Internet Engineering
Czesław Smutnicki
Discrete Mathematics – Discrete Optimization
CONTENTS
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Numerical troubles
Packages
Tools
Useful methods
OPTIMIZATION TROUBLES.
NICE BEGINNINGS OF BAD NEWS
FIND EXTREMES OF THE FUNCTION
2D
1D
DE JONG TEST FUNCTION
OPTIMIZATION TROUBLES. MULTIPLE EXTREMES
GRIEWANGK TEST FUNCTION
FIND EXTREMES OF THE FUNCTION
2D
OPTIMIZATION TROUBLES.
EXPONENTAL NUMBER OF EXTREMES
LANGERMANN TEST FUNCTION
FIND EXTREMES OF THE FUNCTION
2D
OPTIMIZATION TROUBLES.
DECEPTION POINTS
FOX HOLES TEST FUNCTION
FIND EXTREMES OF THE FUNCTION
2D
OPTIMIZATION TROUBLES.
TIME OF CALCULATIONS/COST OF CALCULATIONS
CURSE OF
DIMENSIONALITY
NPHARDNESS
LAB INSTANCE
5..20 VARIABLES

NONLINEAR FUNCTION OF 1980 VARIABLES !!!
INSTANCE FROM PRACTICE
Please wait.
Calculations will
last 3 289 years
!! ?
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
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
1,2
frequence [%]
1,0
ALL
FEAS
0,8
0,6
0,4
0,2
DIST [%]
0,0
0
10
0,25
Goal function values are
distributed normally in the
solution space;
RE 
RANDOM  BEST
BEST
20
30
40
50
60
70
frequence [%}
FEAS
0,20
0,15
0,10
0,05
RE [%]
0,00
0
25
50
75
100
125
150
175
200
OPTIMIZATION TROUBLES. FUR
Example: job-shop scheduling problem
SIMULATION OF GOAL FUNCTION VALUES TOWARDS CENTER OF THE SPACE
90
RE [%]
80
70
60
50
40
30
20
10
DIST [%]
0
1
21
41
61
81
101
121
141
161
181
OPTIMIZATION TROUBLES. ZOOM IN ON THE FUR
Example: job-shop scheduling problem
SIMULATION OF GOAL FUNCTION VALUES TOWARDS CENTER (ZOOM)
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RE [%]
16
14
12
10
8
6
4
2
DIST [%]
0
0,01
0,21
0,41
0,61
0,81
1,01
1,21
1,41
1,61
1,81
OPTIMIZATION TROUBLES.
STONE FOREST
Transformation of a
sample of random
solutions from the 90D
space into 2D space.
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
CURRENT STATE IN DISCRETE OPTIMIZATION
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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
OPTIMIZATION HISTORY/TRENDS
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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
APPROXIMATE METHODS
• 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
* * * * *
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
EVOLUTION: DARWIN’S VIEW.
COMPONENTS
GENOTYPE
CHROMOSOM
MORE …
SOLUTION
FEASIBILITY
REPAIRING
GENE EXPRESSION
FENOTYPE
CODING
CONTROL OF
POPULATION DYNAMICS
SELECTION SCHEME
MATTING POOL
LETHALITY
MUTATION
BIG VALLEY PHENOMENON
INTENSIFICATION
CROSSING OVER
OPERATOR MSXF
EVOLUTION: DARWIN’S VIEW.
COPYING FROM THE NATURE
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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, crossingover, mutation)
adaptive mutation
gene expression
distributed populations
…
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
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
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individual, meme, chromosome, trait
population (structure, size, composition, learning)
crossing-over, mutation, learning
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theoretical properties ?
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.
y  a  R(b  c ), R [0,2]
Separate mechanism
generating
ani offspring by simple copying of the parent. Significant role
i prevents
i
i
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.
ARTIFICIAL IMMUNE SYSTEM
fitness
LIBRARY OF ANTIBODIES
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.
ANT SEARCH.
COOPERATIVE SWARMS
control
system
Pheromone
detectors
ANT
• seeks for food
• leaves pheromone on the trail
• moves at random, but prefers pheromone trails
• pheromone density decreases in time
pheromone
generator
moving
drive
ANT SEARCH.
SEEKING FOODS. DISCOVERING THE PATH
E
E
E
D
D
C
H
B
A
E
D
C
H
B
A
C
H
B
A
A
ANT SEARCH.
PHEROMONE DISTRIBUTION
m
 ij    ijk
k 1
 ij (t  n)    ij (t )   ij
PARTICLE SWARM OPTIMIZATION
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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
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).
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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)
TABU SEARCH
STARTING SOLUTION
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NEIGHBOURHOOD
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SUCCESSIVE NEIGHBOURHOODS
EXPLORED EXHAUSTIVELY
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
ADAPTIVE MEMORY SEARCH
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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
INTELLIGENT WATER DROPS
Based on the dynamic of the river systems, action and reaction, that happen among water drops in
rivers:
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a drop has some (static) parameters, namely velocity, soil;
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these parameters may change during the lifetime (e.g. iterative cost)
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drops flow from a source to destination
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a drop starts with some initial velocity and zero soil
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during the flow, drop removes some soil from the environment
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speed of the drop incereases non-linearly inversely to the amount of soil; path with less soil is
faster than path with more soil
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soil is gathered in the drop and removed from the environment
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drop statistically prefers path with lower soil
SIMULATED ANNEALING.
COOLING SCHEMES
annealing = slow cooling of ferromagnetic or antyferromagnetic solid in order to eliminate
internal stretches
Boltzman (harmonic)
Logarithmic (Hajek lemay)
Geometric
1
Tk    
k
Tk 
Tk 1 

