Nonlinear Programming Formulation of Chance-Constraints...NonlinearProgrammingFormulationof...

Nonlinear Programming Formulation ofChance-Constraints

Andreas Wächterjoint with Alejandra Peña-Ordieres and Jim Luedtke

Department of Industrial Engineering and Management SciencesNorthwestern University

Evanston, IL, [email protected]

US & Mexico Workshop on Optimization and Its ApplicationsHuatulco, Mexico

January 9, 2018

Andreas Wächter Nonlinear Programming Formulation of Chance-Constraints

Problem Statement

minx∈X

f (x)

s.t. Pξ[c(x , ξ) ≤ 0] ≥ 1− α

f (x) : Rn → Rc(x , ξ) : Rn × Ξ→ Rm

I Random variable ξ with support ΞI Assume f (x) and c(x , ξ) are sufficiently smooth for all ξ ∈ ΞI X ⊆ Rn: Captures additional constraints

I Goal: Formulate as continuous NLPI Do not want to assume particular probability distributionI Do not want to assume convexity (or convex approximations)I Want to avoid combinatorial approachI Use powerful NLP algorithms and techniques


Problem Statement

minx∈X

f (x)

s.t. Pξ[c(x , ξ) ≤ 0] ≥ 1− α

f (x) : Rn → Rc(x , ξ) : Rn × Ξ→ Rm

I Random variable ξ with support ΞI Assume f (x) and c(x , ξ) are sufficiently smooth for all ξ ∈ ΞI X ⊆ Rn: Captures additional constraintsI Goal: Formulate as continuous NLP

I Do not want to assume particular probability distributionI Do not want to assume convexity (or convex approximations)I Want to avoid combinatorial approachI Use powerful NLP algorithms and techniques


Constraint in Probability SpaceImpose p(x) = P[c(x , ξ) ≤ 0] ≥ 1− α

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

( - 1) - p(x)

y = 0

c(x , ξ) = x2 − 2 + ξξ ∼ N (0, 1)

IssuesI Linearization is poor approximationI Always nonconvexI Used in [Hu, Hong, Zhang 13], [Bremer, Henrion, Möller 15]


Quantile Formulation

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Nomal CDF

I Let Y be a real-valued random variableI (1− α)-quantile:

QY1−α = inf{y ∈ R : P[Y ≤ y ] ≥ 1− α}

p(x) ≥ 1− α ⇐⇒ q1−α(x) = Qc(x ,ξ)1−α ≤ 0



-3 -2 -1 0 1 2 3

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Nomal CDF

I Let Y be a real-valued random variable (Y = c(x , ξ))I (1− α)-quantile:

Qc(x ,ξ)1−α = inf{y ∈ R : P[c(x , ξ) ≤ y ] ≥ 1− α}

p(x) ≥ 1− α ⇐⇒ q1−α(x) = Qc(x ,ξ)1−α ≤ 0



-3 -2 -1 0 1 2 3

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0

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lative

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Nomal CDF

I Let Y be a real-valued random variable (Y = c(x , ξ))I (1− α)-quantile:

Qc(x ,ξ)1−α = inf{y ∈ R : P[c(x , ξ) ≤ y ] ≥ 1− α}

p(x) ≥ 1− α ⇐⇒ q1−α(x) = Qc(x ,ξ)1−α ≤ 0Andreas Wächter Nonlinear Programming Formulation of Chance-Constraints

Choice of Formulation

p(x) ≥ 1− α ⇐⇒ q1−α(x) ≤ 0

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

( - 1) - p(x)

q1 -

(x)

y = 0

I Quantile formulation more suitable for NLP solver


Sample Average Approximation (SAA)

minx∈X

f (x)

s.t. Pξ[c(x , ξ) ≤ 0] ≥ 1− α

f (x) : Rn → Rc(x , ξ) : Rn × Ξ→ R

I Finite scenario set Ξ̂N = {ξ̂1, . . . , ξ̂N} with ξ̂i ∈ Ξ chosen i.i.d.I For simplicity: ci (x) = c(x , ξ̂i )

SAA approximation of quantileI Ensure that M = d(1− α)Ne constraints are satisfiedI Scenario vector: C(x) = (c1(x), . . . , cN(x))I Order constraint values: c[1](x) ≤ c[2](x) ≤ . . . ≤ c[N](x)I Empirical quantile: q̃1−α(C(x)) = c[M](x) ≤ 0



minx∈X

f (x)

s.t. Pξ[c(x , ξ) ≤ 0] ≥ 1− α

f (x) : Rn → Rc(x , ξ) : Rn × Ξ→ R


SAA approximation of quantileI Ensure that M = d(1− α)Ne constraints are satisfied

