

supy∈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.

8 / 23

Page 14: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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 L1 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 supy∈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([minx∈L

lbx, minx∈L,C(x)≤0

ubx]) ≤ ε.

9 / 23

Page 15: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Main Branch and bound

x

Cx = 0

10 / 23

Page 16: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Main Branch and bound

x

Cx = 0

10 / 23

Page 17: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Main Branch and bound

x

Cx = 0

10 / 23

Page 18: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Main Branch and bound

x

Cx = 0

10 / 23

Page 19: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx,max]

11 / 23

Page 20: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx,max]

11 / 23

Page 21: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx,max]

11 / 23

Page 22: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx,max]

11 / 23

Page 23: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx,max]

11 / 23

Page 24: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx,max]

11 / 23

Page 25: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx,max]

11 / 23

Page 26: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx,max]

11 / 23

Page 27: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx,max]

11 / 23

Page 28: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx,max]

11 / 23

Page 29: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

lbx

[yx,max]

11 / 23

Page 30: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

lbx

[yx,max]

11 / 23

Page 31: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

lbx

[yx,max]

11 / 23

Page 32: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

lbx

[yx,max]

11 / 23

Page 33: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

lbx

[yx,max]

11 / 23

Page 34: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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) = {supyf(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

12 / 23

Page 35: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Main Branch and bound

x

Cx = 0Cx = 0

[fmax(x)], [yx,max]

x1 x2

[fmax(x)], [yx,max] [fmax(x)], [yx,max]

13 / 23

Page 36: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Main Branch and bound

x

Cx = 0Cx = 0

[fmax(x)], [yx,max]

x1 x2

[fmax(x)], [yx,max] [fmax(x)], [yx,max]

13 / 23

Page 37: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Main Branch and bound

x

Cx = 0Cx = 0

[fmax(x)], [yx,max]

x1 x2

[fmax(x)], [yx,max] [fmax(x)], [yx,max]

13 / 23

Page 38: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Main Branch and bound

x

Cx = 0Cx = 0

[fmax(x)], [yx,max]

x1 x2

[fmax(x)], [yx,max] [fmax(x)], [yx,max]

13 / 23

Page 39: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Main Branch and bound

x

Cx = 0Cx = 0

[fmax(x)], [yx,max]

x1 x2

[fmax(x)], [yx,max] [fmax(x)], [yx,max]

13 / 23

Page 40: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Main Branch and bound

x

Cx = 0Cx = 0

[fmax(x)], [yx,max]

x1 x2

[fmax(x)], [yx,max] [fmax(x)], [yx,max]

13 / 23

Page 41: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx1,max]

14 / 23

Page 42: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx1,max]

14 / 23

Page 43: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx1,max]

14 / 23

Page 44: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx1,max]

14 / 23

Page 45: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx1,max]

14 / 23

Page 46: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx1,max]

14 / 23

Page 47: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx1,max]

14 / 23

Page 48: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

[yx1,max]

14 / 23

Page 49: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

lbx

[yx1,max]

14 / 23

Page 50: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

lbx

[yx1,max]

14 / 23

Page 51: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

lbx

[yx1,max]

14 / 23

Page 52: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

lbx

[yx1,max]

14 / 23

Page 53: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

lbx

[yx1,max]

14 / 23

Page 54: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Main Branch and bound

x

Cx = 0Cx = 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

Page 55: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Main Branch and bound

x

Cx = 0Cx = 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]

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

Main Branch and bound

x

Cx = 0Cx = 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]

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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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Page 58: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Examples

Problems Obj. func. x dim y dim Cx CxyArticle example[1] other 2 2 no noArticle example[3] polynomial 1 1 no yesArticle example[3] trigonometric 1 1 no yes

Control rational 3 1 yes noControl rational 4 1 yes noControl rational 2 1 yes noControl rational 4 1 yes noControl rational 4 1 yes no

Risk Management[2] polynomial 2 2 no noRisk Management[2] polynomial 2 2 no noRisk Management[2] polynomial 2 2 no noRisk Management[2] polynomial 2 3 no noRisk 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 maximizationproblem, inclusion properties used → B 1000.

10 bisections are performed in the maximization problem,inheritance properties not used → NH 10.

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

Performance profile: cpu time

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

Performance profile: number of function evaluation

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Page 62: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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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Page 63: Global Optimization of continuous MinMax problem · 2016-10-11 · Min Max problem in controlGlobal optimization for Min max problemsBenchmarkConclusion Global Optimization of continuous

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

Conclusion

Solver for non-convex problems (non-convex objectivefunction and non-convex constraints).

Taking advantage of Inclusion properties save computationtime.

Finding the best number of bisection is difficult.

Next steps:

Test the algorithm on more examples.

Improve convergence time (monotonicity tests, affinearithmetic, ...).

How to find the number of bisection?

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

E. Carrizosa and F. Messine. A branch and bound methodfor global robust optimization. Proc. 12th globaloptimization workshop (Malaga, Spain, September 2014),2014.

B. Rustem and M. Howe. Algorithms for Worst-CaseDesign and Applications to Risk Management. PrincetonUniversity Press, 2002.

M. Sainz, P. Herrero, J. Armengol, and J. Vehı. Continuousminimax optimization using modal intervals. Journal ofMathematical Analysis and Applications, 339(1):18–30,2008.

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