Multifactorial Optimization: Towards Evolutionary Multitasking

1. AUSTRALASIAN CONFERENCE ON ARTIFICIAL LIFE AND COMPUTATIONAL INTELLIGENCE (ACALCI 2016) Multifactorial Optimization: Towards Evolutionary Multitasking talking writing checking texts gesturing presented by Yew-Soon Ong Professor of Computer ScienceChair, School of Computer Engineering Nanyang Technological University watching TV working 2-5 February 2016, Canberra, Australia

2. Towards Evolutionary Multitasking:Outline of Today’s Talk • Background • Foundational works of Evolutionary Multitasking • Multitasking Problem formulation • Novelty of the paradigm • Key ingredients of the proposed algorithm • Some empirical results • Cross-domain multitasking • Conclusions • Q&A

3. An exposition towards Implicit Transfer of knowledgefor multitasking with Evolution Evolutionary Multitasking A. Gupta, Y. S. Ong, L. Feng, “Multifactorial Evolution: Toward Evolutionary Multitasking,” IEEE Transactions in Evolutionary Computation, 2015. Y. S. Ong and A. Gupta, Evolutionary Multitasking: A Computer Science View of Cognitive Multitasking, Cognitive Computation Journal, Invited Keynote Paper, 2016, To Appear.

4. Multitasking Problem Formulation • Consider a situation where K optimization tasks are to be performed simultaneously. • The ith task, denoted Ti, has a scalar objective function Fi : Xi → ℝ to be minimized. • Evolutionary Multitasking builds on the implicit parallelism of population-based search with the aim to simultaneously find • {x1,x2, …. , xK-1, xK} = argmin{F1(x), F2(x), …., FK-1(x), FK(x)}.

5. Benefits of multitasking • A simple optimization task may complement some other, more complex task in a multitasking environment. • Helps accelerate optimization process on complex problems • Transfer of knowledge (in the form of encoded genetic material) across seemingly distinct complex optimization problems. • Improve throughput by minimizing makespan (total time for job completion) over two or more optimization problems.

6. Some Guiding thoughts on effective multitasking in optimization Phenotypes Decoding • Case 1: Complete overlap • Case 2: Partial overlap • Case 3: No overlap y ∈ Y Y: Shared Genotype Space Describing a common representation space is crucial for effective evolutionary multitasking.

7. Case 1: Complete overlap in phenotype space  Tasks are distinguished based on separate environmental/auxiliary variables • Example 1: In aerodynamic wing-shape/airfoil design where the Mach/Reynolds numbers form the environmental variables • Example 2: Where multiple viable strategies are to be analyzed at the same time.

8. Case 2: Partial overlap in phenotype space • Example 1: Manufacturing processes with various recurrent design variables. Airfoil geometry characterized using a 24-parameter Hicks-Henne representation Airfoil geometry characterized using a 22-parameter Hicks-Henne representation Airfoil geometry characterized using a 12-parameter Hicks-Henne representation • Example 2: Complex routing problems with similar geographical distribution of customers

9. Case 3: No overlap in phenotype space  Imagine an evolutionary multitasking engine living in the cloud Cloud computing platforms face a natural phenomenon wherein multiple diverse jobs can be received from multiple users at the same time. An Evolutionary Solver that lives on the cloud to harness the inter-task complementarities so as to efficiently solve Multiple Optimization Problems at once.

10. Multifactorial Optimization (MFO) for Multitasking • {x1,x2, …. , xK-1, xK} = argmin{F1(x), F2(x), …., FK-1(x), FK(x)}. • Each Fiis treated as an additional factor influencing evolution. • The problem is referred to as a K-factorial problem and the formulation is labelled as • Multifactorial Optimization (MFO) A. Gupta, Y. S. Ong, L. Feng, “Multifactorial Evolution: Toward Evolutionary Multitasking,” IEEE Transactions in Evolutionary Computation, 2015.

