Chance constrained optimization - applications, properties ... · PDF fileChance constrained...

Chance constrained optimization - applications,properties and numerical issues

Dr. Abebe Geletu

Ilmenau University of TechnologyDepartment of Simulation and Optimal Processes (SOP)

May 31, 2012

Chance constrained optimization - applications, properties and numerical issues

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This Chancy, Chancy, Chancy WorldLeonard Rastrigin

Uncertainty is the only certainty ...John Allen Paulos, Temple University, Philadelphia


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Topics

Chance constrained optimization problems

History and major contributions

Some application areas

Properties and difficulties of chance constraints

Approximation strategies for chance constraints

A new analytic approximation

Conclusion


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Chance constrained optimization problems

(CCOPT ) minx

E [f (x , ξ)] (1)

s.t.

either Prgi (x , ξ) ≤ 0 ≥ αi , i = 1, . . . ,m. (2)

or Prgi (x , ξ) ≤ 0, i = 1, . . . ,m ≥ α. (3)

f , g : Rn × Rp → R are at least differentiable w.r.t. x ∈ X ⊂ Rn;

x- a vector of deterministic variables;

ξ ⊂ Ω ⊂ Rp- a vector of random variables with joint probabilitydensity function φ(ξ).


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Chance constrained optimization problems...

Pr·,E [·] are probability, expectation operators;

Prg(x , ξ) ≤ 0 ≥ α - chance or probability constraint.12 ≤ α ≤ 1 probability (reliability) level.

• Single chance constraints

Prgi (x , ξ) ≤ 0 ≥ αi , i = 1, . . . ,m.

• Joint chance constraints

Prgi (x , ξ) ≤ 0, i = 1, . . . ,m ≥ α.

• The random vector ξ:can be either a Gaussian or non-Gaussian.(See Geletu et al. 2012 for recent review article.)


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History and major contributionsThe beginning:

Charnes, Cooper & Symonds 1958, 1959.

Major contribution:

Prekopa 1972,1973, 1995, 2001, 2011.

Major references:

Ben-Tal, El Ghaoui & Nemirovski: Robust optimization, 2009.

Birge & Lauveaux: Introduction to stochastic programming, 1997.

Kall & Wallace : Stochastic Programming, 1995.

Kibzun & Kan : Stochastic programming problems. withprobability and quantile functions, 1996.

Marti: Stochastic Optimization Methods, 2005.

Prekopa: Stochastic programming, 1995.

Ruszczynski & Shapiro: Stochastic programming, 2003.


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Some application areasClassical application areas

Water reservoir management

Optimal power flow (OPF)

Financial risk management (using risk metrics like VaR andcVaR)

Reliability based (engineering) design optimization (RBDO)

Modern applications

Control and optimization based on prediction (eg. using weatherforecast data, etc.)

Reliable navigation and obstacle avoidance in unmannedautonomous ground/areal vehicles

Optimal and reliable wind/soalr power generation

Reliability and fault-tolerance in mechatronic systems

etc.Chance constrained optimization - applications, properties and numerical issues

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Some application areas ...


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Some application areas ....


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Some application areas ....Some related terminologies

RobustnessReliability /ZuverlassigkeitRisk metricsRisk managementFault tolerance

Chance Constraints

♣

Worst case scenario

Chance Constraints

F

min PrFailure

max PrReliability

Chance Constraints


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Properties and difficulties of chance constraintsMeaning of chance constraints:• Z := g(x , ξ) is a random variable .• Hence,

Prg(x , ξ) ≤ 0+ Prg(x , ξ) > 0 = 1.

irrespective of the distribution of Z = g(x , ξ).

reliability constraint

For α near 1, Prg(x , ξ) ≤ 0 ≥ α; eg. α = 0.95, α = 0.99.

risk constraint

Prg(x , ξ) > 0 ≤ 1− α; eg. 1− α = 0.05, α = 0.01.

• x is feasible iff Prg(x , ξ) ≤ 0 ≥ α holds with reliability α.• Let p(x) := Pr g(x , ξ) ≤ 0. Then the feasible set of CCOPT is

P := x ∈ X | p(x) ≥ α .Chance constrained optimization - applications, properties and numerical issues

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Properties and difficulties of chance constraintsImportant issues

(A) Continuity (see Kibzun & Kan 1996, Birge & Lauveaux 1997,Prekopa 1995).

