1. Opracowanie językowe dr inż. J. Jarnicki
Internet Engineering
Czesław Smutnicki
Discrete Mathematics – Discrete Optimization

2. CONTENTS
Numerical troubles
Packages
Tools
Useful methods

3. OPTIMIZATION TROUBLES. NICE BEGINNINGS OF BAD NEWS
FIND EXTREMES OF THE FUNCTION 2D 1D
DE JONG TEST FUNCTION

4. OPTIMIZATION TROUBLES. MULTIPLE EXTREMES
GRIEWANGK TEST FUNCTION
FIND EXTREMES OF THE FUNCTION 2D

5. OPTIMIZATION TROUBLES. EXPONENTAL NUMBER OF EXTREMES
LANGERMANN TEST FUNCTION
FIND EXTREMES OF THE FUNCTION 2D

6. OPTIMIZATION TROUBLES. DECEPTION POINTS
FOX HOLES TEST FUNCTION
FIND EXTREMES OF THE FUNCTION

7. OPTIMIZATION TROUBLES. TIME OF CALCULATIONS/COST OF CALCULATIONS
CURSE OF DIMENSIONALITY
Please wait. Calculations will last years
NP-HARDNESS

LAB INSTANCE
5..20 VARIABLES
! ! ?
NONLINEAR FUNCTION OF 1980 VARIABLES !!!
INSTANCE FROM PRACTICE

8. OPTIMIZATION TROUBLES. SIZE OF THE SOLUTION SPACE
The smallest practical instance FT10 of the job-shop scheduling problem (waited 25 years for the solving), consists of 10 jobs, 10 machines, 100 operations; solution space contains 1048 discrete feasible solutions; each solution has dimension 90; the greatest currently used benchmarks have dimension 1980
SOLUTION SPACE
FT 10 corresponds to printed area of 1032 km2 (Jupiter has 1010 km2) if single solution is a dot 0.01 x 0.01 mm
dimension and size

9. OPTIMIZATION TROUBLES. DISTRIBUTION OF THE GOAL FUNCTION VALUES
Example: job-shop scheduling problem; relative Hamming distances DIST between a feasible solution and the „best” solution are distributed normally in the solution space
Goal function values are distributed normally in the solution space;

10. OPTIMIZATION TROUBLES. FUR
Example: job-shop scheduling problem
SIMULATION OF GOAL FUNCTION VALUES TOWARDS CENTER OF THE SPACE

11. OPTIMIZATION TROUBLES. ZOOM IN ON THE FUR
Example: job-shop scheduling problem
SIMULATION OF GOAL FUNCTION VALUES TOWARDS CENTER (ZOOM)

12. OPTIMIZATION TROUBLES. STONE FOREST
Transformation of a sample of random solutions from the 90D space into 2D space.

13. PROPERTIES OF SOLUTION SPACE LANDSCAPE
BIG VALLEY – positive correlation between goal function value and the distance to optimal solution (the best found solution); in the big valley the concentration of local extremes is high. The size of the valley is usually relatively small in relation to the size of the whole solution space.
RUGGEDNESS – measure of diversity of goal function values of related (neighboring) solutions; rruggedness is greater if diversity of the goal function value in the neighborhood of this point is greater; less differentiation of the goal function value means the flat landscape.
THE NUMBER OF LOCAL EXTREMES (peaks) in relation to to the size of the solution space
DISTRIBUTION OF LOCAL EXTREMES experimental
OTHER MEASURES
autocorrelation function, correlation function between random trajectories, landscape statistically isotropic, fractal landscape, correlation between genes (epitasis), correlation of the distance of fitness

14. CURRENT STATE IN DISCRETE OPTIMIZATION
Packages and solvers (LINDO, CPLEX, ILOG, …)
Exact methods (B&B, DP, ILP, BLP, MILP, SUB,…)
Approximate methods (…): heuristics, metaheuristics, meta2heuristics
Quality measures of approximation (absolute, relative, …)
Analysis of quality measure (worst-case, probabilistic, experimental)
Calculation cost (pessimistic, average, experimentally tested)
Approximation schemes (AS, polynomial-time PTAS, fully polynomial-time FPTAS)
Inapproximality
Useful experimental methods (…)
„No free lunch” theorem
Public benchmarks
Parallel and distributed methods: new class of algorithms