I Scenario vector: C(x) = (c1(x), . . . , cN(x))I Order constraint values: c[1](x) ≤ c[2](x) ≤ . . . ≤ c[N](x)I Empirical quantile: q̃1−α(C(x)) = c[M](x) ≤ 0



minx∈X

f (x)

s.t. Pξ[c(x , ξ) ≤ 0] ≥ 1− α

f (x) : Rn → Rc(x , ξ) : Rn × Ξ→ R


SAA approximation of quantileI Ensure that M = d(1− α)Ne constraints are satisfiedI Scenario vector: C(x) = (c1(x), . . . , cN(x))

I Order constraint values: c[1](x) ≤ c[2](x) ≤ . . . ≤ c[N](x)I Empirical quantile: q̃1−α(C(x)) = c[M](x) ≤ 0



minx∈X

f (x)

s.t. Pξ[c(x , ξ) ≤ 0] ≥ 1− α

f (x) : Rn → Rc(x , ξ) : Rn × Ξ→ R


SAA approximation of quantileI Ensure that M = d(1− α)Ne constraints are satisfiedI Scenario vector: C(x) = (c1(x), . . . , cN(x))I Order constraint values: c[1](x) ≤ c[2](x) ≤ . . . ≤ c[N](x)

I Empirical quantile: q̃1−α(C(x)) = c[M](x) ≤ 0



minx∈X

f (x)

s.t. Pξ[c(x , ξ) ≤ 0] ≥ 1− α

f (x) : Rn → Rc(x , ξ) : Rn × Ξ→ R


SAA approximation of quantileI Ensure that M = d(1− α)Ne constraints are satisfiedI Scenario vector: C(x) = (c1(x), . . . , cN(x))I Order constraint values: c[1](x) ≤ c[2](x) ≤ . . . ≤ c[N](x)I Empirical quantile: q̃1−α(C(x)) = c[M](x) ≤ 0


Empirical QuantileI Feasible region for c(x , ξ) = ξ1x1 + ξ2x2 − 1 ξ1, ξ1 ∼ N (0, 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

0.1

0.2

0.3

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0.6

0.7

N = 2000 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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0.7

N = 5000 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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0.7

N = 2000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

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0.7


Empirical Quantile

I Feasible region for c(x , ξ) = ξ1x1 + ξ2x2 − 1 ξ1, ξ1 ∼ N (0, 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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0.7

N = 2000 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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0.7

N = 5000 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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0.5

0.6

0.7

N = 2000

Observations:I Approximation improves as N increasesI Rough boundary of feasible region results in spurious local minimaI A lot of variance for small N as ξi are resampledI q̃1−α(C(x)) is non-differentiable


Variance Reduction using Kernels

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

0

0.5

1

1.5

2

I Y[1], . . . ,Y[N] ordered realizations of a random variable (c(x , ξi ))I Empirical quantiles: For p ∈ [0, 1] define

Q̃Yp = Y[j], where j ∈ {1, ..,N} withj−1N < p ≤

jN

I Kernel smoothing [Parson 79]

QY ,N,h1−α =∫ 1

0Q̃Yp 1hK

(α−p

h

)dp

=N∑

j=1Y[j]

=:wN,hj︷︸︸︷∫ jN

j−1N

1hK

(α−p

h

)dp


Variance Reduction using Kernels

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

0

0.5

1

1.5

2

I Y[1], . . . ,Y[N] ordered realizations of a random variable (c(x , ξi ))I Empirical quantiles: For p ∈ [0, 1] define

Q̃Yp = Y[j], where j ∈ {1, ..,N} withj−1N < p ≤

jN

I Kernel smoothing [Parson 79]

QY ,N,h1−α =∫ 1

0Q̃Yp 1hK

(α−p

h

)dp =

N∑j=1

Y[j]

