Multiobjective Optimization Metrics BiMADS Other methods References Di culty ISingle-objective optimization: Particular case where p= 1. An optimal solution typically consists of a

Introduction Metrics BiMADS Other methods References

Multiobjective Optimization

MTH8418

S. Le Digabel, Polytechnique Montreal

Winter 2019(v4)

MTH8418: Multiobjective 1/37


Plan

Introduction

Metrics

BiMADS

Other methods

References



Introduction

Metrics

BiMADS

Other methods

References



Multiobjective optimization problemI The multiobjective optimization problem (MOP) can be

formally stated asminx∈Ω

F (x)

I whereF : Ω→ R ∪ +∞p

andF (x) =

(f (1)(x), f (2)(x), . . . , f (p)(x)

)I p is the number of objective functions

I Case p = 2: Biobjective optimization problem (BOP)

I The feasible set Ω remains unchanged

I Typically, the different objectives are contradictory: A decreasein one objective causes an increase in the other objectives



Difficulty

I Single-objective optimization: Particular case where p = 1.An optimal solution typically consists of a single vector x ∈ Ω

I Multiobjective optimization: There is usually no such vectorthat simultaneously minimizes all of the p ≥ 2 objectivefunctions

I The solution consists of a set of trade-off solutions in Ω, thePareto solutions

I The methods presented in this lesson construct anapproximation to this set



Pareto notionI Single-objective: u, v ∈ Ω can be trivially ranked by

comparing f(u) and f(v)

I Generalization with p > 1:I u dominates v, denoted u ≺ v, if and only if F (u) ≤ F (v) and

f (q)(u) < f (q)(v) for at least one index q in 1, 2, . . . , pI u is indifferent to v, denoted u ∼ v, if and only if u does not

dominate v, and v does not dominate u

I A point u ∈ Ω is Pareto optimal if and only if there is now ∈ Ω such that w ≺ u

I The set of Pareto optimal solutions is the Pareto set ΩP

I The image of ΩP under the mapping F defines the solution tothe problem and is called the Pareto front FP ⊆ Rp



Pareto front example

Feasible region : Ω ⊂ R3

6f (2)

-f (1)

cx1∈Ω.

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............ cF (x1)c

x2∈Ω.

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F (x2)

sx3∈Ω.

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sF (x3)

sx4∈Ω.

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sF (x4)

F (Ω)

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Image of Ω in objective space R2

Dominance zonefor F (x1)



Individual minima

The individual minima of F are the solutions to thesingle-objective optimization problems

minx∈Ω

f (q)(x) , for q ∈ 1, 2, . . . p



How to choose one solution?

I Can be done visually with p = 2 and some knowledge of theproblem. Large and small slopes should be identified.

I For p ≥ 2, engineers use carpet plots.

I More generally, this is the subject of multicriteria optimization.


http://en.wikipedia.org/wiki/Carpet_plot


The ε-constraint method

I The most commonly used method.

I It transforms objectives into constraints: The original problemwith p objectives becomes a problem with one objective andp− 1 constraints.

I Then, change the bounds on the constraints in order to graspthe Pareto front.



Weighted sums of objectives for BOP (p = 2)Natural single-objective reformulation: Solve

minx∈Ω

αf (1)(x) + (1− α)f (2)(x) (1)

Inconvenient: Some regions of the Pareto front are never optimalfor (1), regardless of α



Application: Constraint sensitivity analysis

I Biobjective optimization can be used in order to conductsensitivity analyses relative to constraints

I The constraint of interest is transformed as an objectivefunction

I The analysis of the approximated Pareto front allows tointerpret the impact of this constraint on the original objective

I Two different tools are available within NOMAD:I A post-optimization analysis. Cheap and rough approximation

of the sensitivities

I An additional biobjective execution. More expensive, but givesa good approximation of the sensitivities



Sensitivity analysis: Example

−0.1 −0.05 0 0.05 0.125.8

25.9

26

26.1

26.2

26.3

26.4

26.5

Sensitivity to x1−2 ) 0

constraint value

obje

ctiv

e fu

nctio

n va

lue



Introduction

Metrics

BiMADS

Other methods

References



Metrics

I [Audet et al., 2018]

I How to compare the approximations to the Pareto frontobtained by different solvers?