15. OPTIMIZATION HISTORY/TRENDS
Priority rules
Theory of NP-completeness
Plynomial-time algorithms
Exact methods (B&B, DP, ILP, BLP,…)
Approximation methods: quality analysis
Approximation schemes (AS, PTAS, FPTAS, …)
Inapproximality theory
Competitive analysis (on-line algorithms)
Metaheuristics
Theoretical foundations of metaheuristics
Parallel metahuristics
Theoretical foundations of parallel metaheuristics

16. APPROXIMATE METHODS METHODS RESISTANT TO LOCAL EXTREMES
constructive/improvement
priority rules
random search
greedy randomized adaptive
simulated annealing
simulated jumping
estimation of distribution
tabu search
adaptive memory search
variable neighborhood search
evolutionary, genetic search
differential evolution
biochemistry methods
immunological methods
ant colony optimization
particle swarm optimization
neural networks
threshold accepting
path search
beam search
scatter search
harmony search
path relinging
adaptive search
constraint satisfaction
descending, hill climbing
multi-agent
memetic search
bee search
intelligent water drops
* * * * *
METHODS RESISTANT
TO LOCAL EXTREMES

17. EVOLUTION: DARWIN’S VIEW. GENETIC ALGORITHMS
GOAL OF THE NATURE? optimization, fitness, continuity preservation,
follow up changes
SUCCESION: genetic material carries data for body construction
EVOLUTION: crossing over, mutation
SELECTION: soft/hard
individual=solution=genotype≠fenotype
individual, gene, chromosome, trait
population (structure, size, composition)
crossing-over (what is the key of progress?)
mutation (insurance?)
sex ?
democracy/elitarism
theoretical properties

18. EVOLUTION: DARWIN’S VIEW. COMPONENTS
GENOTYPE
CHROMOSOM
MORE …
SOLUTION
GENE EXPRESSION
CONTROL OF
POPULATION DYNAMICS
FEASIBILITY
REPAIRING
FENOTYPE
SELECTION SCHEME
CODING
MATTING POOL
LETHALITY
MUTATION
BIG VALLEY PHENOMENON
CROSSING OVER
INTENSIFICATION
OPERATOR MSXF

19. EVOLUTION: DARWIN’S VIEW. COPYING FROM THE NATURE
control of population dynamics/preserving diversity
parents matching strategies: (sharing function to prevent too close relative parents; incest preventing by using Hamming distance to evaluate genotype similarity)
structures of the population (migration, diffusion models)
social behavior patterns (satisfied, glad, disappointed -> clonning, crossing-over, mutation)
adaptive mutation
gene expression
distributed populations …

20. EVOLUTION: DARWIN’S VIEW. MULTISTEP FUSION MSXF
SOURCE SOLUTION (PARENT)
NEIGHBORHOOD OF THE SOURCE
DISTANCE TO TARGET
TRAJECTORY = GOAL ORIENTED PATH
TARGET SOLUTION
(PARENT)
TARGET NEIGHBORHOOD
SUCCESSIVE NEIGHBOURHOODS
SEARCHED IN THE STOCHASTIC WAY
DEPENDING THE DISCTANCE TO TARGET

21. EVOLUTION: LAMARCK/BALDWIN’S VIEW. MEMETIC ALGORITHMS
GOAL OF THE NATURE? optimization, fitness, continuity preservation,
follow up changes, transfer knowledge to successors
SUCCESION: genetic material carries data for body building
plus acquired knowledge
EVOLUTION: crossing over, mutation, learning
SELECTION: soft/hard
individual=solution=memotype≠fenotype
individual, meme, chromosome, trait
population (structure, size, composition, learning)
crossing-over, mutation, learning
theoretical properties ?

