Data correcting algorithms in combinatorial optimization:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Groningen
SOM Research Inst., Groningen Univ.
2002
|
| Schriftenreihe: | Theses on systems, organisation and management
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Zugl.: Groningen, Univ., Diss., 2002 |
| Beschreibung: | XVIII, 208 S. graph. Darst. |
Internformat
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| 100 | 1 | |a Goldengorin, Boris |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Data correcting algorithms in combinatorial optimization |c Boris Goldengorin |
| 264 | 1 | |a Groningen |b SOM Research Inst., Groningen Univ. |c 2002 | |
| 300 | |a XVIII, 208 S. |b graph. Darst. | ||
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| 490 | 0 | |a Theses on systems, organisation and management | |
| 500 | |a Zugl.: Groningen, Univ., Diss., 2002 | ||
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| 650 | 7 | |a Combinatieleer |2 gtt | |
| 650 | 7 | |a Mathematische programmering |2 gtt | |
| 650 | 4 | |a Algorithms | |
| 650 | 4 | |a Combinatorial analysis | |
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Datensatz im Suchindex
| _version_ | 1804130118618054656 |
|---|---|
| adam_text | Contents
1 Introduction 1
1.1 Data Correcting (DC) for Real Valued Functions . . 3
1.2 DC for NP Hard Combinatorial Optimization Prob¬
lems 8
1.3 The DC Approach in Action 12
1.4 Preliminary Computational Experience with ATSP
Instances 19
1.5 Summary 22
1.6 Overview of the Book 23
2 The Binary Knapsack Problem: Solutions with
Guaranteed Quality 27
2.1 Introduction 28
2.2 The a mtl Algorithm 30
2.3 Computational Experiments 33
2.4 Summary 38
3 Preliminaries for the Maximization of Submodular
Functions 49
3.1 Introduction 50
3.2 The Structure of Local and Global Maxima of Sub
modular Set Functions 52
3.3 Excluding Rules: an Old Proof 65
3.4 Preservation Rules: Generalization and a Simple
Justification 68
3.5 The Preliminary Preservation Algorithm (PPA) . . 75
3.6 Non Binary Branching Rules 78
3.7 Summary 82
vii
4 Data Correcting Approach for the Maximization of
Submodular Functions 85
4.1 The Main Idea of the Data Correcting (DC) algo¬
rithm; an extension of the PPA 86
4.2 The DC Algorithm 91
4.3 The Simple Plant Location Problem; an Illustration
of the DC Algorithm 94
4.4 Computational Experiments with the Quadratic
Cost Partition Problem (QCP) and Quadratic Zero
One Optimization Problem: a Brief Review .... 95
4.5 The QCP: Computational Experiments 99
4.6 Remarks for the DC Algorithm 104
4.7 A Generalization of the DC Algorithm: a Multilevel
Search in the Hasse Diagram 107
4.7.1 PPA of order r (PPAr) 107
4.7.2 The Data Correcting Algorithm Based on the
PPAr 114
4.7.3 Computational Experiments for the QCP
with the DCA(PPAr) 117
4.8 Summary 127
5 Pseudo Boolean Approach for the Simple Plant Lo¬
cation Problem 129
5.1 Introduction 130
5.2 A Pseudo Boolean Approach to SPLP 133
5.3 Equivalent Instances of the SPLP 137
5.4 Solving the SPLP 140
5.4.1 Solving SPLP Instances using Polynomially
Solvable Cases 141
5.4.2 Cherenin s Preprocessing Rules 146
5.5 Branch and Peg Algorithms 152
5.6 Summary 161
6 Data Correcting Approach for the Simple Plant Lo¬
cation Problem 165
6.1 Ingredients of Data Correcting for the SPLP .... 166
6.1.1 The Reduction Procedure 169
6.1.2 The Data Correcting Procedure 172
viii
6.2 Computational Experiments 173
6.2.1 Bilde and Krarup Type Instances 177
6.2.2 Galvao and Raggi Type Instances 178
6.2.3 Instances from the OR Library 180
6.2.4 Korkel Type Instances with 65 sites 180
6.2.5 Korkel Type Instances with 100 Sites .... 183
6.3 Summary 186
ix
List of Tables
1.1 8 city ATSP instance 15
1.2 ATSP instance after subtracting the row and column
minima 15
1.3 The corrected matrix at the root node 16
1.4 The DC solutions for the 8 city ATSP 18
2.1 Performance of a MTl on UC knapsack instances
(Data range [1,1000]) 40
2.2 Performance of a MTl on UC knapsack instances
(Data range [1001,2000]) 40
2.3 Performance of a MTl on WC knapsack instances