(B) Differentiability of p(·)Uryasev 1994, Marti 1996.

(C) Convexity of the feasible set PPrekopa 1995, Henrion & Strugarek .

(D) Stability and regularityHenrion and Romisch 1998.

(E) How to determine if a given x is feasible for CCOPT or not?

(F) For a given x , how to determine the valuep(x) = Pr g(x , ξ) ≤ 0?

In general, (E) & (F) are not trivial issues.Chance constrained optimization - applications, properties and numerical issues

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Properties and difficulties of chance constraintsMajor difficulty:the value

p(x) = Prg(x , ξ) ≤ 0 =

∫ξ∈Ω | g(x ,ξ)≤0

φ(ξ)dξ

is difficult to compute. Hence, CCOPT is known to be ahard-problem. But there are somespecial cases:

Example 1:

g(x , ξ) = a>x + b − ξ, ξ ∈ R and ξ ∼ N(µ, σ2). Then

Prg(x , ξ) ≤ 0 = Pra>x + b ≤ ξ = 1− Pra>x + b ≥ ξ

= 1− Pr

ξ − µσ≤(a>x + b

)− µ

σ

= 1− Φ

(a>x + b

).


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Properties and difficulties of chance constraints

Example 1: contd...

Hence, if g(x , ξ) = a>x + b − ξ, ξ ∈ R and ξ ∼ N(µ, σ2), then

Prg(x , ξ) ≤ 0 ≥ α⇔ Φ−1 (1− α)−(a>x + b

)≥ 0.

The expression on the left is an exact analytic representation of thechance constraint.

Example 2:(separable form)

g(x , ξ) = g(x) + c>ξ + b with ξ ∼ N(µ, σ2).

Example 3:(a linear transformation of ξ)

g(x , ξ) = (Ax + b) ξ + b with ξ ∼ N(µ, σ2).


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Properties and difficulties of chance constraints...

An analytic representation can lead to a nonlinear optimizationproblem.

Example 4 (Theorem 6.2., P. 83, B. Liu 2002)

Let ξ = (ξ1, . . . , ξn, ξn+1) and g(x , ξ) = x1ξ1 + . . .+ xnξn − ξn+1. Ifξ1, . . . , ξn, ξn+1 are independently normally distributed randomvariables, then

Prg(x , ξ) ≤ 0 ≥ α

⇔n∑

i=1

E [ξi ]xi+Φ−1(α)

√√√√ n∑i=1

V [ξi ]x2i + V [ξn+1] ≤ E [ξn+1)],

where Φ is the standard normal distribution.


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Properties and difficulties of chance constraints...• The above result can be extended to

Pr

w0(x) +

p∑k=1

ξkwk(x) ≤ 0

≥ α. (4)

Garnier et al. 2008 (Linearization)

If ξ = (ξ1, . . . , ξp) is a normal multivariate random vector with mean0 and small variance, then

g(x , ξ) ≈ g(x , 0) +

p∑i=1

∂g

∂ξi(x , 0)ξ,

which leads to (??)

• In general, there is no closed-form analytic representation.• Nemirovski 2012 studies tractability of (??) for non-Gaussian ξ.


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Approximation strategies for chance constraintoptimization

It remains to use approximation methods.Approximation strategies

(I) Back-mapping

(II) robust optimization

(III) sample average approximation


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I. Back-mapping (projection) method

Idea of back-projection (Wendt, Li and Wozny 2002):Find a monotonic relation between Z = g(x , ξ) and some randomvariable ξj ; i.e. verify theoretically (see Geletu et al. 2010) orexperimentally that there is a real-valued function ϕ such that, forany x ∈ X ,

Z = ϕx(ξj);

ϕx(·) is strictly increasing (ξj ↑ Z ) or decreasing (ξj ↓ Z ).

⇒ ξj = ϕ−1x (Z ).

• ξ ↑ Z ⇒ Pr g(x , ξ) ≤ 0 = Prξj ≤ ϕ−1

x (0).

• ξ ↓ Z ⇒ Pr g(x , ξ) ≤ 0 = Prξj ≥ ϕ−1

x (0).


