### High-Order Pareto Frontier Approximation and

```High-order Pareto frontier approximation
and visualization:
30 years of experience and new trends
Abstract of the paper at MCDM 2011
Alexander V. Lotov
Dorodnicyn Computing Center of Russian
Lomonosov Moscow State University
Plan of the talk
1. Few words concerning Pareto frontier visualization
2. Interactive Decision Maps (IDM) technique for
visualization of the high-order Pareto frontier
3. Few words about application of IDM in the convex case
4. IDM in the non-linear non-convex case:
using of hybrid (classic&genetic) optimization
5. Application of IDM in non-linear non-convex problems:
a) study of the cooling equipment for continuous steel casting
b) designing the release rules for the Baikal Lake Basin
c) development of efficient strategies for AIDS treatment
6. New studies: linearization of response surface,
identification of math models
This job was carried out by a large group of
specialists and students that includes
V.Bushenkov, O.Chernykh,
G.Kamenev, D.Gusev,
L.Bourmistrova, R. Efremov,
V.Berezkin, A.Pospelov,
and many others.
Few words concerning Pareto
frontier visualization
Notation
Let X be the feasible set in decision space,
z=(z1, z2,…, zm)= f(x) be the criteria vector,
where f(x) is the vector of objective functions.
Then, Z=f(X) = feasible set in objective space R m
Pareto domination (minimization case)
z'  z' '  z'i  z' 'i , i  1,2,...,m; z'  z' ' 
Non-dominated (efficient, Pareto) frontier
P(Z )  z  Z : z' Z : z'  z  emptyset
Feasible set in objective space
z2
Z=f(X)
z1
Non-dominated (Pareto) frontier is visible
z2
Z=f(X)
z1
Pareto frontier methods
(a posteriori preference) methods
Pareto frontier methods consist in
1) approximating the Pareto frontier and
2) informing the Decision Maker about it.
In contrast to preference-oriented methods,
Pareto frontier methods do not require the
concerning his/her preferences, but only
inform the DM.
Two basic ways for informing the DM
• By providing a list of the objective
(criterion) points that belong to the
Pareto frontier
• By visualization of the Pareto frontier
Selecting from a large list of objective
points (more than one or two dozens)
with more than two criteria turned out to
be too complicated for a human being.
See, for example, the paper
• Larichev O. Cognitive Validity in
Design of Decision-Aiding
Techniques. Journal of Multi-Criteria
Decision Analysis, 1992, v.1, n 3.
Visualization of the Paretoo frontier can
help
”A picture is worth a thousand words”
Prof. A.Wierzbicki proved that
a picture is worth 10 thousand words
Tradeoff information is important for DM
z2
f(x*)
f(x1)
f(x2)
z1
We develop the Pareto frontier
visualization methods
for the high-order problems (m > 2)
Two problems must be solved
• How to approximate the Pareto frontier?
• How to inform the stakeholders about the
Pareto frontier?
Interactive Decision Maps (IDM)
technique solves these problems in highorder MOO problems.
Its generic ideas were formulated in 1980
(Lotov, A.V. On the concept of the GRS
and its constructing for linear controlled
systems. Sov. Phys. Dokl., American
Istitute of Physics, 1980, 25(2), 82–84)
The IDM technique is based on
approximating the feasible objective set Z
or
its Edgeworth-Pareto Hull (EPH),
that is
Zp  Z
m
 R
Sometimes the EPH is called the Free Disposal Hull.
It holds
P(Z p )  P(Z )
z2
P(Z)
f(X)
Zp
z1
Visualization by usung the IDM tech
In this talk, we restrict with visualization of the
EPH.
The IDM tech is based on the display of biobjective slices of the EPH.
Decision map is a collection of overlapped biobjective slices in the case m=3.
If for m>3, the IDM technique displays decision
maps interactively. Decision maps can be rearranged, animated, zoomed, etc. by the user.
This option is based on the approximating the
EPH, which has to be completed in advance.
Example of the Pareto frontier
display for m=5 in the convex case
Another example of the Pareto frontier
display for m=5 in the convex case
by using the matrix of decision maps
Application of the IDM tech:
Feasible Goals Method (FGM).
It is the IDM-supported goal method, in
the framework of which the goal is
identified at the Pareto frontier.