(Data range [1,1000]) 41
2.4 Performance of a MTl on WC knapsack instances
(Data range [1001,2000]) 41
2.5 Performance of a MTl on SC knapsack instances
(Data range [1,1000]) 42
2.6 Performance of a MTl on SC knapsack instances
(Data range [1001,2000]) 43
2.7 Performance of a MTl on ISC knapsack instances
(Data range [1,1000]) 44
2.8 Performance of a MTl on ISC knapsack instances
(Data range [1001,2000]) 45
2.9 Performance of a MTl on ASC knapsack instances
(Data range [1,1000]) 46
2.10 Performance of a MTl on ASC knapsack instances
(Data range [1001,2000]) 47
3.1 The data of the SPLP 79
4.1 The comparison of computational results 100
xi
4.2 The distribution of the diagonal dominances 103
4.3 Threshold QCP instances solved by the DC algo¬
rithm within 10 min 106
4.4 Average calculation times (in seconds) against pre¬
scribed accuracies of 0% and 5% for instances of the
QCP with 100 500 vertices and densities 10% 100%
within 10 min 121
5.1 Number of instances in each set solved within 600
CPU seconds 156
5.2 The average number of subproblems generated by
the algorithms 156
5.3 The average execution times required by the algo¬
rithms 158
5.4 Computational experience with the instances in the
OR Library. 159
5.5 Number of subproblems generated in the MBnP al¬
gorithm 163
6.1 Number of free locations after preprocessing SPLP
instances in the OR Library 175
6.2 Number of non zero nonlinear terms in the Hammer
function after preprocessing SPLP instances in the
OR Library. 176
6.3 Comparison of bounds used with the DCA on
Korkel type instances with m = n = 65 176
6.4 Description of the instances in Bilde and Krarup
(1977) 177
6.5 Results from Bilde and Krarup type instances. . . . 178
6.6 Description of the instances in Galvao and Raggi
(1989) 179
6.7 Results from Galvao and Raggi type instances. . . . 180
6.8 Results from OR Library instances 181
6.9 Description of the fixed costs for instances in
Korkel (1989) 182
6.10 Costs of solutions output by the DCA on Korkel type
instances with 65 sites 182
6.11 Execution times for the DCA on Korkel type in¬
stances with 65 sites 183
xii
6.12 Costs of solutions output by the DCA on Korkel type
instances with 100 sites 184
6.13 Execution times for the DCA on Korkel type in¬
stances with 100 sites 185
xiii
List of Figures
1.1 A DC algorithm for a real valued function 5
1.2 A general function / 6
1.3 Illustrating the DC approach on / 7
1.4 A patching operation 11
1.5 A Data correcting algorithm for combinatorial opti¬
mization problems with a minimization objective. . 14
1.6 The DC Algorithm tree with non disjoint subproblems. 17
1.7 The DC Algorithm tree with disjoint subproblems. 18
1.8 Accuracy achieved versus a for ftv instances. ... 20
1.9 Variation of execution times versus a for ftv instances. 20
1.10 Accuracy achieved versus a for rbg instances. ... 21
1.11 Variation of execution times versus a for rbg instances. 21
2.1 Pseudocode of q MTI 33
3.1 The Hasse diagram of {1,2,3,4} 58
3.2 Example of local maxima {1,2}, {1,2,3}, {1,3},
{2,3}, {3}, and the global maximum {4} on the
Hasse diagram 59
3.3 Example of the chain 0 C {2} C {2,4} C {1,2,4} C
{1,2,3,4} in the Hasse diagram of {1,2,3,4}. ... 60
3.4 Example of a nondecreasing (nonincreasing) function
on the chain in the Hasse diagram of {1,2,3,4}. . . 60
3.5 A quasiconcave behaviour of a submodular function
on the chain with a local maximum L (Cherenin s
theorem) 61
3.6 Lower local maxima: {1,2}, {3}; upper local maxi¬
mum: {1,2,3}; SDC (shadowed); global maximum:
{4} 62
xv
3.7 The behaviour of a submodular function on a chain
with lower and upper local maxima (Khachaturov s
theorem) 63
3.8 Example of Prime Excluding Rules 68
3.9 A representation of the upper partition of the inter¬
val [S, T] = [0, {1, 2,3,4}] with Q S = {1, 2,3}. . . 70
3.10 A representation of the lower partition by Q = {1,2}
for the interval [S, T] = [0, {1, 2, 3,4}] with T Q =
{3,4} 71
3.11 The non overlapping representation of the lower
partition by parallel intervals [{3}, {1,2,3}] and
[{4}, {1,2,3,4}] 72
3.12 The nonoverlapping representation of the upper
