global optimization of continuous minmax problem · 2016-10-11 · min max problem in controlglobal...

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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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.

u yre+

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What is control?

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What is control?

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What is control?

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What is control?

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What is control?

K Gu yr

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Frequency constraint

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

Pulsation (rad/s)

Magnitude

Bode Diagrams

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

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

Cx = 0

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Cx = 0

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Cx = 0

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Cx = 0

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Secondary Branch and Bound algorithm: maximization

f(x, .)

Cxy(x, .)

lbxlbx

[yx,max]

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f(x, .)

Cxy(x, .)

lbxlbx

[yx,max]

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f(x, .)

Cxy(x, .)

lbxlbx

[yx,max]

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f(x, .)

Cxy(x, .)

lbxlbx

[yx,max]

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f(x, .)

Cxy(x, .)

lbxlbx

[yx,max]

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f(x, .)

Cxy(x, .)

lbxlbx

[yx,max]

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f(x, .)

Cxy(x, .)

lbxlbx

[yx,max]

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f(x, .)

Cxy(x, .)

lbxlbx

[yx,max]

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f(x, .)

Cxy(x, .)

lbxlbx

[yx,max]

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f(x, .)

Cxy(x, .)

lbxlbx

[yx,max]

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f(x, .)

Cxy(x, .)

[yx,max]

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f(x, .)

Cxy(x, .)

[yx,max]

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f(x, .)

Cxy(x, .)

[yx,max]

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f(x, .)

Cxy(x, .)

[yx,max]

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f(x, .)

Cxy(x, .)

[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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Cx = 0Cx = 0

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

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

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Cx = 0Cx = 0

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

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Cx = 0Cx = 0

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

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Cx = 0Cx = 0

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

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Cx = 0Cx = 0

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

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Cx = 0Cx = 0

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

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Main Branch and bound with Inclusion properties

f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

lbxlbx

[yx1,max]

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f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

lbxlbx

[yx1,max]

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f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

lbxlbx

[yx1,max]

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f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

lbxlbx

[yx1,max]

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f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

lbxlbx

[yx1,max]

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f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

lbxlbx

[yx1,max]

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f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

lbxlbx

[yx1,max]

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f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

lbxlbx

[yx1,max]

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f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

[yx1,max]

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f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

[yx1,max]

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f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

[yx1,max]

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f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

[yx1,max]

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f(x, .)

Cxy(x, .)

f(x1, .)

Cxy(x1, .)

[yx1,max]

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Cx = 0Cx = 0

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

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

x11 x12

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

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Cx = 0Cx = 0

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

x11 x12

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Cx = 0Cx = 0

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

x11 x12

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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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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...

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