Various Topics Outline 1. Dynamic (time-varying) Optimization - - PDF document

HEURISTIC OPTIMIZATION

Various Topics

Outline

• 1. Dynamic (time-varying) Optimization Problems
• 2. Stochastic Optimization Problems
• 3. Continuous (real-parameter) Optimization Problems
• 4. SLS Algorithms Engineering

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Dynamic (time-varying) optimization problems

I in many problems data, objectives, or constraints change over

time

I as a result, a candidate solution to a problem may (need to)

adapt while implementing it

I in dynamic optimization problems, a dynamic (i.e.

time-varying) problem is solved online

I large variety of different problem characteristics depending on

how and when changes are considered

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DOPs: classification

I time-linkage: does future behavior of the problem depend on

current solution?

I predictability: are changes predictable? I detectability: are changes visible or detectable? I recurrency: are changes cyclic / recurrent? I changes: which are the problem data / information that

changes? (objectives? number, domain, type of decision variables? constraints? instance data?)

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DOP example: dynamic travelling salesman problem (DTSP)

I various DTSP formulations are possible I time-varying travel costs

I edge weights may change e.g. mimicking traffic jams etc.

I time-varying customers (nodes)

I occasionally some nodes disappear / appear and, thus,

modified instances are obtained

I instances parameterized by frequency and amount of changes

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Tackling DOPs

I detecting changes? I restarting algorithms after changes?

I easy, straightforward choice I may be effective if change is very strong I however, it may (i) waste computation resources, (ii) may lead

to very different solutions after change (even if change is small)

I other approaches to adapt algorithms to specificities of the

dynamic problems

I uses of memory to store useful information / promising

solution components

I adaptation of parameters or neighborhoods I increasing diversity by ewn solutions (e.g. random immigrants) I prediction of changes and pro-active actions Heuristic Optimization 2016 6

Tackling DOPs .. cont’d

I periodic reoptimization

I periodically, a static problem instance is solved either when

available data changes or at fixed intervals of time

I can rely on known effective algorithms for static problems I but requires optimization before updating solutions

I continuous reoptimization

I perform optimization throughout the day I maximizes computational capacity I however, solutions may change at any time Heuristic Optimization 2016 7

Performance evaluation for DOPs

I two main aspects

I convergence speed I quality of obtained solutions

I a large number of performance measures w.r.t. measuring

quality and convergence speed related behavior have been proposed

I unification possible e.g. by using hypervolume of dominated

time/cost tradeoff

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Stochastic optimization problems

I stochastic optimization concerns the study and solution of

• ptimization problems that involve uncertainty

I part of the information about problem data is partially

unknown

I knowledge about the probability distribiton of unknown is

assumed

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Modeling approaches to uncertainty

I how is uncertainty modeled?

I prefect knowledge of data 7! classical deterministic

• ptimization

I by means of random variables with known distributions I fuzzy sets / quantities I interval values without known distribution I no knowledge 7! online optimization Heuristic Optimization 2016 10

Modeling approaches to uncertainty

I dynamicity of the model? I i.e. time when uncertain information is revealed w.r.t. time

when decision needs to be taken

I distinguish time before actual realization of random variables

and time after random variables are revealed

I a priori optimization versus decision in stages I two-stage optimization problems: first stage decision is done (a

priori solution) and corrective actions can be made once random realizations are known

I also known as simple recourse model Heuristic Optimization 2016 11

Domain of stochastic (combinatorial)

• ptimization

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Formalization of stochastic combinatorial optimization problem

I problems that can be described as

Min F(x) = E ⇥ f (x, ω) ⇤ , subject to x 2 S,

I x is a solution I S is the set of feasible solutions I E is the mathematical expectation I f is the cost function I ω is a multi-variate random variable, hence f (x, ω) makes the

cost function a random variable

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Probabilistic TSP (PTSP)

I complete graph G = (V , A, C, P) with

I set of nodes V I set of edges A I C cost-matrix for travel costs between pairs of nodes I probability vector P that for each node i specifies its

probability pi of requiring visit.

I i.e. ω here is a n-variate Bernoulli distribution I realization: a binary vector of size n

I 1: node requires visit I 0: node is to be skipped (no visit)

I homogeneous PTSP: pi = p : 8i 2 V I heterogenous PTSP: otherwise

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Probabilistic TSP (2)

a priori optimization

I Stage 1: determine permutation of all nodes

a priori solution

I . . . realization of random variable becomes available . . . I Stage 2: determine actual tour by skipping nodes not to be

visited a posteriori solution

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Applying metaheuristics to SOPs

I typically involves computation / approximation of expected

value of the objective function

I three main possibilities

I closed-form expressions available to compute exact expected

value

I ad hoc and fast approximation if computation is too expensive I estimation of expected values by simulation

excursion: efficient local search for PTSP

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Stochastic vs. dynamic problems