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Back projection ....

Abbildung: Back Projection of chance constraints

• Requires a global implicit function theorem (see Geletu et al. 2011).Chance constrained optimization - applications, properties and numerical issues

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I. Back-projection ...

Now, for ξj ↑ Z , (CCOPT) is equivalent to

(CCOPT ) minx

E [f (x , ξ)]

s.t. p(x) = Prξj ≤ ϕ−1

x (0)≥ α,

u ∈ U .

p(x) =

∫ +∞

−∞. . .

∫ +∞

−∞

∫ ϕ−1x (0)

−∞φ(ξ)dξ (5)

∇p(x) =

∫ +∞

−∞. . .

∫ +∞

−∞∇xϕ

−1x (0)φ(ξ)dξ (6)


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I. Back-projection ...

Advantages:

usable if monotonic relations are easy to find;

provides direct representation of chance constraints.

Disadvantages:

monotonic relations may not exist;

monotonic relations can be difficult to verify.


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II. Sample average approximation (SAA)

Shapiro 2003, Pagnoncelli et al. 2009.• Define

I(0,+∞] (g(x , ξ)) =

0, if g(x , ξ) > 01, if g(x , ξ) ≤ 0.

• Determine samples ξ1, . . . , ξN ⊂ Ω (e.g. low discrepancysequences like Fourer, Sobol or Niederreiter sequences).• Replace the chance constrains with

pN(x) =1

N

N∑k=1

I(−∞,0]

(g(x , ξk)

)≥ α.

(pN(x) =Relative-frequency count for the satisfaction of g(x , ξ) ≤ 0. )


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II. Sample average approximation...

Advantages:

(SAA) avoids computation of multidimensional integrals;

Convexity structures are preserved.

Disadvantages:

SAA leads to a non-smooth optimization problem.

Feasibility of the obtained solution to the (CCOPT) isguaranteed only when N →∞.


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III. Robust optimization technique

Robust optimization considers the (worst-case) problem

(RO) minx

E [f (x , ξ)]

s.t. g(x , ξ) ≤ 0, ξ ∈ Ω,

x ∈ X ,

where g(x , ξ) ≤ 0 is required to be satisfied for as many realizationsof ξ from Ω as possible.

Randomized solution based on scenario generation:• Generate independent identically distributed random samplesξ1, . . . , ξN from Ω (Monte-Carlo method).


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III. Robust optimization technique ...• Solve the optimization problem

(NLP)RO minx

1

N

N∑k=1

f (x , ξk)

s.t. g(x , ξk) ≤ 0, k = 1, . . . ,N;

x ∈ X .

Theorem (Califore & Campi 2005)

Suppose α ∈ (0, 1) and f (·, ξ) is convex w.r.t. x ∈ Rn. If the numberof random samples ξ1, . . . , ξN

N ≥ 2n

(1− α)ln

(1

1− α

)+

(2

1− α

)ln

(1

α

)+ 2n,

then the optimal solution obtained from (NLP)RO is an optimalsolution of (RO) with reliability α.


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III. Robust optimization technique ...

Advantages:

there is no need to compute integrals;

the problem (NLP)RO simple to implement and solve;

it also preserves convexity structures.

Disadvantages:

solution of (NLP)RO may not be feasible to the (CCOPT);

for a higher reliability level α, very large number of scenariosξ1, . . . , ξN are required.

Recent suggestion: scenarios reduction methods:Henrion, Kuchler & Romisch 2009; Campi & Garetti 2011.


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A new analytic approximation strategyLet

h(x , ξ) =

0, if g(x , ξ) ≤ 01, if g(x , ξ) > 0.

This impliesPrg(x , ξ) > 0 = E [h(x , ξ)]

⇒

Prg(x , ξ) ≤ 0 ≥ α⇔ E [h(x , ξ)] ≤ 1− α.

A general idea of analytic approximation (Geletu et al. 2012):Find a continuous (possibly smooth) parametric function ψ(τ, ·) sothat

E [h(x , ξ)] ≤ ψ(τ, x),

where τ > 0.Chance constrained optimization - applications, properties and numerical issues

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A new analytic approximation strategy...Hence

M(τ) = x | ψ(τ, x) ≤ 1− α ⊂ P := x | E [h(x , ξ)] ≤ 1− α.


