A constrained optimization problem under uncertainty Raluca Andrei, - - PowerPoint PPT Presentation

A constrained optimization problem under uncertainty

Raluca Andrei, Gert de Cooman, Erik Quaeghebeur & Keivan Shariatmadar

CMU Games & Decision Meeting

March 3rd, 2010

Toy problem: two-component massless rod

Y1 Y2 a L l1 = (1−x)L l2 = xL

Toy problem: two-component massless rod, tensile load

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Toy problem: two-component massless rod, tensile load

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that d2 < D.

Two-component massless rod, tensile load: FE analysis

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that d2 < D.

Two-component massless rod, tensile load: FE analysis

FE analysis 3 nodes, boundary conditions

c1 +c2 −c2 −c2 c2 d1 d2

• =

F

• ,

ci = Yia

li .

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that d2 < D.

Two-component massless rod, tensile load: FE analysis

FE analysis 3 nodes, boundary conditions

c1 +c2 −c2 −c2 c2 d1 d2

• =

F

• ,

ci = Yia

li .

Solution solving the system (analytically) gives

d1 = FL

a 1−x Y1 ,

d2 = d1 + FL

a x Y2 .

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that d2 < D.

Two-component massless rod, tensile load: FE analysis

FE analysis 3 nodes, boundary conditions

c1 +c2 −c2 −c2 c2 d1 d2

• =

F

• ,

ci = Yia

li .

Solution solving the system (analytically) gives

d1 = FL

a 1−x Y1 ,

d2 = d1 + FL

a x Y2 .

Goal Maximize x under the constraint that 1−x

Y1 + x Y2 < Da FL.

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that d2 < D.

Two-component rod, tensile load: design optimization

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that 1−x

Y1 + x Y2 < Da FL.

Two-component rod, tensile load: design optimization

Precisely known elastic moduli Y1 and Y2 This problem is

◮ a classical constrained optimization problem; ◮ considered ‘solved’.

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that 1−x

Y1 + x Y2 < Da FL.

Two-component rod, tensile load: design optimization

Precisely known elastic moduli Y1 and Y2 This problem is

◮ a classical constrained optimization problem; ◮ considered ‘solved’.

Uncertainty about elastic moduli Y1 and Y2 This problem is

◮ a constrained optimization problem under uncertainty; ◮ not well-posed as such.

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that 1−x

Y1 + x Y2 < Da FL.

Two-component rod, tensile load: design optimization

Precisely known elastic moduli Y1 and Y2 This problem is

◮ a classical constrained optimization problem; ◮ considered ‘solved’.

Uncertainty about elastic moduli Y1 and Y2 This problem is

◮ a constrained optimization problem under uncertainty; ◮ not well-posed as such.

Approach:

◮ reformulate as a well-posed decision problem; ◮ solve the decision problem, i.e.,

derive a classical constrained optimization problem.

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that 1−x

Y1 + x Y2 < Da FL.

Overview

Toy problem General problem formulation Uncertainty models Optimality criteria Probabilistic and indeterminacy aspects of uncertainty Objective Results Application: bridge design for vehicle-pillar collisions

A constrained optimization problem under uncertainty

Goal Maximize f(x) under the constraint that xRY.

x optimization variable (values in X ) f objective function (from X to R) Y random variable (realizations y in Y ) R relation on X ×Y .

A constrained optimization problem under uncertainty

Goal Maximize f(x) under the constraint that xRY.

x optimization variable (values in X ) f objective function (from X to R) Y random variable (realizations y in Y ) R relation on X ×Y .

Decision problem Find the optimal decisions x:

◮ associate a utility function with every decision z:

Gz(y) = f(z)IzR +LIzR =

• f(z),

zRy, L, zRy,

with penalty value L < inff;

f(x) x z f(z) Gz(y) y zR zR L f(z)

A constrained optimization problem under uncertainty

Goal Maximize f(x) under the constraint that xRY.

x optimization variable (values in X ) f objective function (from X to R) Y random variable (realizations y in Y ) R relation on X ×Y .

Decision problem Find the optimal decisions x:

◮ associate a utility function with every decision z:

Gz(y) = f(z)IzR +LIzR =

• f(z),

zRy, L, zRy,

with penalty value L < inff;

f(x) x z f(z) Gz(y) y zR zR L f(z)

◮ choose an optimality criterion, e.g., maximinity, maximality.

Uncertainty models

Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions Gz on Y for all z in X .

Uncertainty models

Random variable Y Formal model for the uncertainty about y in Y . Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions Gz on Y for all z in X .

Uncertainty models

Random variable Y Formal model for the uncertainty about y in Y . Lower and upper expectation With (almost) all typical uncertainty models correspond lower and upper expectation operators (E and E), or (almost) equivalently, a set of linear expectation operators M :

EM (G) := infE∈M E(G), EM (G) := supE∈M E(G), ME := {E : E ≥ E}.

Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions Gz on Y for all z in X .

Uncertainty models

Random variable Y Formal model for the uncertainty about y in Y . Lower and upper expectation With (almost) all typical uncertainty models correspond lower and upper expectation operators (E and E), or (almost) equivalently, a set of linear expectation operators M :

EM (G) := infE∈M E(G), EM (G) := supE∈M E(G), ME := {E : E ≥ E}.

Examples

◮ probabilities (measures, PMF, PDF, CDF); ◮ upper and/or lower of the above

(inner/outer measures, Choquet capacities, p-boxes);

◮ intervals, vacuous expectations: EA(G) := infy∈A G(y); ◮ possibility distributions, belief functions, . . . ◮ convex mixtures of the lot (e.g., contamination models).

Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions Gz on Y for all z in X .

Optimality criteria: maximizing expected utility generalized

Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions Gz on Y for all z in X .

Optimality criteria: maximizing expected utility generalized

Maximinity Worst-case reasoning; optimal x maximize the lower (minimal) expected utility (P(A) := E(IA)):

E(Gx) = supz∈X E(Gz) = supz∈X E

• f(z)IzR +LIzR
• = L+supz∈X
• f(z)−L
• P(zR).

Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions Gz on Y for all z in X .

Optimality criteria: maximizing expected utility generalized

Maximinity Worst-case reasoning; optimal x maximize the lower (minimal) expected utility (P(A) := E(IA)):

E(Gx) = supz∈X E(Gz) = supz∈X E

• f(z)IzR +LIzR
• = L+supz∈X
• f(z)−L
• P(zR).

Maximality Optimal x are undominated in pairwise comparisons with all

• ther decisions:

0 ≤ infz∈X E(Gx −Gz) = infz∈X E

• f(x)−f(z)
• IxR∩zR +
• f(x)−L
• IxR∩zR +
• L−f(z)
• IxR∩zR
• .

Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions Gz on Y for all z in X .

Optimality criteria: maximizing expected utility generalized

Maximinity Worst-case reasoning; optimal x maximize the lower (minimal) expected utility (P(A) := E(IA)):

E(Gx) = supz∈X E(Gz) = supz∈X E

• f(z)IzR +LIzR
• = L+supz∈X
• f(z)−L
• P(zR).

Maximality Optimal x are undominated in pairwise comparisons with all

• ther decisions:

0 ≤ infz∈X E(Gx −Gz) = infz∈X E

• f(x)−f(z)
• IxR∩zR +
• f(x)−L
• IxR∩zR +
• L−f(z)
• IxR∩zR
• .

Others Maximaxity, E-admissibility, interval dominance Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions Gz on Y for all z in X .

Probabilistic and indeterminacy aspects of uncertainty

Example X = Y := R, R :=≤.

f(x) x y Ry Ry supf|Ry gy(x) = Gx(y) x y L supgy

Probabilistic and indeterminacy aspects of uncertainty

Example X = Y := R, R :=≤.

f(x) x supf|Ry x1 y1 x2 x3 x4 x5 x6 gy(x) = Gx(y) x y L supgy

Indeterminacy Assume y can be either y1 or y2, but nothing more is known.

gy1(x) x y1 L supgy1 x1 x2 gy2(x) x x1 y2 L supgy2 x2

Gx3(yi) i L supgy1 supgy2 1 2 Gx1(yi) i 1 2 Gx4(yi) i 1 2 Gy1(yi) i 1 2 Gx5(yi) i 1 2 Gx2(yi) i 1 2 Gx6(yi) i 1 2

Probabilistic and indeterminacy aspects of uncertainty

Example X = Y := R, R :=≤.

f(x) x supf|Ry x1 y1 x2 x3 x4 x5 x6 gy(x) = Gx(y) x y L supgy

Probabilistic Assume that y1 and y2 are equally likely.

gy1+gy2 2

(x) x y1 y2 sup

gy1+gy2 2

= supgy1 supgy2 x1 x2 L

E(Gx3) L supgy1 supgy2 E(Gx1) E(Gx4) E(Gy1) E(Gx5) E(Gx2) E(Gx6)

Objective, deliverables, and a disclaimer

Research objective decision problem solutions for combinations

• f various uncertainty models and optimality criteria.

Deliverables A solution toolbox for a specific, but quite general class

• f decision problems under uncertainty.

Objective, deliverables, and a disclaimer

Research objective decision problem solutions for combinations

• f various uncertainty models and optimality criteria.

Deliverables A solution toolbox for a specific, but quite general class

• f decision problems under uncertainty.

Disclaimer No reduction in the computational complexity;

• ne faces

◮ an optimization problem to find

the uncertainty-independent constraints,

◮ the resulting classical constrained optimization problem.