I S: Set of solvers; P: Set of problems

I To draw performance and data profiles, we need aperformance measure tp,s > 0 for each p ∈ P and s ∈ S

I Fp,s: Approximated Pareto front determined by the solvers ∈ S for problem p ∈ P

I Fp: Approximated Pareto front for problem p. Obtained by∪s∈SFp,s and by removing the dominated points



Purity metric

I Purity metric:

purityp,s =|Fp,s ∩ Fp||Fp,s|

∈ [0; 1]

I The higher the better

I Take tp,s = 1/purityp,s if purityp,s 6= 0, +∞ otherwise

I Problem: The purity is equal to one (i.e. perfect) for a solverthat gives only one non-dominated solution



Largest holeI These measures compute the spread of an approximated

Pareto front with the maximum size of the “holes” in thefront. We need |Fp,s| > 1

I tp,s = Γp,s = maxq∈1,2,...,p

(max

i∈1,2,...,|Fp,s|

δ

(q)i

)where δ

(q)i

represents the distance between the ith point of Fp,s and itsclosest neighbor, in terms of f (q)

I HRS (Hole Relative Size): tp,s = maxi∈1,2,...,|Fp,s|

di/d

where

di represents the distance between the ith point of Fp,s and

its closest neighbor, and d =∑|Fp,s|

i=1 di/|Fp,s|

I Standard deviation: tp,s =

√√√√ |Fp,s|∑i=1

(di−d)2

|Fp,s|−1



Progress measures (1/2)

I These measures are focused on the convergence of themethods. Useful for plotting simplified data profiles

I Progress for objective q ∈ 1, 2, . . . , p at evaluation k:

prog(q)k = log

√f(q)1

f(q)k

where f(q)k represents the best value

obtained after the kth evaluation, in terms of f (q)

I We need feasible starting solutions, and all objective valuesneed to be > 0

I We could consider tp,s = maxq∈1,2,...,p

prog(q)k for s and p



Progress measures (2/2)

I Number of non-dominated points at each evaluation: For kthe number of evaluation or a group of n+ 1 evaluations,consider |Fp,s|

I Or consider the number of new non-dominated pointsbetween two values of k

I Number of waves: Consider all the solutions produced bysolver s on problem p. Recursively remove the non-dominatedpoints, and W is the number of times that this operation isnecessary to consider all the points. The more W is close to1, the better is s



Generational Distance (GD)

I Measures a distance between Fp,s and Fp

I GDp,s =

√∑|Fp,s|i=1 d2i,p|Fp,s|

I di,p represents the distance between the ith point in Fp,s andthe closest point of Fp

I The standard deviation of the GD measures the deformationof the front obtained by s ∈ S compared to the global

approximation: STDGDp,s =∑|Fp,s|

i=1 (di,p−GDp,s)2

|Fp,s|−1

I Maximum Pareto Front Error: MEp,s = maxi∈1,2,...,|Fp,s|

di,p



Hypersurface

I Consider tp,s = HSp,s =Sp,s

Sp

I Sp,s represents the surface under the plot of Fp,s and Sp thesurface under the plot of Fp

I Not easy to generalize/compute for p > 2



Introduction

Metrics

BiMADS

Other methods

References



BiMADS: Series of single-optimization executionsI [Audet et al., 2008]

I Based on a single-objective optimization algorithm: MADS

I MADS is launched on a series of subproblems

I Constraints are handled by MADS with EB/PB/PEBtechniques

I Each subproblem is obtained by a single-objectivereformulation that is not based on weights

I The solutions of each of these subproblems produces a localapproximation of the Pareto set

I The set of undominated solutions produces an approximationof the entire Pareto set

I BiMADS is implemented in NOMAD


https://www.gerad.ca/nomad


I Reference point in the objective space: r ∈ R2

I Reformulated objective:

φr(F (x)) :=

−

p∏q=1

(rq − f (q)(x))2 if F (x) ≤ r,p∑q=1

((f (q)(x)− rq)+

)2otherwise

I When minimized on x ∈ Ω, starting from “F−1(r)”, itpotentially generates a solution that dominates r

6f (2)

-f (1)

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φr<0

φr<0

φr=0

φr>0

r. .......................... ........................ ...................... ..................... ...................

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6f (2)

-f (1)

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r

I Every Pareto solution can be obtained with some r.





φr(F (x)) :=

−

p∏q=1

(rq − f (q)(x))2 if F (x) ≤ r,p∑q=1

((f (q)(x)− rq)+

)2otherwise


6f (2)

-f (1)

...........................................

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φr<0

φr<0

φr=0

φr>0

r. .......................... ........................ ...................... ..................... ...................

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6f (2)

-f (1)

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r






φr(F (x)) :=

−

p∏q=1

(rq − f (q)(x))2 if F (x) ≤ r,p∑q=1

((f (q)(x)− rq)+

)2otherwise


6f (2)

-f (1)

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φr<0

φr<0

φr=0

φr>0

r. .......................... ........................ ...................... ..................... ...................