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A new analytic approximation strategy...

• Requirements on ψ(τ, x):

(a) for each fixed τ , ψ(τ, x) should be easy to compute.

(b) For each fixed x

infτ>0

ψ(τ, x) = E [h(x , ξ)]

• (b) ⇒ supτ M(τ) = P.Question:

How to construct such a function ψ(τ, x)?

Basically, ψ(τ, x) = E [Ψ(τ, g(x , ξ))], for Ψ(τ, s) : R+ × R→ R.


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Some suggestions:

Ψ1(τ, s) = exp(τs), τ > 0 (Pinter 1984)

Ψ2(τ, s) = exp(τ−1s), τ > 0 (Nemirovski and Shapiro 2006,Nemirovski 2011).

Ψ3(τ, s) = τ + 11−α [s − τ, 0]+, τ > 0 (Rockafellar and Uryasev

2000), where [s − τ, 0]+ = maxs − τ, 0.A new analytic approximation function (Geletu et al. 2012)

Ψ4(τ, s) =1 + m1τ

1 + m2τe− 1

τs

with 0 < m2 ≤ m1 and τ > 0.


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Abbildung: Comparison of Ψ2 and Ψ4


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Theorem (A. Geletu 2012)

1 Let τ > 0. Then

Ψ4(τ, s) > 0 for any value of s;Ψ4(τ, s) ≥ 1 for s ≥ 0.

2 For 0 < τ ≤ 12 , then

the function Ψ4(·, s) non-decreasing w.r.t. τ ;

limτ0+

Ψ4(τ, s) =

1, if s ≥ 00, if s < 0

Hence, for any s ∈ R : limτ0+ |Ψ4(τ, s)− I[0,+∞)(s)| = 0.

Hence,

limτ0+

ψ(τ, x) = limτ0+

E [Ψ4(τ, g(x , ξ))] = limτ0+

E [h(x , ξ)]


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Instead of CCOPT solve the problem

(NLP)τ minx

E [f (x , ξ)] (7)

s.t. (8)

x ∈ M(τ) = x ∈ X | ψ(τ, x) ≤ 1− α,

for each τ with the same objective function E [f (x , ξ)] as in (CCOPT).


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Theorem (Geletu et al. 2012)

If τk any sequence, such that τk 0+ and X is a compact set,then

for each τk , the feasible set M(τk) is compact.

limk→+∞M(τk) = P. (in fact, limk→+∞H(M(τk),P) = 0 )

If x∗ ∈ X is a limit point of a sequence xk of optimal solutionsof (NLP)τk , then x∗ is a solution of CCOPT.

Computational advantage:Solution of NLPτ for a sufficiently small τ provides a goodapproximation for CCOPT.


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Advantages:

CCOPT becomes more tractable

NLPτ can be solved using state-of-the-art optimization solvers

solution of NLPτ is always feasible to the CCOPT

Disadvantages:

Objective and constraints may require the evaluation of highdimensional integrals

convexity structures of CCOPT may not be preserved in the newapproximation

currently, the approximation works only for single chanceconstraints


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Conclusions

Analytic approximation facilitates tractability of nonlinear chanceconstrained optimization problems.

For fast processes, linearization and analytic approximation mightbe imperative.

Large-scale problems require faster evaluation of multidimensionalprobability integrals. This can be achieved through: efficient numerical integration techniques (like QMC,

sparse-grid, etc. , integration ) the design of new numerical integration technique parallel computation, etc.


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References:

Andrieu, L.; Henrion, R.; Romisch, W. A model for dynamicchance constraints in hydro power reservoir management.European j. Oper. Res. 207, 579 – 589, 2010.Liu, B. Theory and practice of uncertain programming.Physica-Verlag, Heidelberg, 2002.Ben-Tal A.; Nemirovski, A. Robust convex optimization. Math.of OR 23, 769 - 805.Birge, J. R.; Lauveaux, F. Introduction of stochasticprogramming. Springer Verlag, New York, 1997.Califore, G.; Campi, M. C. Uncertain convex programs:randomized solutions and confidence levels. Math. Prog. Ser. A,102, 25 – 46, 2005.Campi, M. C.; Garetti, S. A sampling-and-discarding-approach tochance-constrained optimization: feasibility and optimality. J.Optim. Theory and Appl. 148, 257 – 280, 2011.


