set-based minmax robust efficiency for uncertain multi...

Jonas Ide Minmax Robust Efficiency September 12, 2014 1 / 29

Set-based minmax robust efficiencyfor uncertain multi-objective

optimization

Jonas Ide

joint work withMatthias Ehrgott and Anita Schobel

University of Gottingen

September 12, 2014at the Workshop on Recent Advances in Multi-Objective Optimization, Vienna

IntroductionMulti-objective optimizationRobust optimizationRobust multi-objective optimization

Set based minmax robust efficiency

Calculating minmax robust efficient solutionsWeighted sum scalarizationε-constraint-methodApproach via the objective-wise worst caseExamples of minmax robust efficient sets

Conclusion & Outlook

Definition (Multi-objective optimization problem)Given a feasible set X ⊂ Rn and a function f : Rn → Rk , a multi-objectiveoptimization problem is given by

min f (x)s.t. x ∈ X

In multi-objective optimization one searches for the set of nondominated points f (x)with x ∈ X , i.e., where there is no x ′ ∈ X \ {x} such that fi (x ′) ≤ fi (x) for alli = 1, . . . , k.The according solution x is called efficient.

f (X )

Definition (Multi-objective optimization problem)Given a feasible set X ⊂ Rn and a function f : Rn → Rk , a multi-objectiveoptimization problem is given by

min f (x)s.t. x ∈ X

In multi-objective optimization one searches for the set of nondominated points f (x)with x ∈ X , i.e., where there is no x ′ ∈ X \ {x} such that fi (x ′) ≤ fi (x) for alli = 1, . . . , k.The according solution x is called efficient.

f (X )

UncertaintiesI In application of mathematical optimization input data often uncertain or not

(entirely) known beforehand

I Uncertainties can often be described by a set of possible scenarios U

Definition (Uncertain (single objective) optimization problem)Given: uncertainty set U , feasible set X ⊂ Rn, objective function f : Rn × U → R .

Uncertain optimization problem P(U): Family of (deterministic) optimization problems

P(ξ) min f (x , ξ)s.t. x ∈ X ,

where ξ ∈ U .

The question arises:When to call a solution to this family of optimization problems robust optimal?

Different concepts of robustness for single objectiveoptimization problems

I Minmax robustness (Soyster, 1973, Ben-Tal & Nemirovski, 1998):minx∈X

supξ∈U

f (x , ξ)

I many more (e.g., Ben-Tal et al., 2009, Goerigk & Schobel, 2013)

supξ∈U

f (x , ξ)

supξ∈U

f (x , ξ)

supξ∈U

f (x , ξ)

RobustOptimization

Multi-ObjectiveOptimization

Robust Multi-Objective

Optimization

I Both robust and multi-objective optimization important in research andreal-world applications

I Connection of these two topics quite new (e.g., Kuroiwa & Lee, 2012; Witting,2012)

I Some other works available as pre-prints (e.g., Doolittle et al., 2012; Kuhn etal., 2012)

RobustOptimization

Optimization

RobustOptimization

Optimization

RobustOptimization

Optimization

RobustOptimization

Optimization

Definition (Uncertain multi-objective problem)Given an uncertainty set U , a feasible set X ⊂ Rn and a function f : Rn × U → Rk ,an uncertain multi-objective problem P(U) is given by the family of all problems

P(ξ) min f (x , ξ)s.t. x ∈ X

with ξ ∈ U .

The question arisesWhen do we call a solution x ∈ X robust efficient?

Definition (Uncertain multi-objective problem)Given an uncertainty set U , a feasible set X ⊂ Rn and a function f : Rn × U → Rk ,an uncertain multi-objective problem P(U) is given by the family of all problems

P(ξ) min f (x , ξ)s.t. x ∈ X

with ξ ∈ U .

The question arisesWhen do we call a solution x ∈ X robust efficient?

Hedging against a worst case

What is a worst case for the uncertain multi-objective problem P(U) given by

P(ξ) min

supξ∈U

f (x , ξ)

s.t. x ∈ X

with ξ ∈ U?

Hedging against a worst caseWhat is a worst case for the uncertain multi-objective problem P(U) given by

P(ξ) min

supξ∈U

f (x , ξ)

s.t. x ∈ X

with ξ ∈ U?

Hedging against a worst caseWhat is a worst case for the uncertain multi-objective problem P(U) given by

P(ξ) min supξ∈U

f (x , ξ)

s.t. x ∈ X

with ξ ∈ U?

Interpreting the supremum as a set

f (x3,U)

f (x1,U)

f (x2,U)

f (x4,U)

f (x5,U)

Which of these solutions do we call minmax robust efficient?

supξ∈U

f (x3,U)

supξ∈U

f (x1,U)

supξ∈U

f (x2,U)

supξ∈U

f (x4,U)

supξ∈U

f (x5,U)

Which of these solutions do we call minmax robust efficient?

We will call those x ∈ X minmax robust efficient, where f (x ,U) is nondominated.

Definition (Robust efficiency)Given an uncertain multi-objective problem P(U) we call a solution x ∈ X minmaxrobust efficient,

if there is no x ′ ∈ X \ {x} such that

f (x ′,U) ⊆ f (x ,U)− Rk≥

Definition (Robust efficiency)Given an uncertain multi-objective problem P(U) we call a solution x ∈ X minmaxrobust strictly efficient,

f (x ′,U) ⊆ f (x ,U)− Rk=

Definition (Robust efficiency)Given an uncertain multi-objective problem P(U) we call a solution x ∈ X minmaxrobust weakly efficient,

f (x ′,U) ⊆ f (x ,U)− Rk>

The orange, blue, green and purple solutions are minmax robust strictly efficient, thered one is not even minmax robust weakly efficient.

