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

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Min Max problems

Min Max problems appear in:

Robust control

Game theory

Risk management

Every problem involving uncertainty

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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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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+

−

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What is control?






K

Gu

yre+

−

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What is control?






K

Gu y

re+

−

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What is control?






K

Gu yr

e+

−

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What is control?






K

Gu yr

e+

−

4 / 23


What is control?






K Gu yr

e+

−

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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

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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


Plan



3 Benchmark

4 Conclusion

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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.

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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.

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Main Branch and bound

x

Cx = 0

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x

Cx = 0

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x

Cx = 0

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x

Cx = 0

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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



f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx,max]

11 / 23



f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx,max]

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f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx,max]

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f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx,max]

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f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx,max]

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f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx,max]

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f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx,max]

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f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx,max]

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f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx,max]

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f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

lbx

∪

[yx,max]

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f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

lbx

∪

[yx,max]

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f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

lbx

∪

[yx,max]

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f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

lbx

∪

[yx,max]

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f(x, .)

y

Cxy(x, .)

y

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

ubx

lbx

lbx

∪

[yx,max]

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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

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x

Cx = 0Cx = 0

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

x1 x2

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

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x

Cx = 0Cx = 0

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

x1 x2


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x

Cx = 0Cx = 0

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

x1 x2


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x

Cx = 0Cx = 0

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

x1 x2


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x

Cx = 0Cx = 0

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

x1 x2


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x

Cx = 0Cx = 0

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

x1 x2


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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]

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f(x, .)

y

Cxy(x, .)

y

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx1,max]

14 / 23



f(x, .)

y

Cxy(x, .)

y

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx1,max]

14 / 23



f(x, .)

y

Cxy(x, .)

y

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx1,max]

14 / 23



f(x, .)

y

Cxy(x, .)

y

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx1,max]

14 / 23



f(x, .)

y

Cxy(x, .)

y

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx1,max]

14 / 23



f(x, .)

y

Cxy(x, .)

y

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx1,max]

14 / 23



f(x, .)

y

Cxy(x, .)

y

f(x1, .)

f(x1, .)

y

Cxy(x1, .)

y

ubx

lbx

ubx

lbx

ubx

lbxlbx

∪

[yx1,max]

14 / 23



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



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



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



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



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



x

Cx = 0Cx = 0

x1

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

x2


[fmax(x1)], [yx1,max]

x11 x12

[fmax(x1)], [yx1,max] [fmax(x1)], [yx1,max]

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x

Cx = 0Cx = 0

x1

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

x2



x11 x12


15 / 23



x

Cx = 0Cx = 0

x1

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

x2



x11 x12


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Plan



3 Benchmark

4 Conclusion

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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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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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Performance profile: cpu time

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Performance profile: number of function evaluation

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Plan



3 Benchmark

4 Conclusion

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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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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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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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