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6f (2)

-f (1)

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r




Reference point selection

I Use the ordering propertyinherent to p = 2 to computegaps between 3 successiveundominated solutions in theobjective space

I Choose r with the largest gap

I Associate a weight to r todecrease the probability ofchoosing it again and preventstalling when the Pareto front isdiscontinuous



BiMADS: successive MADS runs

6f (2)

-

f (1)

c

I Initialization:Solve min

x∈Ωf (q)(x) for q ∈ 1, 2




6f (2)

-

f (1)

cs cc c c


x∈Ωf (q)(x) for q ∈ 1, 2




6f (2)

-

f (1)

cs cc c ccc

c cccs


x∈Ωf (q)(x) for q ∈ 1, 2




6f (2)

-

f (1)

cs cc c ccc

c cccs

s s s ss s s


x∈Ωf (q)(x) for q ∈ 1, 2

I Main iterations:I Reference point determination:

Use the set of feasible ordered undominated points generatedso far to generate a reference point r



BiMADS: successive MADS runs6f (2)

-

f (1)

cs cc c ccc

c cccs

s s s ss s s

s s s ss s s

r. ............................... ............................. ........................... ......................... ....................... ..................... ...................

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x∈Ωf (q)(x) for q ∈ 1, 2



I Single-objective minimization:Solve the subproblem min

x∈Ωφr(F (x))




-

f (1)

cs s s ss s s

s s s ss s s

r. ............................... ............................. ........................... ......................... ....................... ..................... ...................

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c csc ccc c cI Initialization:

Solve minx∈Ω

f (q)(x) for q ∈ 1, 2I Main iterations:

I Reference point determination:Use the set of feasible ordered undominated points generatedso far to generate a reference point r


x∈Ωφr(F (x))




-

f (1)

cs cs cs ssccccc c cc

cc

s cscs


x∈Ωf (q)(x) for q ∈ 1, 2




x∈Ωφr(F (x))




-

f (1)

cs cs cs ssccccc c cc

cc

s cscsr. .................................... ................................. ............................... ............................. .......................... ........................ ..................... ................... ................ .............. ............. .............

.............. ................................. .............................. ............................ ......................... ...................... ................... ................. .............. ........... ..............


x∈Ωf (q)(x) for q ∈ 1, 2




x∈Ωφr(F (x))



MultiMADSI [Audet et al., 2010]I Based on the natural boundary intersection (NBI) framework,

and the convex hull of individual minima

I Consider the simplex g(1), g(2), g(3) obtained from theindividual minima



MultiMADS: Example of solution



Convergence analysisI MADS solves the single objective subproblems

I These solutions x are such that if φr(F (.)) is Lipschitz near x,then φr(F (x); d) ≥ 0 for every direction d in the hypertangentcone THΩ (x) to the domain Ω at x

I Every Pareto point is the optimal solution of a reformulation

I Let x ∈ Ω be a refining point produced by MADS on asingle-objective subproblem for some r ∈ Rp. If F is Lipschitznear x, then for any direction d ∈ THΩ (x), there exists a

q ∈ 1, 2, . . . , p such that(f (q)

)(x; d) ≥ 0

I When the functions are regular, it means that moving in afeasible direction deteriorates at least one objective: This is atradeoff solution (Pareto)



Introduction

Metrics

BiMADS

Other methods

References



Direct Multisearch (DMS)I [Custodio et al., 2011]

I Native adaptation of GPS to the unconstrained multiobjectivecase (p ≥ 2 and use of the EB)

I Intensification with a poll step in which the acceptationcriteria are based on the Pareto dominance

I Diversification with a search step

I Convergence based on the Clarke derivatives

I Differences with biMADS and multiMADS:I BiMADS is a framework using MADS in a subproblem while

DMS is a native multiobjective methodI At each step, DMS tries to improve the entire front, while

biMADS focuses on a specific part of it



NSGA-III NSGA-II: Non-dominated Sorting Genetic Algorithm, for BOP

(p ≥ 2) [Deb et al., 2002]

I Constraints are treated with the inclusion of the violation inthe dominance relation

I Each objective parameter is treated separately

I Mutation and crossover are performed on the population

I Selection based on “non-dominated sorting” (intensification),and “crowded-distance sorting” (diversification)

I Heuristic: No guarantee on the quality of the approximatedPareto front

I From the same team: Archive-based Micro Genetic Algorithm(AMGA) [Tiwari et al., 2008]



Multiobjective solversI NOMAD (p = 2)

I NSGA-II: Several implementations can be found:I MATLAB version

I C versions

I AMGA2 [Tiwari et al., 2011]