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References:

Celik, A. N.; Makkawi, A.; Muneer, T. Critical evaluation of windspeed frequency distribution functions. Journal of Renewable andSustainable Energy, 2, 013102, 2010.Charnes, A.; Cooper, W. Chance constrained programming.Management Science, 6, 73 – 79, 1959.Charnes, A.; Cooper, W.; Symonds, G. H. Cost horizons andcertainty equivalence: An approach in stochastic programming ofheating oil. Management Science, 4, 235 – 263, 1958.Geletu, A.; Kloppel, M.; Zhag, H.; Li, P. Advances andapplications of chance-constrained approaches to systemsoptimization under uncertainty. Inter. J. of Sys. Sc., 2012,accepted.Geletu, A.; Hoffmann, A.; Kloppel, M.; Li, P. Monotony analysisand sparse-grid integration for nonlinear chance constrainedprocess optimization. Eng. Optim., 43(10), 1019 – 1041, 2011.


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References:

Henrion, R. Some remarks on value-at-risk optimization. Inter. J.Man. Sc. and Eng. Man., 1, 111 – 118, 2006.

Henrion, R.; Kuchler, C.; Romsch, W. Scenario reduction instochastic programming with respect to discrepancy distances.Comput. Optim. Applic., 43, 67 – 93, 2009.

Henrion, R.; Strugarek, C. Convexity of chance constraints withindependent random variables. Comput. Optim. Applic. 41,262-276, 2008.

Kibzun, A.I.; Kan, Y.S. Stochastic programming problems: withprobability and quantile functions. John Wiley & Sons Ltd.,Chichester, England, 1996.


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References:

Kloppel, M.; Geletu, A.; Hoffmann, A.; Li, P. Using sparse-gridmethods to improve computation efficiency in solving dynamicnonlinear chance-constrained optimization problems. Ind. Eng.Chem. Res., 50, 5693 – 5704, 2011.

Marti, K. Differentiation formulas for probability functions: Thetransformation method. Math. Prog. 75, 201 – 220, 1996.

Nemirovski, A. On safe tractable approximation of chanceconstraints. EJOR, 219, 707 – 718, 2012.

Nemirovski, A.; Shapiro, A. Convex approximation of chanceconstrained programs. SIAM J. Optim., 17, 969 – 996, 2006.

Ouarda, T. B. M. J.; Labadie, J. W. Chance-constrained optimalcontrol for multireservoir system optimization and risk analysis.Stochastic Environmental Research and Risk Analysis, 15,185–204, 2001.


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References:

Pagnoncelli, B. K.; Ahmed, S.; Shapiro, A. Sample averageapproximation method for chance constrained programming:theory and applications. JOTA, 142, 399 – 416, 2009.

Pinter, J. Deterministic approximation of probability inequalities.Oper. Res. 33, 219 – 239, 1989.

Prekopa, A.; Yoda, K.; Subasi, M. M. Uniform quasi-convexity inprobabilistic constrained stochastic programming. OR Letters, 39,188 – 192, 2011.

Prekopa, A. On the concavity of multivariate probabilitydistribution functions. OR Letters, 29, 1 – 4, 2001.

Prekopa, A. Stochastic programming. Kluwer AcademicPublishers, Dordrecht, The Netherlands, 1995.

Rockafellar, R. T.; Uryasev, S. Optimization of conditionalvalue-at-risk. J. of Risk, 2, 21–41, 2000.


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References:

Wendt, M.; Li, P.; Wozny, G. Nonlinear chance-constrainedprocess optimization under uncertainty. Ind. Eng. Chem. Res., 41,3621 – 3629, 2002.

Wozabal, D. Value-at-Risk optimization using the difference ofconvex algorithm. OR Spectrum, Published Online, 2010.

Wozabal, D.; Hochreiter, R.; Pflug, G. Ch. A difference of convexformulation of value-at-risk constrained optimization. J. Math.Prog. and OR, 59, 377–400, 2010.

Uryasev, S. Derivatives of probability functions and someapplications. Ann Oper Res, 56, 287–311, 1995.


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