Results: probabilities

Optimal decision when Y is described by a probability P. Maximizing expected utility filler text

◮ General case:

argsupz∈X (f(z)−L)P(zR).

◮ Example: X = Y := R, R :=≤.

argsupz∈R

• f(z)−L
• 1−F(z)
• ,

where FY(x) := P(R≤x) = 1−P(x ≤) is a continuous CDF .

Results: vacuous models

Optimal decision when Y is described by a vacuous lower expectation relative to A ⊆ Y . Maximinity

◮ General case:

argsupz∈RA f(z), RA :=

y∈A Ry. ◮ Example: X = Y := R, R :=≤, A := [a,b].

argsupz≤a f(z).

Results: vacuous models

Optimal decision when Y is described by a vacuous lower expectation relative to A ⊆ Y . Maximinity

◮ General case:

argsupz∈RA f(z), RA :=

y∈A Ry. ◮ Example: X = Y := R, R :=≤, A := [a,b].

argsupz≤a f(z).

Maximality

◮ General case:

x ∈ RA

such that

f(x) = supz∈RA f(z), RA :=

y∈A Ry. ◮ Example: X = Y := R, R :=≤, A := [a,b].

x ≤ b

such that

f(x) ≥ supz≤a f(z).

Results: possibility distributions

Optimal decision when Y is described by a possibility distribution π on Y ; P(A) := 1−supy∈Y \A π(y). Maximinity

◮ General case:

argsupz∈X

• f(z)−L
• 1−supy∈zR π(y)
• .

◮ Example: X = Y := R, R :=≤, continuous π with

minimal mode c ∈ R.

argsupz<c

• f(z)−L
• 1−π(z)
• .

Bridge design: vehicle colliding into pillar

Vehicle parameters mass m, stiffness k, initial speed v0, average deceleration a, and swerve angle α. Bridge parameters pillar design loads F= (longitud.) and F⊥ (perpendic.). What is the optimal lateral distance x between the vehicle and curb that ensures structural integrity?

veh.: m, k pillar

x α v0 a F⊥ F= Fveh.

Bridge design: vehicle colliding into pillar

Vehicle parameters mass m, stiffness k, initial speed v0, average deceleration a, and swerve angle α. Bridge parameters pillar design loads F= (longitud.) and F⊥ (perpendic.). What is the optimal lateral distance x between the vehicle and curb that ensures structural integrity?

veh.: m, k pillar

x α v0 a F⊥ F= Fveh.

Structural integrity constraint

• Fveh. cosα ≤ F=,
• Fveh. sinα ≤ F⊥,
• Fveh. =
• mk(v2

0 −2ax/sinα).

Bridge design: optimization problem under uncertainty

Goal Choose an optimal x under the constraint that

• Fveh. cosα ≤ F= and
• Fveh. sinα ≤ F⊥.

m, k x α v0 a F⊥ F= Fveh.

Bridge design: optimization problem under uncertainty

Objective function Based on dimensions-dependent building costs:

f(x) := −45B

• (L1 +2d)2 +2L2

2

• ,

x [m] 10 20 30 40 50 −log10|f(x)| −6 −6.5 −7

where B = 14, L1 = 33, L2 = 15 for a typical 3-span bridge. Goal Choose an optimal x under the constraint that

• Fveh. cosα ≤ F= and
• Fveh. sinα ≤ F⊥.

m, k x α v0 a F⊥ F= Fveh.

Bridge design: optimization problem under uncertainty

Objective function Based on dimensions-dependent building costs:

f(x) := −45B

• (L1 +2d)2 +2L2

2

• ,

x [m] 10 20 30 40 50 −log10|f(x)| −6 −6.5 −7

where B = 14, L1 = 33, L2 = 15 for a typical 3-span bridge. Penalty value L was difficult to assess, so a number of values between −106.3 and −108.6 were tried. Parameters k = 300 [kN/m]; Y = (m,v0,a,α), independent product. Goal Choose an optimal x under the constraint that

• Fveh. cosα ≤ F= and
• Fveh. sinα ≤ F⊥.

m, k x α v0 a F⊥ F= Fveh.

Bridge design: uncertainty models for the parameters

Mass m [t]

◮ Lorry: normal with mean 20, standard deviation 12,

and realistic range [12,40].

◮ Car: vacuous in the interval [.5,1.6].

Initial velocity v0 [km/h] blabla

◮ Highway: 80, 10, [50,100]. ◮ Urban: 40, 8, [30,70]. ◮ Courtyard: lognormal 15, 5, [5,30]. ◮ Parking: lognormal 5, 5, [5,20].

Average deceleration a [m/s2] Lognormal 4, 1.3, [1,5]. Swerve angle α [° ] Normal 30, 3, [8,45].