Objective (criterion) tradeoff information helps the
decision maker to identify the preferable nondominated objective point (goal) consciously.
It is important that the goal located at the Pareto
frontier is feasible.
Due to it, the associated decision does exist and can
be computed.
Application of the IDM in the
convex case
In the convex case polyhedral
approximation of the EPH is used.
Method and applications are described
in
Lotov A.V., Bushenkov V.A., and Kamenev G.K.
Interactive Decision Maps.
Approximation and Visualization of
Pareto Frontier.
Real-life applications of FGM and its
modification (Reasonable Goals Method) in
the convex case
DSS for Water Quality Planning (Russian Federal Programme
“Revival of the Volga River”)
Searching for trans-boundary air pollution control strategies (jointly
with M.Pohjola and V.Kaitala, Finland)
Exploration of pollution abatement cost in the Electricity Sector –
Israeli case study (Ministry of National Infrastructures of Israel,
D.Soloveichik et al.).
Web-based Participatory Decision Support for Integrated River Basin
Planning (jointly with J.Dietrich and A.H. Schumann, Germany)
Water quality planning in rivers of Cataluña (A.L.Udias Moinelo and
R.Efremov, Spain, A.Pospelov, Russia)
problems
•
•
•
•
•
Development of smart response strategies related to global
climate change
Environmentally sound agricultural planning in the Netherlands
(jointly with S.Orlovski and P.˚van Walsum from IIASA,
Austria)
Allocation of sea oil platforms and planning the oil fields
development (jointly with R.Efremov, Spain, A.Barron
Alcantara, Mexico)
E-democracy: web-based participatory decision support (jointly
with Efremov R., Rios-Insua D.)
Etc.
IDM in the non-linear
non-convex case
The main problems
that arise in the non-linear case:
1. non-convexity of the EPH;
2. time-consuming processes of global scalar
optimization, i.e. computing the support
function may require too much time or may
be impossible.
Approximation of the non-convex EPH
Approximation for visualization
The EPH is approximated by the set T* that is the
union of the non-negative cones
zR
m

with apexes in a finite number of points z of the set
Z=f(X). Collection of such points z is called the
approximation base and is denoted by T.
Important! Multiple slices of such an approximation
can be computed and displayed fairly fast.
Visualization example for 8 criteria
Approximation of the EPH
We concentrated our efforts at the methods for
approximating the EPH in MOO problems
with criterion functions given by black-box
models (say, FEM/FDM modules, or different
simulation modules). Thus, the Lipschitz
constants are unknown (or may not exist at
all) and cannot be used in the methods.
Actually, the only feasible operation with
the module is variation of its inputs and
collecting the related outputs.
We have developed hybrid methods for
approximating the EPH for black-box
modules that include:
a) random (Monte Carlo) search;
simulation-based optimization;
c) importance sampling (squeezing
the search region);
d) semi-genetic algorithms.
Statistical tests were developed that
provide the basis for the stopping rules.
Local simulation-based optimization
Simulation provides an opportunity to approximate
the gradient of a scalar function. It means that it is
possible to use various effective gradient-based
methods (for example, methods of conjugated
gradients) to find local maximum (or minimum) of a
scalar function. These methods can be used for
‘improving’ a random decision x by moving the
associated criterion point f(x) in the direction of the
Pareto frontier.
Two-phase method: combination of
random search and local optimization
Combination of random search and local optimization is
often used in scalar optimization. The simplest methods
of this kind are multi-start methods, more complicated
methods have been proposed, too. Methods of this kind
are known as the two-phase methods.
We apply two-phase methods as the basic tool for
approximation of the Pareto frontier. In our methods the
scalarizing function is not given. Several concepts for
adaptive selecting of scalarizing functions were proposed
by us.
Three-phase method
The three-phase methods include squeezing of the
search region. The methods for adaptive squeezing
were proposed.
Plastering (semi-genetic) method
Plastering method that has some properties of genetic
algorithms (as cross-over and selecting of nondominated decisions) is used at the very end of the
approximation process.
Approximating the EPH using parallel
computing
The proposed methods have the form that can
be used in parallel computing (clusters,
supercomputers, grids, etc.) – it is sufficient
to separate simulation and Pareto frontier
computing.
Even cloud computing can be used since
methods are nor sensitive to a partial loss of
the results of simulation.