=:wN,hj︷︸︸︷∫ jN

j−1N

1hK

(α−p

h

)dp


Kernel Estimation of QuantilesI Approximate q1−α(C(x)) = QY1−α with Yi = c(x ; ξi ) by kernel

estimate

qN,h1−α(C(x)) :=N∑

i=1wN,hi c[i](x) ≤ 0

I The weights depend only on N and h and can be precomputed

I Epanechnikov kernel: K (u) = 34(1− u2)1|u|

Kernel Estimation of QuantilesI Approximate q1−α(C(x)) = QY1−α with Yi = c(x ; ξi ) by kernel

estimate

qN,h1−α(C(x)) :=N∑

i=1wN,hi c[i](x) ≤ 0

I The weights depend only on N and h and can be precomputedI Epanechnikov kernel: K (u) = 34(1− u

2)1|u|

Feasible Region of Kernel-Quantile Formulationc(x , ξ) = ξ1x1 + ξ2x2 − 1 ξ1, ξ1 ∼ N (0, 1) h ∈ {0.05, 0.01}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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0.7

N = 2000 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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N = 5000 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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N = 2000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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0.6

0.7


Feasible Region of Kernel-Quantile Formulationc(x , ξ) = ξ1x1 + ξ2x2 − 1 ξ1, ξ1 ∼ N (0, 1) h ∈ {0.05, 0.01}

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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N = 2000 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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N = 5000 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

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0.7

N = 2000Observations:

I “Looks” smoother and convexI Reduces variance as ξi are resampledI Large h creates biasI qN,h1−α(C(x)) has more points of non-differentiabilityI Usually “solved” well with Knitro, but no proper termination


Smoothed Empirical CDF

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Nomal CDF

I Quantile QYp = inf{y ∈ R : Φ(y) ≥ p} where Φ(y) is cdf of Y

I Y1, . . . ,YN (unordered) realizations of a random variableI Empirical cdf:

Φ̃(y) = 1N

N∑i=1

1≥0(y − Yi )I Smoothed cdf [Azzalini 81]: (K�(t) ≈ 1≥0(t) differentiable)

ΦN,�(y) = 1N

N∑i=1

K�(y − Yi )



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c(x, )

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istr

ibution F

unction

Smoothing of empirical CDF

Empirical CDF

I Quantile QYp = inf{y ∈ R : Φ(y) ≥ p} where Φ(y) is cdf of YI Y1, . . . ,YN (unordered) realizations of a random variableI Empirical cdf:

Φ̃(y) = 1N

N∑i=1

1≥0(y − Yi )

I Smoothed cdf [Azzalini 81]: (K�(t) ≈ 1≥0(t) differentiable)

ΦN,�(y) = 1N

N∑i=1

K�(y − Yi )



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c(x, )

0

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0.4

0.6

0.8

1

Cum

ula

tive D

istr

ibution F

unction


Empirical CDF

epsilon = 0.1

epsilon = 0.5

I Quantile QYp = inf{y ∈ R : Φ(y) ≥ p} where Φ(y) is cdf of YI Y1, . . . ,YN (unordered) realizations of a random variableI Empirical cdf:

Φ̃(y) = 1N

N∑i=1

1≥0(y − Yi )I Smoothed cdf [Azzalini 81]: (K�(t) ≈ 1≥0(t) differentiable)

ΦN,�(y) = 1N

N∑i=1

K�(y − Yi )


Differentiable Quantile Estimates

-2 -1.5 -1 -0.5 0

c(x, )

0

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0.6

0.8

1

Cum

ula

tive D

istr

ibution F

unction


Empirical CDF

epsilon = 0.1

epsilon = 0.5

I Yi = c(x , ξi )I Smoothed cdf: ΦN,�(y) = 1N

∑Ni=1 K�(y − ci (x))

I Resulting quantile estimate q̂N,�p (C(x)) is the solution y∗ of

p = ΦN,�(y)

I Implicit function theorem guarantees that q̂N,�p (C(x)) exists and isdifferentiable

(for p ∈ {0, 1N ,2N , . . . ,

N−1N })



-2 -1.5 -1 -0.5 0

c(x, )

0

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0.6

0.8

1

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ula

tive D

istr

ibution F

unction


Empirical CDF

epsilon = 0.1

epsilon = 0.5


∑Ni=1 K�(y − ci (x))


p = 1N

N∑i=1

K�(y − ci (x))

+ 12N

I Implicit function theorem guarantees that q̂N,�p (C(x)) exists and isdifferentiable

(for p ∈ {0, 1N ,2N , . . . ,

N−1N })



-2 -1.5 -1 -0.5 0

c(x, )

