Evolutionary MOO : A Distributed Computing Approach

Search-based SE: without search, you won’t find a thing.“Engineering is optimization and optimization is search.”

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Evolutionary Multi-Objective Optimization : A Distributed

Computing Approach

- Rahul Krishna (rkrish11)- George Mathew (george2)

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Multi-Objective ProblemPareto Frontier : State of solutions in which it is impossible to make any one individual better off without making at least one individual worse off.

Pareto Point : A point that lies on the pareto frontier.

Feasible Point : A satisifiable solution for the problem but not necessarily the optimum one.

Infeasible Point: A solution outside the pareto frontier

Utopia Point: The ideal theoretical solution we would love to reach but practically its not possible. At this point all the objectives are optimal without any conflicts amongst them.

Image from Search Based SE class NCSU Fall - 2014 https://github.com/timm/sbse14/wiki/DiffEvol 2

https://github.com/timm/sbse14/wiki/DiffEvol


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DTLZ2● Set of multiobjective mathematical test problems created by Kalyanmoy Deb, Lothar

Thiele, Marco Laumans and Eckhart Zitzler.● Decisions : DTLZ2 has 30 decisions where each decision ranges between 0 and 1

● Objectives : 3 objectives defined as follows

● Optimal solutions : Ideal decisions xi = 0.5.Ideal objectives should satisfy the equation

Reference: Kalyanmoy Deb et.al, Scalable multi-objective optimization test problems. CEC ’02. Proceedings of the 2002 Congress3


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Evolutionary Algorithm

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Differential Evolution (DE)

● Stochastic evolutionary optimization technique.● Iteratively approximates the shape of the Pareto

Frontier● Advantages:

○ Simple and computationally inexpensive○ High dimensional problems can be handled easily○ Solutions are very stable

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DE (Algorithm)

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Geometric Active Learner(GALE)● Near linear time MOEA● Builds piecewise approximation to the best solutions of the

pareto frontier● Based on WHERE which is a recursive clustering based on

dimensionality reduction.● Advantages:

○ Less number of computations○ Adept at handling objective functions that are non-

differentiable, non-linear, multidimensional or multi-constraint problems

○ Concise representation of problem space7


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GALE(Algorithm)● Repeat for n generations● Cluster data based on WHERE

○ Pick point X from cluster. Then pick point East furthest from X and point West furthest from East. Let c be distance between East and West.

○ For every other point in cluster, compute a and b as distance of point from East and West respectively.

○ Compute the projection x as followsx = (a2 + c2 - b2) / 2c

○ Split the cluster at median value of x and repeat for each sub cluster● Select the best point(East/West) from the non-dominated leaf cluster. and mutate towards it.● The best points after each generation represents the non dominated solutions

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Why Parallelization?• Multi-objective problems are complex and usually resource

intensive.• Exhaustive search consumes prohibitively large amounts of

memory and time• Meta-heuristics:

• Approximate solutions fast.• Reasonably accurate

• However, scaling to solve real world is still hard• Distributed computing approaches offer significant speed up.

• Distribute the evaluation over several “nodes”.• In theory:

• The solutions must be robust, and • Be of better quality. 9


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EvaluationRuntime — Time taken to run the algorithm in the parallelized version versus the serial version. This can be measured using a profiler.

Image courtesy : Kalyanmoy Deb et.al., A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II

Solution Quality

● Convergence ○ Accuracy of the obtained

solutions. ○ It represents the hypervolume

between the obtained solutions and Pareto frontier.

● Diversity ○ Spread of the proposed solutions. ○ Ideally the solutions should be

well distributed across the Pareto frontier.

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Convergence: ● Find a set of H optimal solutions.● For each solution, compute the minimum

euclidean distance from each of the solutions to a point on the Pareto Frontier.

● The average of these distances represent convergence.

Diversity: ● di represents the distance between

consecutive solutions.● represents the mean of all di.● df and dl are the distances between the

extreme solutions and the boundary solutions

Measures

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Framework● Python

○ Better support for efficient computation frameworks: numpy, scipy, etc.○ Quick prototyping and benchmarking.

● Open-MPI○ An open source Message Passing Interface with a python wrapper. ○ The Open MPI Project is actively developed and maintained by a

consortium of academic, research, and industry partners.

● HPC○ The henry2 shared memory linux cluster at NCSU. ○ Up to 16 shared memory processor cores and up to 128GB of memory

accessible through a dedicated queue.

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Challenges

● Identifying dependencies: Since the meta-heuristics were designed to work synchronously, a key challenge would be to identify the modules that are independent, so that they may be executed concurrently.

● Communication: In order to reduce runtime, it is pertinent that the communication between the slaves and the master be limited. In addition, any data transfer needs to be minimized.

● Quality and Runtime Trade-off: The increased speed of parallel algorithms comes at the cost of reduced quality when compared to the serial version of the algorithm.

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Thank You

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