Two-platform implementation
IDM applications in the non-linear
non-convex MOO problems
a) Multi-objective study of the
cooling equipment in continuous
casting of steel
The research was carried out jointly with
K.Miettinen and several other researchers
Criteria
J1 is the original single optimization criterion: deviation
from the desired surface temperature of the steel strand must
be minimized.
J2 to J5 are the penalty criteria introduced to
describe violation of constraints imposed on :
-surface temperature (J2 );
-gradient of surface temperature along the strand
(J3);
-on the temperature after point z3 (J4); and
-on the temperature at point z5 (J5).
J2 to J5 were considered in this study.
Description of the module
FEM/FDM module was developed by researchers
Properties of the model: 325 control variables that
describe intensity of water application.
b) Application of the non-linear
IDM in the design of the release
rules for the hydro power stations
in the Baikal Lake basin
(encouraged by the group of
Prof. R. Soncini-Sessa from
Politecnico di Milano)
Scheme of the Baikal Lake basin
The problem of release rules design for a
Water managers develop some rules, which
relate the flow through a dam (release) to the
current level of the reservoir.
The parameters of the release rules must be
specified to satisfy the requirements of the
industrial, agricultural and municipal users as
well as environmental requirements.
The main concept in the design of
the release rules
The approach is based on direct application of the
historical inflow data. Then, any particular release
rule can be simulated by using a water flow model
combined with historical data on inflow through
the time period under study (about 100 years). By
this, performance indicators for any particular
release rule can be estimated.
Optimization criteria are based on
deviations from :
• Required level of electricity production;
• Satisfaction of environmental requirements
to water use of Baikal;
• Absence of floods;
• etc.
Thus, the objective values must be
minimized.
Completed study:
Pareto frontier for
Irkutsk hydropower station
The Baikal Lake basin
Irkutsk HPS
The decision rule was described by
132 parameters, eight objectives
were used.
After about one hour of computing
at a supercomputer an
approximation of the EPH was
constructed.
Visualization
• After a few steps of visual study of the EPH in general,
it was decided that four objective must have their
minimal values y3=y4=y6=y7=0.
• The rest of objectives are:
y1 is the volume of water released without producing
energy
y2 is the violation of the NO FLOOD requirement
y5 is the violation of a high requirement on energy
production
y8 is the violation of the requirement to have the level of
the Baikal Lake in a corridor set fixed by government
Decision map 1
Decision map 2
Decision map 3
Decision map 4
Decision map 5
Selected objective points
•
•
•
•
•
•
y1=0.86, y2=0.00, y5=0.77, y8=0.93
y1=0.77, y2=0.22, y5=0.85, y8=0.84
y1=0.85, y2=0.11, y5=0.42, y8=0.88
y1=0.91, y2=0.00, y5=0.42, y8=0.91
y1=0.92, y2=0.11, y5=0.34, y8=0.90
y1=0.89, y2=0.00, y5=0.33, y8=0.93
Value Path
Search for effective strategies for
HIV therapy
The model
• n is the number of healthy lymphocytes of CD4+ type,
• I1 is the number of lymphocytes infected by the natural
HIV virus,
• I2 is the number of lymphocytes infected by the mutated
HIV virus,
• h is the concentration of drug in blood,
• u is the intensity of treatment: 0<u<U
Objectives
• y1 is the drug application during one year
• y2 is the maximal concentration of drug in the
body
• y3 is the number of infected lymphocytes at the
very end of the time-period
• y4 is the maximal number of infected
lymphocytes during the time-period
• y5 is the number of healthy lymphocytes at the
very end of the time-period
• y6 is the maximal number of healthy
lymphocytes during the time-period
One of the decision maps
One of the Pareto optimal therapy
strategies
The result of the selected Pareto optimal
therapy
New developments:
linearization of response
surface and identification of
math models
Linearization of response surface
in Water Quality Rehabilitation
Details described in:
A. Castelletti, A. Lotov, R. Soncini-Sessa.
Visualization-based multi-criteria improvement of
environmental decision-making using linearization
of response surfaces. Environmental Modelling
and Software, v.25, 2010, pp. 1552-1564.
Examples of graphs of error function 
of parameter λ


а)
l

б)
в)
l
l
Our Web site
• http://www.ccas.ru/mmes/mmeda/
```