0

0.2

0.4

0.6

0.8

1

Cum

ula

tive D

istr

ibution F

unction


Empirical CDF

epsilon = 0.1

epsilon = 0.5


∑Ni=1 K�(y − ci (x))


p = 1N

N∑i=1

K�(y − ci (x)) +12N

I Implicit function theorem guarantees that q̂N,�p (C(x)) exists and isdifferentiable (for p ∈ {0, 1N ,

2N , . . . ,

N−1N })


Final Estimate

1. Kernel estimate using empirical quantiles c[i](x)

qN,h1−α(C(x)) =N∑

i=1wN,hi c[i](x)

2. Smoothed empirical quantile q̂N,�p (C(x)), solution of

p = 1N

N∑i=1

K�(y − ci (x)) +12N

3. Final estimate

qN,h,�1−α (C(x)) :=N∑

i=1wN,hi q̂

N,�iN

(C(x))


Feasible Region with Smoothing (N = 100)Smoothing with =1, 0.5, 0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Kernel h=0.01 with smoothing = 0.06

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

x1

0

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0.5

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0.7

x2

Smoothing with =1, 0.5, 0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

x1

0

0.1

0.2

0.3

0.4

0.5

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0.7

x2

Kernel h=0.01 with smoothing = 0.06

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

x1

0

0.1

0.2

0.3

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0.5

0.6

0.7

x2


Example: Portfolio Optimization

maxx∈Rn

E[rT x ] = µT x

s.t. Pr [rT x ≥ −0.05] ≥ 0.95n∑

i=1xi = 1, x ≥ 0

I r ∼ N (µ,Σ), fixed µ ∈ R100 and Σ ∈ R100×100 pos. def.I 100 replications

Iterations with Knitro

N h min iter mean iter max iter500 0.05 15 26.7 54

10,000 0.005 17 29.3 45


Portfolio Optimization

100 500 1000 5000 10000

Number of samples

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

P(c

(x,

))

h = 5.00e-02

100 500 1000 5000 10000

Number of samples

-3

-2

-1

0

1

2

3

Re

lative

fu

nctio

n e

rro

r (%

)

10-3 h = 5.00e-02



100 500 1000 5000 10000

Number of samples

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

P(c

(x,

))

h = 1.00e-02

100 500 1000 5000 10000

Number of samples

-5

-4

-3

-2

-1

0

1

2

3

Re

lative

fu

nctio

n e

rro

r (%

)

10-3 h = 1.00e-02



100 500 1000 5000 10000

Number of samples

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

P(c

(x,

))

h = 1.00e-03

100 500 1000 5000 10000

Number of samples

-3

-2

-1

0

1

2

3

Re

lative

fu

nctio

n e

rro

r (%

)

10-3 h = 1.00e-03



100 500 1000 5000 10000

Number of samples

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

P(c

(x,

))

h = 1.00e-04

100 500 1000 5000 10000

Number of samples

-3

-2

-1

0

1

2

3

Re

lative

fu

nctio

n e

rro

r (%

)

10-3 h = 1.00e-04


Joint Chance Constraints

I Original Problem

minx∈X

f (x)

s.t. Pξ[c(x , ξ) ≤ 0] ≥ 1− α

I Scenario vector

C(x) = (c(x , ξ1), . . . , c(x , ξN))

I NLP formulation

minx∈X

f (x)

s.t. qN,h,�1−α (C(x)) ≤ 0

q(Ĉ(x)) := qN,h,�1−α (Ĉ(x))

Note: Constraint no longer differentiable



I Original Problem

minx∈X

f (x)

s.t. Pξ[cj(x , ξ) ≤ 0, j = 1, . . . ,m] ≥ 1− α

I Scenario vector

C(x) = (c(x , ξ1), . . . , c(x , ξN))

I NLP formulation

minx∈X

f (x)

s.t. qN,h,�1−α (C(x)) ≤ 0

q(Ĉ(x)) := qN,h,�1−α (Ĉ(x))




I Original Problem

minx∈X

f (x)

s.t. Pξ[ĉ(x , ξ) ≤ 0] ≥ 1− αĉ(x , ξ) := maxj cj(x , ξi )

I Scenario vector

C(x) = (c(x , ξ1), . . . , c(x , ξN))

I NLP formulation

minx∈X

f (x)

s.t. qN,h,�1−α (C(x)) ≤ 0

q(Ĉ(x)) := qN,h,�1−α (Ĉ(x))




I Original Problem

minx∈X

f (x)


I Scenario vector

Ĉ(x) = (ĉ(x , ξ1), . . . , ĉ(x , ξN))