Properties

I For |U| = 1 these minmax robust efficiency definitions reduce to the definitionof efficiency

I For k = 1 the definition of minmax robust weakly efficiency reduces to thedefinition of minmax robust optimality

Question

I How to calculate robust efficient solutions?

I First idea: Find solutions by solving a robust single-objective problem

I Second idea: Find solutions by solving a deterministic multi-objective

problem

Properties

Question

problem

Properties

Question

problem

Properties

Question

problem

Properties

Question

problem

Properties

Question

I How to calculate robust efficient solutions?I First idea: Find solutions by solving a robust single-objective problem

problem

Properties

Question

I How to calculate robust efficient solutions?I First idea: Find solutions by solving a robust single-objective problem

problem

TheoremIf x ∈ X is the unique minimizer of

supξ∈U

k∑i=1

λi fi (x , ξ)

over X for some λ ∈ Rk≥, x is minmax robust strictly efficient.

TheoremIf maxξ∈U

∑ki=1 λi fi (x , ξ) exists for all x ∈ X and x ∈ X is a minimizer of

maxξ∈U

k∑i=1

λi fi (x , ξ)

over X for some λ ∈ Rk≥, then x is minmax robust weakly efficient.

The purple solution is minmax robust strictly efficient.

The orange solution is minmax robust strictly efficient.

The blue solution is minmax robust strictly efficient.

The green minmax robust strictly efficient solution is no optimal solution for anyscalarization problem.

TheoremIf x ∈ X is the unique minimizer of

supξ∈U

k∑i=1

λi fi (x , ξ)

over X for some λ ∈ Rk≥, x is minmax robust strictly efficient.

TheoremIf maxξ∈U

∑ki=1 λi fi (x , ξ) exists for all x ∈ X and x ∈ X is a minimizer of

maxξ∈U

k∑i=1

λi fi (x , ξ)

over X for some λ ∈ Rk≥, then x is minmax robust weakly efficient.

Definition

εCP(U)(ε, i) min supξ∈U

fi (x , ξ)

s.t. fj (x , ξ) ≤ εj ∀ j 6= i , ∀ ξ ∈ Ux ∈ X

TheoremGiven a problem P(U).

a) If x ∈ X is the unique optimal solution to εCP(U)(ε, i) for some i, then it isminmax robust strictly efficient.

b) If x ∈ X is an optimal solution to εCP(U)(ε, i) for some i and maxξ∈U

fi (x , ξ) exists

for all x ∈ X , then x is minmax robust weakly efficient.

Definition

fi (x , ξ)

s.t. fj (x , ξ) ≤ εj ∀ j 6= i , ∀ ξ ∈ Ux ∈ X

fi (x , ξ) exists

Definition

fi (x , ξ)

s.t. fj (x , ξ) ≤ εj ∀ j 6= i , ∀ ξ ∈ Ux ∈ X

fi (x , ξ) exists

ε1 = 4.5ε1 = 1

The green solution minimizes f2(x , ξ) over {x ∈ X : f1(x , ξ) ≤ 4.5 ∀ξ ∈ U},the purple solution minimizes f2(x , ξ) over {x ∈ X : f1(x , ξ) ≤ 1 ∀ξ ∈ U},

ε2 = 2.5

The blue solution minimizes f1(x , ξ) over {x ∈ X : f2(x , ξ) ≤ 2.5 ∀ξ ∈ U}The orange solution cannot be found with the ε-constraint method.

DefinitionWe formulate a new problem

OWC minx∈X

f owcU (x)

f owcU (x) :=

supξ∈U

f1(x , ξ)

supξ∈U

f2(x , ξ)

...supξ∈U

fk (x , ξ)

Theorem

(1) If x ∈ X is a strictly efficient solution for (OWC), then it is minmax robuststrictly efficient for P(U).

(2) If maxξ∈U

fi (x , ξ) exists for all i = 1, . . . , k and x ∈ X and x is weakly efficient for

(OWC), it is minmax robust weakly efficient for P(U).

f owcU (x3)

f owcU (x1)

f owcU (x2)

f owcU (x4)

f owcU (x5)

The strictly efficient solutions of (OWC) are the purple, green and blue solutions.The orange solution cannot be found this way.

OWC minx∈X

f owcU (x)

f owcU (x) :=

supξ∈U

f1(x , ξ)

supξ∈U

f2(x , ξ)

...supξ∈U

fk (x , ξ)

Theorem

(2) If maxξ∈U

OWC minx∈X

f owcU (x)

f owcU (x) :=

supξ∈U

f1(x , ξ)

supξ∈U

f2(x , ξ)

...supξ∈U

fk (x , ξ)

Theorem

(2) If maxξ∈U

The strictly efficient solutions of (OWC) are the purple, green and blue solutions.The orange solution cannot be found this way.

ε-constraint method

Approximation of the minmax robust efficient set of the ε-constraint method(left to right: ε = 1, 0.5, 0.1)

weighted sum and ε-constraint scalarization

Minmax robust efficient solutions obtained by weighted sum (black) and ε-constraint(grey) scalarization (left linear, right quadratic)

ε-constraint method

Summary

I Introduced (set-based) minmax robust efficiency

I Presented algorithms for calculating minmax robust efficient solutions

I Investigated differences between the scalarization techniques

Further researchI Investigated connection to set-valued optimization

I Applied minmax robust efficiency in practice

Future workI Evaluate practical value

I Other solution techniques?

Thank you for your attention!

Summary

Further research

I Investigated connection to set-valued optimization

Summary

Future work

I Evaluate practical value

Summary

set-based minmax robust efficiency for uncertain multi...

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