22. DIFFERENTIAL EVOLUTION
Differential evolution is a subclass of genetic search methods. Democracy in creating successors with using crossover and mutation in GS has been replaced in DE by directed changes to fathom solution space. DE starts from the random population of individuals (solutions). In each iteration something similar to mutation and crossover is performed, however in completely different way than in GS.
For each solution x from the space, an offspring y is generated as the trial solution being the extension of a selected random solution a and two directional solutions b and c (analogy to parents) selected at random. Generation is based on linear combination with some random parameters.
Separate mechanism prevents generating an offspring by simple copying of the parent. Significant role plays the mutation, which due to specific strategy, is self-adaptive and goal-oriented with respect to the direction, scale and range. If the trial solution is better, it is accepted; otherwise it is released. Iterations are repeated until the fixed a priori number of iterations has been reached, or stagnation has been detected. The method owns some specific tuned parameters: differential weight, crossover probability, … selected experimentally.

23. ARTIFICIAL IMMUNE SYSTEM
LIBRARY OF ANTIBODIES
fitness
recombination
antibody = solution
antigen = problem or instance
Antigen (invasive protein) represents new problem to solve or new (or temporary) constraints set for the solution of already solved problem. Variety of possible antigens is huge, frequently infinite. Moreover, sequence of presented antigens is not known a priori.
Antibody (protein blocking antigent, directed against intruder) corresponds to an algorithm which produces a solution to the problem. Variety of antibodies is usually small, however mechamisms exist of their aggregation and recombination in order to produce new antibodies with various properties. Patterns of antibodies are collected in the library, which constitutes memory of the system.
Matching (fitness) is the selection of antibody for the antigen. Matching is ideal, if the antibody allow us to generate solution of the problem which is globally optimal under given constraints. Otherwise, certain defined measure is used to evaluate quality of the maching. Bad maching forces the system to seek for new types of antibodies, usually by using evolution.

24. ANT SEARCH. COOPERATIVE SWARMS
control system
pheromone generator
Pheromone detectors
moving drive
ANT
seeks for food
leaves pheromone on the trail
moves at random, but prefers pheromone trails
pheromone density decreases in time

25. ANT SEARCH. SEEKING FOODS. DISCOVERING THE PATHB C H E A D D B C H E A D B C H E A

26. ANT SEARCH. PHEROMONE DISTRIBUTION

27. PARTICLE SWARM OPTIMIZATION
swarm is a large set of individuals (particles) moving together
each individual performs the search trajectory in the solution space
trajectories are distributed, correlated and take into account experiences of individuals
location of the individual (solution) is described by the location vector x, changes of location is described by velocity vector v
velocity equation containts an inertiA term and two directional terms weighted by using some random parameters
location of the individual depends on: recent (previous) position, experience (best location up to now), location of the leader of the swarm,
the best up to now solution form the most promising direction of the search

28. BEE SEARCH
waggle dance = distribution of knowledge
bee trajectory = solution
hive
bee
flowers & nectar
nectar amount = goal function
visited site = neighborhood
Neighborhood search combined with random search and supported by cooperation (learning).
bee swarm collects honey in hive
each bee performs the random path (solution) to the search region of nectar
selected elite bees in hive perform „waggle dance” in order to inform other bees about promising search regions (direction, distance, quality)

29. TABU SEARCH STARTING SOLUTION
human thinking in the process of seeking a solution
the method „best in local neighborhood”
repeated from the best recently found
forbidding the return to solutions already visited to prevent cyclic (wandering around); short term memory
NEIGHBOURHOOD
SUCCESSIVE NEIGHBOURHOODS
EXPLORED EXHAUSTIVELY

30. ADAPTIVE MEMORY SEARCH
gathering data in human brain during the process of seeking a solution
the method „best” in the current heighbourhood (a few solution relatively close to the current)
repetition from the best recently found; intensification of the search
operational (short term) memory: prohibition of coming back to solutions already visited to prevent wandering
tactic memory: set direction of the search
strategic memory: selection of search regions (basins of attraction); diversification
recency based, frequency based memory