I NLP formulation

minx∈X

f (x)

s.t. qN,h,�1−α (Ĉ(x)) ≤ 0

q(Ĉ(x)) := qN,h,�1−α (Ĉ(x))

Note: Constraint no longer differentiableAndreas Wächter Nonlinear Programming Formulation of Chance-Constraints


I Original Problem

minx∈X

f (x)


I Scenario vector

Ĉ(x) = (ĉ(x , ξ1), . . . , ĉ(x , ξN))

I NLP formulation

minx∈X

f (x)

s.t. q(Ĉ(x)) ≤ 0q(Ĉ(x)) := qN,h,�1−α (Ĉ(x))

Note: Constraint no longer differentiableAndreas Wächter Nonlinear Programming Formulation of Chance-Constraints

Exact Penalty Function

minx∈X

f (x)

s.t. q(Ĉ(x)) ≤ 0

I Exact penalty function:

φρ(x) = f (x) + ρmax{q(Ĉ(x)), 0} (+penalty for X )

I Model of scenario vector Ĉ(x) = (. . . ,maxj cj(x , ξi ), . . .)

mĈ (xk , d) =(. . . ,max

j{cj(xk , ξi ) +∇cj(xk , ξi )T d}, . . .

)I Model of penalty function

mφ(xk , d) =f (xk) +∇f (xk)T d + 12dT Bkd+

ρmax{q(Ĉ(xk)) +∇q(Ĉ(xk))T (mĈ (xk , d)− Ĉ(xk)), 0}



minx∈X

f (x)

s.t. q(Ĉ(x)) ≤ 0


φρ(x) = f (x) + ρmax{q(Ĉ(x)), 0} (+penalty for X )I Model of scenario vector Ĉ(x) = (. . . ,maxj cj(x , ξi ), . . .)

mĈ (xk , d) =(. . . ,max


)

I Model of penalty functionmφ(xk , d) =f (xk) +∇f (xk)T d + 12d

T Bkd+ρmax{q(Ĉ(xk)) +∇q(Ĉ(xk))T (mĈ (xk , d)− Ĉ(xk)), 0}



minx∈X

f (x)

s.t. q(Ĉ(x)) ≤ 0


φρ(x) = f (x) + ρmax{q(Ĉ(x)), 0} (+penalty for X )I Model of scenario vector Ĉ(x) = (. . . ,maxj cj(x , ξi ), . . .)

mĈ (xk , d) =(. . . ,max


)I Model of penalty function




Trust Region S`1QP Algorithm



I Can show: |φρ(xk)−mφ(xk , d)| = O(‖d‖2)I Standard S`1QP-type trust region algorithm can be applied

I QP Subproblem

mind ,z,t

f (xk) +∇f (xk)T d + 12dT Bkd + ρ t

s.t. q(Ĉ(xk)) +∇q(Ĉ(xk))T Z ≤ t [Z = (z1, . . . , zn)]cj(xk , ξi ) +∇cj(xk , ξi )T d ≤ zi for all i , j

t ≥ 0, xk + d ∈ X̃ , ‖d‖∞ ≤ ∆k

I Size of QP proportional to number of nonzeros in ∇q(Ĉ(xk)), not N


Trust Region S`1QP Algorithm



I Can show: |φρ(xk)−mφ(xk , d)| = O(‖d‖2)I Standard S`1QP-type trust region algorithm can be appliedI QP Subproblem

mind ,z,t

f (xk) +∇f (xk)T d + 12dT Bkd + ρ t

s.t. q(Ĉ(xk)) +∇q(Ĉ(xk))T Z ≤ t [Z = (z1, . . . , zn)]cj(xk , ξi ) +∇cj(xk , ξi )T d ≤ zi for all i , j

t ≥ 0, xk + d ∈ X̃ , ‖d‖∞ ≤ ∆k

I Size of QP proportional to number of nonzeros in ∇q(Ĉ(xk)), not N


Summary

I Goal: Formulate chance-constrained problem as NLPI Constrain the quantile q1−α(x) ≤ 0

I Better linearization than p(x) ≥ (1− α)I Sample-based empirical quantile

I No need to know probability distributionI Variance reduction using kernel

I Improves approximation of feasible region, “smoother boundary”I Quantile estimates using smoothed empirical cdf

I Constraint becomes differentiableI Joint chance constraints

I Exact merit functionI S`1QP-type trust region algorithm


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