11. Some definitions in Multifactorial Evolution • For every individual pi in a population P we define: • (Factorial rank): Factorial rank rijis the rank of pion task Tj, relative to all other individuals in P • (Scalar fitness):Scalar fitness φi of pi is based on its best rank over all tasks; i.e. φi = 1/min{ ri1, ri2, …, riK}. • (Skill factor): Skill factor τi of pi is the one task, amongst all other tasks in MFO, with which the individual is associated. This may be defined as τi = argminj{rij}.

12. The Multifactorial Evolutionary Algorithm (MFEA) A Unified representation Initial population, P, with skill factor Evaluate P for skill factor • An algorithm inspired by the biological concept of multifactorial inheritance. • Gene-cultural interaction forms the crux of evolutionary algorithms • Assortative mating • Vertical cultural transmission Condition satisfied? Y N Over Offspring C = Assortative Crossover + Mutation Assign skill factor to C “Multifactorial Evolution: Toward Evolutionary Multitasking,” IEEE Transactions in Evolutionary Computation, 2015. Evaluate C for skill factor with LS Combine P+C and perform selection (elitist)

13. Important Ingredient: Population initialization with A Common chromosome representation • With K optimization tasks to be performed simultaneously, the dimensionality of the ithtask is given by Di. • Accordingly, we define a unified search space with dimensionality (Dmultitask) equal to maxi{Di}. • During the population initialization step, every individual is assigned a vector of Dmultitask random keys that lie in the fixed range [0, 1]. • While addressing task Ti, we consider the Random-key chromosome representation, with Di denoting the first of the random-keys of the chromosome. We do not append, we unify

14. Random-Key Encoding + Decoding Exemplar & Multitasking Problem Task 2: 6-D TSP Task 1: 12-D Knapsack Sample 12-D Chromosome Binary Decoding 12-D Knapsack Solution Rank 1 2 … 6 Sequence-Based Decoding 6-D TSP Solution

15. Assortative Mating The principle of Assortative Mating suggests that biological entities prefer to mate with those sharing similar characteristics or similar cultural backgrounds. In the MFEA, the above is realized with individuals preferring to mate with those possessing the same skill factor.

16. One-Point Crossover in multitasking good block for T2 Improved Child for T1 Improved Child for T2 inferior block for T1 good block for T1 Implicit genetic transfer good block for T2 good block for T1 crossover Parent 1 with skill factor T1 Parent 2 with skill factor T2 unified search space search space for task T1 search space for task T2 Knapsack Problem Set-Covering Problem

17. Simulated Binary Crossover: Multitasking in Continuous Spaces Vertical Knowledge Transmission Region with skill factor T2 C2 P2 SBX crossover P1 C2 P2 C1 Imitates Implicit genetic transfer (P1+P2)/2 = (C1+C2)/2 P1 C1 Imitates Region with skill factor T1 Hypothetical 2-D Unified Search Space

18. Vertical Cultural Transmission One of the most prevalent forms of Vertical Cultural Transmission is Offspring Imitating Parents In the MFEA, the above is realized with offspring imitating the skill factor of any one parent at random.

19. Highlighting the distinction between Multitasking and Multiobjective Optimization

20. MFO vs MOO As both paradigms are involved with optimizing a set of objective functions, a conceptual overlap may be seen to exist between them. However, • Evolutionary multitasking aims to leverage the implicit parallelism of population-based search to exploit latent complementarities betweendistinct tasks. • Multi-objectiveoptimization deals with efficiently resolving conflicts among competing objectives of the same task.

21. Distinctive selection pressure: An example • As per MOO {p2, p3, p4, p5} belong to the first non-dominated front and are always preferred over {p1, p6} which belong to the second front. • As per MFO {p2, p5} are favored over {p1, p6} which are in turn favored over {p3, p4}.