I many problems domains where stochastic problems arise can

also be modeled as dynamic problems

I advantage of stochastic problems is that assumed distribution

• f data may be useful to generate realistic solutions that are

more easily adapted to practical situations

I in a sense, stochastic information is used to define “policies” I however, computation of objective function is more

demanding and there is a necessity to assess probability distributions from data or (subjective) expertise

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Continuous (real-parameter) optimization problems

see excerpt of slides from Anne Auger of her course on derivative-free optimization

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SLS algorithms

I among most successful techniques for tackling hard problems I prominent in computing science, operations research and

engineering

I range from simple constructive and iterative improvement

algorithms to general-purpose methods (“metaheuristics”)

I widely studied, thousands of publications I sub-areas have become established fields (evolutionary

algorithms, swarm intelligence) SLS algorithms are by now a well established tool for solving theoretically and practically relevant search problems

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SLS algorithms

Current deficiencies

I few general guidelines of how to design efficient SLS

algorithms; application often considered an art

I high development times and expert knowledge required I shortcomings in experimental methodology I relationship between problem / instance characteristics and

performance not well understood

I enormous gap between theory and practice

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Which metaheuristic for which problem?

Metaheuristics network

I collaborative research project among four academic and one

industry partner

I initial structure of research

I work on a common set of problems I each lab implements its favorite metaheuristic and one more I compare performance of SLS algorithms to allow insights into

which metaheuristic strategies are the most successful for specific problems

I ideal case: matching between problems / instance classes and

success of metaheuristics

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Which metaheuristic for which problem?

Insights from Metaheuristics Network

I success with SLS algorithms rather due to

I level of expertise of developer and implementer I time invested in designing and tuning the SLS algorithm I creative use of insights into algorithm behaviour and interplay

with problem characteristics

I fundamental are issues like choice of underlying

neighbourhoods, efficient data structures, creative use of algorithm components; to a less extent strictly following the templates of specific SLS methods (“metaheuristics”)

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SLS algorithm engineering

Main GOAL: develop a sound methodology for the design, implementation, and analysis of stochastic local search algorithms

I devise principled procedures that lead to (sufficiently) high

performing SLS algorithms

I exemplary step-wise engineering procedure

I get insight into the problem being tackled I implement basic constructive and local search procedures I starting from these add complexity (simple SLS methods) I add advanced concepts like perturbations, population I if needed: iterate through these steps

bottom-up approach: add complexity step-by-step

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Algorithm engineering

Algorithm engineering (AE)

I process of designing, analyzing, implementing, tuning, and

experimentally evaluating algorithms [Demetrescu et al. 2003]

I is conceived as an extension of traditional (rather theoretical)

research in algorithmics

SLS algorithm engineering

I analogous high-level process to AE I but much more difficult because

I problems tackled are highly complex (NP-hard) I stochasticity of algorithms makes analysis harder I many more degrees of freedom Heuristic Optimization 2016 28

SLS algorithm engineering: tools

Tools

I tools are needed to assist development process I several tools to support specific tasks are available

I software frameworks, statistical tools, experimental design,

search space analysis, data structures, etc.

I missing: integration into an SLS engineering process

Practical GOAL: make available a complete set of procedures to assist the design process of SLS algorithms

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SLS algorithm engineering: knowledge

Knowledge / expertise

I raise awareness about important knowledge in SLS algorithms

and problems

I computer science basics (especially algorithmics and AI) I statistical methodologies I general-purpose SLS methods as well as basic techniques

(constructive heuristics, iterative improvement)

I problems, their features and characteristics and classical

solution techniques

I relationship between algorithm performance and problem

features

Pedagogical GOAL: define a curriculum for SLS; provide complete case-studies of SLS algorithm development

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SLS algorithm engineering

SLS Appli- cations Computer Science Operations Research Statis- tics SLS

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SLS algorithm engineering: applications

Impact on applications

I SLS algorithms have a very wide range of applications (from

bioinformatics over telecommunications and engineering to business administration)

I advancements of methodological aspects have the high

potential to have strong repercussion in many application fields Marketing GOAL: make researchers and practitioneers aware of the high importance of SLS algorithms

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SLS algorithm engineering: science

Scientific issues

I provable properties and guarantees I understanding parameter responses and dependencies I understand the relationship between performance, instance

features and SLS algorithm components

I motivate principled decisions in SLS design

Scientific GOAL: provide helpful insights into SLS behavior to inform SLS engineering

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Interplay with SLS science

Synergies

I SLS engineering can leverage understanding of SLS behavior I SLS science can inform SLS engineering

Intersection

I sound empirical analysis techniques I in-depth experimental studies

principled SLS research = SLS science + SLS engineering

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SLS algorithm engineering: structuring

Benefits to SLS research

I research in SLS very much scattered into different directions I SLS engineering offers orientation by defining important areas

such as

I methodological developments I development of algorithmic techniques (large-scale

neighbourhoods, ACO, VND, ..)

I development of tools (R, F-races, EasyLocal++, etc) I systematic, in-depth experimental studies

Structuring GOAL: give a framework for research efforts in SLS

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