ln (k  2)
Tk   (a k )
Tk
T0

1   Tk 1  k  1 T0
k  0,1, ....
SIMULATED ANNEALING.
AUTOTUNING
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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
max  maxxY maxx'N ( x) K ( x' )  K ( x)
T0 
  max
ln p
Tk 1 
Tk
k 
1  k Tk
ln(1   )
3 k
p  0.9
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
T (t  1) podgrzewan ie  T (t ) 
R

T (t  1) studzenia   T (t ) podgrzewan ia
R  [0,  ]   [1, 2, ...N ]
  (0,  ]
DISCRETE OPTIMIZATION. SOLUTION SPACE PROPERTIES
DISTANCE MEASURES IN THE SOLUTION SPACE
Move type
measure
receipt
mean
A
DA (, )
number of inversion
in -1 o 
n( n  1)
4
S
I
DI (, )
DS (, )
n minus the number
of cycles in -1 o 
n  Hn
variance
n( n  1)( 2n  5)
72
Hn  H
complexity
O(n2 )
O(n)
( 2)
n
n minus the lenght of
the maximal increasing
subsequence in -1 o 
n2 n
1
3
(n )
O(n log n)
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
180
RE [%]
4
160
y
2
140
0
120
-2
100
-4
80
-6
60
40
-8
start
BIG VALLEY
-10
best
DIST [%]
-12
20
0
x
-14
0
5
10
15
20
25
30
-18
-14
-10
-6
-2
2
6
10
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
RANDOM SEARCH TRAJECTORY
best
RAN
RE [%]
40
RAN
y
0
35
-2
30
25
-4
20
-6
15
-8
10
-10
5
DIST [%]
0
0
2
4
6
8
10
12
-12
-14
-18
x
-16
-14
-12
-10
-8
-6
-4
-2
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
40
SA
RE [%]
SA
y
0
35
-2
30
-4
25
-6
20
15
-8
10
-10
5
-12
DIST [%]
0
0
2
4
6
8
10
12
x
-14
-18
-16
-14 -12
-10
-8
-6
-4
-2
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
TS
RE [%]
0
35
TS
y
-2
30
-4
25
-6
20
15
-8
10
-10
5
DIST [%]
0
0
2
4
6
8
10
12
-12
-14
-18
x
-16
-14
-12
-10
-8
-6
-4
-2
PARALLEL OPTIMIZATION: NEW CLASS OF ALGORITHMS
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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
PARALLEL OPTIMIZATION: FESTIVAL OF APPROACHES
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SIMULATED ANNEALING:
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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:
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Single thread, conventional GA, parallel calculation of the goal function value; small grain;
theory of convergence
Single thread, parallel evaluation of population;
Multiple independent threads; coarse grain
Multiple cooperative threads, distributed subpopulations: migration, diffusion, island models
…
Thank you for your attention
DISCRETE MATHEMATICS
Czesław Smutnicki