22. The Boon of Multitasking A. Gupta, Y. S. Ong, L. Feng, “Multifactorial Evolution: Toward Evolutionary Multitasking,” IEEE Transactions in Evolutionary Computation, 2015. http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=7161358&url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel7%2F4235%2F4358751%2F07161358.pdf%3Farnumber%3D7161358 Web site dedicated to Evolutionary Multi-tasking with Source Codes: http://www.cil.ntu.edu.sg/mfo/home.htm Numerical examples in continuous optimization

23. Multi-tasking with Continuous and Continuous Tasks (I) (a) Multimodal and Unimodal with overlapping optima Comparison of convergence trends of MFO vs SOO. 1-D slice of the functions A 2-D view of Rastrigin’s function A 2-D view of the sphere function

24. Multi-tasking with Continuous and Continuous Tasks (II) (b) Multimodal and Unimodal with shifted optima Comparison of convergence trends of MFO vs SOO. 1-D slice of the functions A 2-D view of Rastrigin’s function A 2-D view of the shifted sphere function

25. Multi-tasking with Continuous and Continuous Tasks (III) • (c) Multimodal and Multimodal with overlapping optima Comparison of convergence trends of MFO vs SOO. 1-D slice of the functions A 2-D view of Rastrigin’s function A 2-D view of Ackley’s function

26. Multi-tasking with Continuous and Continuous Tasks (IV) • (c) Multimodal and Multimodal with shifted optima Comparison of convergence trends of MFO vs SOO. 1-D slice of the functions A 2-D view of Rastrigin’s function A 2-D view of shifted Ackley’s function

27. Diverse Problem Types Cross-domain Multitasking

28. Numerical results for Cross-domain Multitasking (a) Continuous and Discrete problems solved together (QAP, KP, Rastrigin’s function) , 3 Tasks optimized together Example of task-wise positive genetic transfer during multitasking. A. Gupta, Y. S. Ong, L. Feng, “Multifactorial Evolution: Toward Evolutionary Multitasking,” IEEE Transactions in Evolutionary Computation, 2015.

29. Numerical results for cross-domain multitasking (b) Continuous and Discrete problems solved together (KP, CVRP, Ackley’s function) , 3 Tasks optimized together Showcasing the possibility of negative genetic transfer (for Task 1) during multitasking. A. Gupta, Y. S. Ong, L. Feng, “Multifactorial Evolution: Toward Evolutionary Multitasking,” IEEE Transactions in Evolutionary Computation, 2015.

30. Multitasking in Multi-objective optimization

31. Sample results  We have designed a multi-objective version of the MFEA called MO-MFEA. In the special case of 1 task, the MO-MFEA takes the exact form of NSGA-II Significant Speedup also in Multi-Tasking Multi-Objective Problems

32. Some Real-World Illustrations

33. In Composite Manufacturing Injection/Compression Liquid Composite Molding (I/C-LCM) Resin Transfer Molding (RTM)

34. In Multi-UAV Path-Planning Mission 2 Mission 1

35. In last-mile logistics (on the Cloud)  A new routing problem can be introduced into a multitasking engine at different intermediate stages of solving an ongoing routing task. "City Vehicle Routing Problem (City VRP): A Review", IEEE Transactions on Intelligent Transportation Systems, 2015

36. Summarizing the results • The numerical measure of the “cumulative complementarity between functions” is shown to have qualitative agreement with the observed performance of the MFEA. In other words, large complementarity corresponds to faster convergence, while a drop in complementarity results in convergence deceleration. • Thus, the measure provides a deeper understanding of the underlying complementarity between tasks that gets harnessed during the process of multitasking. • With the aid of this understanding, new evolutionary multitasking engines can be developed that can adapt to varying levels of inter-task complementarity in multitasking environments by fully exploiting available synergies.

37. Conclusions We have presented a novel evolutionary multitasking paradigm as a new concept within the purview of Evolutionary Computation From evolutionary multitaskingemerges the scope for implicit genetic transfer across diverse problems, which potentially leads to accelerated optimization of complex functions. Finally, guided by a deeper understanding of inter-task complementarity, we envision the ideal evolutionary multitasking engine of the future to be a complex adaptive system with worst case performance approaching that of standard serial evolutionary optimizers of the present day.