partition by the parallel intervals [{1, 2}, {1, 2,4}],
[{l},{l,3,4}],and[0,{2,3,4}] 73
3.13 The Dichotomy (Preliminary Preservation) Algorithm 76
3.14 The idea of the Dichotomy algorithm: z({l,3}) = 4
is the global maximum for all submodular functions
from the subclass of V 77
3.15 The SPLP example: illustration of non binary
branching rule 81
4.1 Procedure DC 91
4.2 The recursive solution tree for £0 = 0 94
4.3 The recursive solution tree for £0 = 2 95
4.4 Average calculation time in sees against the density
d (case: m = 80, e0 = 0) 101
4.5 Average calculation time in sees against the number
of vertices m (case: d = 0.3, £q = 0) 101
4.6 Average time in sees against prescribed accuracy Eq
(case: m = 80,d = 0.2) 102
4.7 7max as percentage of the value of a global minimum. 102
4.8 Average time in sees against diagonal dominance
(cases: m = 40, d = 0.3,0.4,..., 1.0) 103
4.9 dist( I ,n/2) for instances of the QCP with 50 100
vertices and densities 10% 100% 122
xvi
4.10 Natural logarithm of the average calculation time (in
seconds) for instances of the QCP with 50 100 ver¬
tices and densities 10% 100% 123
4.11 Average calculation times (in seconds) for values of
r for instances with size 100 and densities 70 100%. 124
4.12 The number of generated subproblems against the
level r for instances of the QCP with 100 vertices
and densities 70% 100% 125
4.13 The average calculation time (in seconds) against the
level r for instances of the QCP with 200 vertices
and density 100% 126
5.1 Handling polynomially solvable special cases 142
5.2 Pseudocode for RECOGNIZE 144
5.3 Pseudocodes for BnB and BnP algorithms 157
5.4 Pseudocode for the Khachaturov Minoux bound. . . 158
5.5 Performance of BnP algorithms using BnB algorithm
as a basis 160
6.1 Performance of the DCA for Korkel type instances
with 65 sites 184
6.2 Performance of the DCA for Korkel type instances
with 100 sites 185
xvii
|
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| spelling | Goldengorin, Boris Verfasser aut Data correcting algorithms in combinatorial optimization Boris Goldengorin Groningen SOM Research Inst., Groningen Univ. 2002 XVIII, 208 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Theses on systems, organisation and management Zugl.: Groningen, Univ., Diss., 2002 Algoritmen gtt Combinatieleer gtt Mathematische programmering gtt Algorithms Combinatorial analysis Mathematical optimization Kombinatorische Optimierung (DE-588)4031826-6 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Kombinatorische Optimierung (DE-588)4031826-6 s DE-188 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010409838&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Goldengorin, Boris Data correcting algorithms in combinatorial optimization Algoritmen gtt Combinatieleer gtt Mathematische programmering gtt Algorithms Combinatorial analysis Mathematical optimization Kombinatorische Optimierung (DE-588)4031826-6 gnd |
| subject_GND | (DE-588)4031826-6 (DE-588)4113937-9 |
| title | Data correcting algorithms in combinatorial optimization |
| title_auth | Data correcting algorithms in combinatorial optimization |
| title_exact_search | Data correcting algorithms in combinatorial optimization |
| title_full | Data correcting algorithms in combinatorial optimization Boris Goldengorin |
| title_fullStr | Data correcting algorithms in combinatorial optimization Boris Goldengorin |
| title_full_unstemmed | Data correcting algorithms in combinatorial optimization Boris Goldengorin |
| title_short | Data correcting algorithms in combinatorial optimization |
| title_sort | data correcting algorithms in combinatorial optimization |
| topic | Algoritmen gtt Combinatieleer gtt Mathematische programmering gtt Algorithms Combinatorial analysis Mathematical optimization Kombinatorische Optimierung (DE-588)4031826-6 gnd |
| topic_facet | Algoritmen Combinatieleer Mathematische programmering Algorithms Combinatorial analysis Mathematical optimization Kombinatorische Optimierung Hochschulschrift |
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