HEURISTIC OPTIMIZATIONstuetzle/Teaching/HO/Slides/Lecture12.pdf · Heuristic Optimization 2018 12....

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

Heuristic Optimization 2018 2

Dynamic (time-varying) optimization problems

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

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 di↵erent problem characteristics depending onhow and when changes are considered


DOPs: classification

I time-linkage: does future behavior of the problem depend oncurrent 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 thatchanges? (objectives? number, domain, type of decisionvariables? constraints? instance data?)


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 tra�c 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


Tackling DOPs

I detecting changes?I restarting algorithms after changes?

I easy, straightforward choiceI may be e↵ective if change is very strongI however, it may (i) waste computation resources, (ii) may lead

to very di↵erent solutions after change (even if change is small)

I other approaches to adapt algorithms to specificities of thedynamic problems

I uses of memory to store useful information / promisingsolution components

I adaptation of parameters or neighborhoodsI increasing diversity by new solutions (e.g. random immigrants)I prediction of changes and pro-active actions


Tackling DOPs .. cont’d

I periodic reoptimizationI periodically, a static problem instance is solved either when

available data changes or at fixed intervals of timeI can rely on known e↵ective algorithms for static problemsI but requires optimization before updating solutions

I continuous reoptimization

I perform optimization throughout the dayI maximizes computational capacityI however, solutions may change at any time


Performance evaluation for DOPs

I two main aspects

I convergence speedI quality of obtained solutions

I a large number of performance measures w.r.t. measuringquality and convergence speed related behavior have beenproposed

I unification possible e.g. by using hypervolume of dominatedtime/cost tradeo↵


Stochastic optimization problems

I stochastic optimization concerns the study and solution ofoptimization problems that involve uncertainty

I part of the information about problem data is partiallyunknown

I knowledge about the probability distribution of unknowns isassumed


Modeling approaches to uncertainty

I how is uncertainty modeled?I prefect knowledge of data 7! classical deterministic

optimizationI by means of random variables with known distributionsI fuzzy sets / quantitiesI interval values without known distributionI no knowledge 7! online optimization


Modeling approaches to uncertainty

I dynamicity of the model?

I i.e. time when uncertain information is revealed w.r.t. timewhen decision needs to be taken

I distinguish time before actual realization of random variablesand time after random variables are revealed

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

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

I also known as simple recourse model


Domain of stochastic (combinatorial)optimization


Formalization of stochastic combinatorial optimization problem

I problems that can be described as

Min F (x) = E⇥f (x ,!)

⇤, 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 ! is a multi-variate random variable, hence f (x ,!) makes thecost function a random variable


Probabilistic TSP (PTSP)

I complete graph G = (V ,A,C ,P) withI set of nodes VI set of edges AI C cost-matrix for travel costs between pairs of nodesI probability vector P that for each node i specifies its

probability pi of requiring visit.

I i.e. ! here is a n-variate Bernoulli distribution

I realization: a binary vector of size nI 1: node requires visitI 0: node is to be skipped (no visit)

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

I heterogenous PTSP: otherwise


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 bevisited a posteriori solution


Applying metaheuristics to SOPs

I typically involves computation / approximation of expectedvalue of the objective function

I three main possibilitiesI closed-form expressions available to compute exact expected

valueI ad hoc and fast approximation if computation is too expensiveI estimation of expected values by simulation

excursion: e�cient local search for PTSP


Stochastic vs. dynamic problems

I many problems domains where stochastic problems arise canalso be modeled as dynamic problems

I advantage of stochastic problems is that assumed distributionof data may be useful to generate realistic solutions that aremore easily adapted to practical situations

I in a sense, stochastic information is used to define “policies”

I however, computation of objective function is moredemanding and there is a necessity to assess probabilitydistributions from data or (subjective) expertise


Continuous (real-parameter) optimizationproblems

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


SLS algorithms

I among most successful techniques for tackling hard problems

I prominent in computing science, operations research andengineering

I range from simple constructive and iterative improvementalgorithms to general-purpose methods (“metaheuristics”)

I widely studied, thousands of publications

I sub-areas have become established fields (evolutionaryalgorithms, swarm intelligence)

SLS algorithms are by now a well established tool for solvingtheoretically and practically relevant search problems




SLS algorithms

Current deficiencies

I few general guidelines of how to design e�cient SLSalgorithms; application often considered an art

I high development times and expert knowledge required

I shortcomings in experimental methodology

I relationship between problem / instance characteristics andperformance not well understood

I enormous gap between theory and practice


Which metaheuristic for which problem?

Metaheuristics network

I collaborative research project among four academic and oneindustry partner

I initial structure of researchI work on a common set of problemsI each lab implements its favorite metaheuristic and one moreI compare performance of SLS algorithms to allow insights into

which metaheuristic strategies are the most successful forspecific problems

I ideal case: matching between problems / instance classes andsuccess of metaheuristics


Which metaheuristic for which problem?

Insights from Metaheuristics Network

I success with SLS algorithms rather due toI level of expertise of developer and implementerI time invested in designing and tuning the SLS algorithmI creative use of insights into algorithm behaviour and interplay

with problem characteristics

I fundamental are issues like choice of underlyingneighbourhoods, e�cient data structures, creative use ofalgorithm components; to a less extent strictly following thetemplates of specific SLS methods (“metaheuristics”)


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 (su�ciently) highperforming SLS algorithms

I exemplary step-wise engineering procedureI get insight into the problem being tackledI implement basic constructive and local search proceduresI starting from these add complexity (simple SLS methods)I add advanced concepts like perturbations, populationI if needed: iterate through these steps

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


Algorithm engineering

Algorithm engineering (AE)

I process of designing, analyzing, implementing, tuning, andexperimentally evaluating algorithms [Demetrescu et al. 2003]

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


I analogous high-level process to AEI but much more di�cult because

I problems tackled are highly complex (NP-hard)I stochasticity of algorithms makes analysis harderI many more degrees of freedom


SLS algorithm engineering: tools

Tools

I tools are needed to assist development processI 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 toassist the design process of SLS algorithms


SLS algorithm engineering: knowledge

Knowledge / expertise

I raise awareness about important knowledge in SLS algorithmsand problems

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

(constructive heuristics, iterative improvement)I problems, their features and characteristics and classical

solution techniquesI relationship between algorithm performance and problem

features

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



SLS

Appli-cations

ComputerScience

OperationsResearch

Statis-tics

SLS


SLS algorithm engineering: applications

Impact on applications

I SLS algorithms have a very wide range of applications (frombioinformatics over telecommunications and engineering tobusiness administration)

I advancements of methodological aspects have the highpotential to have strong repercussion in many applicationfields

Marketing GOAL: make researchers and practitioneers aware of thehigh importance of SLS algorithms


SLS algorithm engineering: science

Scientific issues

I provable properties and guarantees

I understanding parameter responses and dependencies

I understand the relationship between performance, instancefeatures and SLS algorithm components

I motivate principled decisions in SLS design

Scientific GOAL: provide helpful insights into SLS behavior toinform SLS engineering


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


SLS algorithm engineering: structuring

Benefits to SLS research

I research in SLS very much scattered into di↵erent directionsI SLS engineering o↵ers orientation by defining important areas

such asI methodological developmentsI 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 e↵orts in SLS


HEURISTIC OPTIMIZATIONstuetzle/Teaching/HO/Slides/Lecture12.pdf · Heuristic Optimization 2018 12....

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