I DMS: MATLAB version (by email)

I DFL toolbox:I DFMO: Linesearch, constraints, FORTRANI MODIR: DIRECT, constraints, FORTRANI MOIF: Implicit filtering, bounds, MATLAB


https://www.gerad.ca/nomad

http://www.mathworks.com/matlabcentral/fileexchange/31166-ngpm-a-nsga-ii-program-in-matlab-v1-4

http://www.iitk.ac.in/kangal/codes.shtml

http://www.iitk.ac.in/kangal/codes.shtml

http://www.mat.uc.pt/dms/

http://www.dis.uniroma1.it/~lucidi/DFL/

http://www.iasi.cnr.it/~liuzzi/DFL/index.php/list3/15-multiobjective-optimization/21-dfmo

http://www.iasi.cnr.it/~liuzzi/DFL/index.php/list3/15-multiobjective-optimization/20-modir

http://www.iasi.cnr.it/~liuzzi/DFL/index.php/list3/15-multiobjective-optimization/19-moif


Introduction

Metrics

BiMADS

Other methods

References



References I

Audet, C., Bigeon, J., Cartier, D., Le Digabel, S., and Salomon, L. (2018).

Performance indicators in multiobjective optimization.Technical Report G-2018-90, Les cahiers du GERAD.

Audet, C., Dennis, Jr., J., and Le Digabel, S. (2012).

Trade-off studies in blackbox optimization.Optimization Methods and Software, 27(4–5):613–624.(Constraint Sensitivity Analysis).

Audet, C., Savard, G., and Zghal, W. (2008).

Multiobjective Optimization Through a Series of Single-Objective Formulations.SIAM Journal on Optimization, 19(1):188–210.(biMADS).

Audet, C., Savard, G., and Zghal, W. (2010).

A mesh adaptive direct search algorithm for multiobjective optimization.European Journal of Operational Research, 204(3):545–556.(multiMADS).

Collette, Y. and Siarry, P. (2002).

Optimisation multiobjectif.Eyrolles.(metrics).



References II

Custodio, A., Madeira, J., Vaz, A., and Vicente, L. (2011).

Direct multisearch for multiobjective optimization.SIAM Journal on Optimization, 21(3):1109–1140.(DMS, metrics, profiles).

Das, I. and Dennis, Jr., J. (1998).

Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear MulticriteriaOptimization Problems.SIAM Journal on Optimization, 8(3):631–657.(NBI).

Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. (2002).

A fast and elitist multiobjective genetic algorithm: NSGA-II.IEEE Transactions on Evolutionary Computation, 6(2):182–197.

E.F.Campana, M.Diez, G.Liuzzi, S.Lucidi, R.Pellegrini, V.Piccialli, F.Rinaldi, and A.Serani (2018).

A Multi-objective DIRECT algorithm for ship hull optimization.Computational Optimization and Applications, 71(1):53–72.(MODIR).

G.Cocchi, G.Liuzzi, A.Papini, and M.Sciandrone (2018).

An implicit filtering algorithm for derivative-free multiobjective optimization with box constraints.Computational Optimization and Applications, 69(2):267–296.(MOIF).



References III

Laumanns, M., Zitzler, E., and Thiele, L. (2000).

A Unified Model for Multi-Objective Evolutionary Algorithms with Elitism.In Congress on Evolutionary Computation, volume 1, pages 46–53, Piscataway, New Jersey, USA.(metrics).

Liuzzi, G., Lucidi, S., and Rinaldi, F. (2016).

A Derivative-Free Approach to Constrained Multiobjective Nonsmooth Optimization.SIAM Journal on Optimization, 26(4):2744–2774.(DFMO).

Tiwari, S., Fadel, G., and Deb, K. (2011).

AMGA2: improving the performance of the archive-based micro-genetic algorithm for multi-objectiveoptimization.Engineering Optimization, 43(4):377–401.

Tiwari, S., Koch, P., Fadel, G., and Deb, K. (2008).

AMGA: An Archive-based Micro Genetic Algorithm for Multi-objective Optimization.In Proceedings of the 10th Annual Conference on Genetic and Evolutionary Computation, GECCO ’08,pages 729–736, New York, NY, USA. ACM.

Zitzler, E., Deb, K., and Thiele, L. (2000).

Comparison of Multiobjective Evolutionary Algorithms: Empirical Results.Evolutionary Computation, 8(2):173–195.(metrics).


Documents

Multiobjective Optimization Metrics BiMADS Other methods References Di culty ISingle-objective optimization: Particular case where p= 1. An optimal solution typically consists of a