31. INTELLIGENT WATER DROPS
Based on the dynamic of the river systems, action and reaction, that happen among water drops in rivers:
a drop has some (static) parameters, namely velocity, soil;
these parameters may change during the lifetime (e.g. iterative cost)
drops flow from a source to destination
a drop starts with some initial velocity and zero soil
during the flow, drop removes some soil from the environment
speed of the drop incereases non-linearly inversely to the amount of soil; path with less soil is faster than path with more soil
soil is gathered in the drop and removed from the environment
drop statistically prefers path with lower soil

32. SIMULATED ANNEALING. COOLING SCHEMES
annealing = slow cooling of ferromagnetic or antyferromagnetic solid in order to eliminate internal stretches
Boltzman (harmonic)
Logarithmic (Hajek lemay)
Geometric

33. SIMULATED ANNEALING. AUTOTUNING
Random starting solution
Sequence of k trial moves in the space
K steps in each fixed temperature
Starting temperature adjusted automatically
Adaptive speed of cooling
p  0.9

34. SIMULATED JUMPING
annealing by successive heating and cooling, in order to eliminate internal stretches of the spin-glass solid (mixed ferromagnetic and antyferromagnetic material); the aim is to penetrate high barriers that exist between domains

35. DISCRETE OPTIMIZATION. SOLUTION SPACE PROPERTIES
DISTANCE MEASURES IN THE SOLUTION SPACE
Move type A S I
DA (, )
DS (, )
DI (, )
measure
number of inversion
in -1 o 
n minus the number of cycles in -1 o 
n minus the lenght of the maximal increasing
subsequence in -1 o 
receipt
mean
variance
complexity

36. SELECTED INSTANCES. BIG VALLEY
There exists strong correlation between quality of the function value (RE) and distance to the best solution (DIST); this correlation is preserved after transformation of the solution to x/y coordinates
start
best
BIG VALLEY

37. SELECTED METHODS. RANDOM SEARCH
Random search offers slow convergence to the good solution because it doesn’t use any information about structure of the solution space
start
best
RANDOM SEARCH TRAJECTORY

38. SELECTED METHODS. SIMULATED ANNEALING
Simulated annealing offers moderate speed of convergence to the good solution; it is much more similar to the random search than to goal-oriented search
start
best
SIMULATED ANNEALING TRAJECTORY

39. SELECTED METHODS. TABU SEARCH
Tabu search offers quick convergence to the good solution; this is the fast descent method supported by adaptive memory
start
best
TABU SEARCH TRAJECTORY

40. PARALLEL OPTIMIZATION: NEW CLASS OF ALGORITHMS
Theoretical models of parallel calculation: SISD, SIMD, MISD, MIMD
Theoretical models of memory access: EREW, CREW, CRCW
Parallel calculation environments: hardware, software, GPGPU
Shared memory programming: Pthreads (C), Java threads, Open MP (FORTRAN, C, C++)
Distributed memory programing, message-passing, object-based, Internet computing: PVM, MPI, Sockets, Java RMI, CORBA, Globus, Condor
Measures of quality of parallel algorithms: runtime, speedup, effciency, cost
Single/multiple searching threads; granularity
Independent/cooperative search threads
Distributed (reliable) calculations in the net

41. PARALLEL OPTIMIZATION: FESTIVAL OF APPROACHES
SIMULATED ANNEALING:
Single thread, conventional SA, parallel calculation of the goal function value; fine grain; theory of convergence
Single thread, pSA, parallel moves, subset of random trial solutions selected in the neighborhood, parallel evaluation of trial solutions; theory of convergence
Exploration of equilibrium state at fixed temperature in parallel
Multiple independent threads; coarse grain
Multiple cooperative threads; coarse grain
GENETIC SEARCH:
Single thread, conventional GA, parallel calculation of the goal function value; small grain; theory of convergence
Single thread, parallel evaluation of population;
Multiple cooperative threads, distributed subpopulations: migration, diffusion, island models …

42. Thank you for your attention
DISCRETE MATHEMATICS
Czesław Smutnicki