Bridge design: maximinity results for different vehicle types

Lorry – Highway F= = 1000, F⊥ = 500 Lorry – Urban F= = 500, F⊥ = 250 Lorry – Courtyard F= = 150, F⊥ = 75 Car – Courtyard F= = 50, F⊥ = 25 Car – Parking F= = 40, F⊥ = 25

Bridge design: maximinity results for different vehicle types

Lorry – Highway F= = 1000, F⊥ = 500

35 40 45 50 55 60 98 98.5 99 99.5 100 50.3 99.0001 98.1721 98.5572 99.227 d P PdR −− d 35 40 45 50 55 60 −4 −3 −2 −1 x 10

8

35.9595 41.5309 45.0725 47.1102 48.5619 49.6973 51.2302 52.535 d* L Optimum d −− L

Lorry – Urban F= = 500, F⊥ = 250 Lorry – Courtyard F= = 150, F⊥ = 75 Car – Courtyard F= = 50, F⊥ = 25 Car – Parking F= = 40, F⊥ = 25

Bridge design: maximinity results for different vehicle types

Lorry – Highway F= = 1000, F⊥ = 500, x = 42 for L = −108. Lorry – Urban F= = 500, F⊥ = 250

15 20 25 30 35 40 98.8 99 99.2 99.4 99.6 99.8 19.475 99.0041 98.7599 98.8408 99.1185 d P PdR −− d 15 20 25 30 35 40 −9 −8 −7 −6 x 10

7

18.5773 18.8581 18.9849 19.5008 19.5465 19.6586 19.9693 d* L Optimum d −− L

Lorry – Courtyard F= = 150, F⊥ = 75 Car – Courtyard F= = 50, F⊥ = 25 Car – Parking F= = 40, F⊥ = 25

Bridge design: maximinity results for different vehicle types

Lorry – Highway F= = 1000, F⊥ = 500, x = 42 for L = −108. Lorry – Urban F= = 500, F⊥ = 250, x = 20 for L = −108. Lorry – Courtyard F= = 150, F⊥ = 75

1 2 3 4 5 6 7 8 9 10 98.5 99 99.5 100 3.466 99.0141 98.3186 99.2004 99.5384 99.6724 99.8277 d P PdR −− d 1 2 3 4 5 6 7 8 9 10 −4 −3 −2 −1 x 10

7

3.1246 3.5988 3.9557 4.1817 4.2804 4.4038 4.4873 4.5854 d* L Optimum d −− L

Car – Courtyard F= = 50, F⊥ = 25 Car – Parking F= = 40, F⊥ = 25

Bridge design: maximinity results for different vehicle types

Lorry – Highway F= = 1000, F⊥ = 500, x = 42 for L = −108. Lorry – Urban F= = 500, F⊥ = 250, x = 20 for L = −108. Lorry – Courtyard F= = 150, F⊥ = 75, x = 3.6 for L = −107. Car – Courtyard F= = 50, F⊥ = 25

1 2 3 4 5 6 7 8 9 10 98.5 99 99.5 100 4 99.003 98.3963 99.1384 99.3944 99.604 99.7395 d P PdR −− d 1 2 3 4 5 6 7 8 9 10 −3 −2 −1 x 10

7

1.8831 3.5675 4.1307 4.4471 4.6828 4.8298 5.0429 5.1969 d* L Optimum d −− L

Car – Parking F= = 40, F⊥ = 25

Bridge design: maximinity results for different vehicle types

Lorry – Highway F= = 1000, F⊥ = 500, x = 42 for L = −108. Lorry – Urban F= = 500, F⊥ = 250, x = 20 for L = −108. Lorry – Courtyard F= = 150, F⊥ = 75, x = 3.6 for L = −107. Car – Courtyard F= = 50, F⊥ = 25, x = 4.0 for L = −107. Car – Parking F= = 40, F⊥ = 25

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 96 97 98 99 100 1.502 99.004 95.8297 99.426999.6452 d P PdR −− d 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −3 −2 −1 x 10

7

0.91312 1.7052 1.8759 1.997 1.9941 1.9994 1.9986 1.9997 d* L Optimum d −− L

Bridge design: maximinity results for different vehicle types

Lorry – Highway F= = 1000, F⊥ = 500, x = 42 for L = −108. Lorry – Urban F= = 500, F⊥ = 250, x = 20 for L = −108. Lorry – Courtyard F= = 150, F⊥ = 75, x = 3.6 for L = −107. Car – Courtyard F= = 50, F⊥ = 25, x = 4.0 for L = −107. Car – Parking F= = 40, F⊥ = 25, x = 1.8 for L = −107.

Bridge design: maximality results for different vehicle types

Car – Courtyard F= = 50, F⊥ = 25, x ∈ [2.4,4.0] for L = −12·106. Car – Parking F= = 40, F⊥ = 25, x ∈ [0.4,1.7] for L = −7·106.