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TRANSCRIPT
ISBN : 978-979-95093-9-0
PROCEEDINGS
155 2013
International Seminar on Sciences 2013 "Perspectives on Innovative Sciences"
Bogar 15-17 November 2013
IPB International Convention Center
Published by
Faculty of Mathematics and Natural Sciences Bogar Agricultural University
Copyright© 2014 Faculty of Mathematics and Natural Sciences, Bogor Agricultural University Proceedings of International Seminar on Sciences 2013 "Perspectives on Innovative Sciences" Bogor 15-17 November 2013. Published by: FMIPA-IPB, Jalan Meranti Kampus IPB Dramaga, Bogor 16680 Telp/Fax: 0251 -8625481/8625708 http://fmipa.ipb.ac.id ix + 395 pages
ISBN: 978-979-95093-9-0
ii
Boa rd of Editors
PROCEEDINGS INTERNATIONAL SEMINAR ON SCIENCES 2013
Chief Editor: Endar H. Nugrahani
Managing Editor: lndahwati Nisa Rachmania
Managing Team: Wisnu Ananta Kusuma Ali Kusnanto
International Scientific Committee:
Manabu D. Yamanaka (Kobe University, Japan)
Kanaya (Nara Institute of Science and Technology, NAIST, Japan)
Ken Tanaka (Toyama University, Japan) Daniel Oosgood (Columbia University, USA)
Emmanuel Paradis (lnstitut de Recherche pour le Developpement, IRD, France)
Rizaldi Boer (Begor Agricultural University, Indonesia)
Antonius Suwanto (Begor Agricultural University, Indonesia)
National Reviewer:
Kiagus Dahlan
Tania June Sri Sugiarti Miftahudin
Anja Meryandini I mas Sukaesih Sitanggang Farit Mochamad Afendi
Paian Sianturi Husin Alatas Heru Sukoco
Charlena Suryani
iii
FOREWORD
The International Seminar on Sciences 2013, which had the main theme "Perspectives on Innovative
Sciences", was organized on November 151h -1 ih, 2013 by the Faculty of Mathematics and Natural
Sciences, Bogor Agricultural University. This event aimed at sharing knowledge and expertise, as well
as building network and collaborations among scientists from various institutions at national and
international level.
Scientific presentations in this seminar consisted of a keynote speech, some invited speeches, and
about 120 contributions of oral and poster presentations. Among the contributions, 66 full papers
have been submitted and reviewed to be published in this proceeding. These papers were clustered in
four groups according to our themes:
A. Sustainability and Science Based Agriculture
B. Science of Complexity
C. Mathematics, Statistics and Computer Science
D. Biosciences and Bioresources
In this occasion, we would like to express our thanks and gratitude to our distinguished keynote and
invited speakers: Minister of Science and Technology, Prof. Manabu D. Yamanaka (Kobe University,
Japan), Prof. Kanaya (Nara Institute of Science and Technology, NAIST, Japan), Prof. Ken Tanaka
(Toyama University, Japan), Emmanuel Paradis, PhD. (lnstitut de Recherche pour le Developpement,
IRD, France), Prof. Dr. Ir. Rizaldi Boer, MS (Bogor Agricultural University), and Prof. Dr. Ir. Antonius
Suwanto, M .Sc. (Bogar Agricultural University).
We would like also to extend our thanks and appreciation to all participants and referees for the
wonderful cooperation, the great coordination, and the fascinating efforts. Appreciation and special
thanks are addressed to our colleagues and staffs who help in editing process. Finally, we
acknowledge and express our thanks to all friends, colleagues, and staffs of the Faculty of
Mathematics and Natural Sciences IPB for their help and support.
Bogar, March 2014
The Organizing Committee
International Seminar on Sciences 2013
iv
Table of Content
Page
Board of Editors 111
Foreword i \I
Table of Content v
A. Sustainability and Science Based Agriculture
Development of a natural rubber dryer Based on muh1 energy Didin Suwardin, Afrizal 3 resources (biomass. solar and wind) Vachlepi. Mili Pubaya, Sherly
Hanifarianty
2 Characterization ofHDTMABr-modified Na1ural Zeolite and its Budi Riza Putra. Laiifah K 7
Application in Cr(VI) Adsorption Darusman, Eti Rohaeti
3 Potency of Andrographis paniculata, Tinospora crispa, and Wulan Tri Wahyuni, Latifah K 17
Combination Extract as Darusman. Rona Jutama a-Glucosidase Inhibitor and Chromatographic Fingerprint Profile of the Extracts
4 Utilization of Frond Palm Oil as Second Generation Bioethanol Deliana Dahnum, Dyah Styarini, 21
Production using Alkaline Pretreatment and Separated Hydrolysis Sudiyarmanto, Muryanto. and Fermentation Method Haznan Abimanyu
5 Pretreatment of Grass Biomass with Biological Process for Desy Kumiawati, Muhamad 27
Efficient Hydrolysis Natsir, Rahmi Febrialis and Prima Endang Susilowati
6 Alkaloid Compounds from Oil-Free Mahogany Seed (Swietenia Sri Mursiti, Sabirin Matsjeh, 31 macrophylla, King) and Hypoglycemia Effect of Mahogany Seed Jumina, and Mustafa
on The Rat (Rattus novergicus)
7 Utilization Of Vetiver Roots Waste Product as Strong, Low Galuh Suprobo, Tatang 43 Density, and Eco Friendly Material Pot Gunawan, Cynthia Andriani, Rio
Candra lslami
8 Green Products from Wastewater of Tempe Industry Susanti Pudji Hastuti, Yofi 47 Bramantya Adi, Bary Fratarna. Samuel Arunglabi, Dewi KAK Hastuti, and Santoso Sastrodiharjo
9 Saccharification of Oil Palm Empty Fruit Bunch After Alkaline Muryanto, Eka Triwahyuni, 55 Pretreatment Followed by Electron Beam Irradiation for Ethanol Yanni Sudiyani Production
10 Isolation and Screening of Endophytic Bacteria from Bark of Raru Wasinton Simanjuntak, Heri 61 Plant (Tarrietia ribiginosa) and Their Potential for Bioetahnol Satria, and Nurul Utami Production
v
II Tht: Effect of Hypenension I lerbs Fonnula to Thi.' Kidne) Agus Triyono. Saryanto 67 Functions
12 The Use of Activated Carbon from Ointaro Fruit-Shell (Cerbera Armi Wulanawati, Kamella 71 manghas) as an Adsorbent to Increase Water Quality Gustina and Djeni Hendra
13 Analysis of Active Compounds from Mangosteen Rind (Garcinia Fachrurrazie, Hall) Noviardi 77 mangostana L.) by Binding Affinity to The Androgen Receptor as Anti-Prostate Cancer Drug Candidates
14 Antioxidant Activity from Formula of Jati Belanda (Guazuma S) aefudin. Sulistiyani, Edy 81 ulmifolia Lamk.), Jambu Biji (Psidium guajava Linn.). and Salam Djauhari Purwakusumah (Eugenia polyanrha Wight.) Leaves Extract
15 Diversity of Bacterial Mercury Reductase Resistance (mer A) from Prima Endang Susilowati, Sapto 87 Bombana Gold Mine Raharjo, Rachmawati Rusdin.
Muzuni
16 Brake Fem (Pteris vittata) as a Prospective Heavy Metal Mochamad Taufiq Ridwan. Rike 91 Accumulator: Utilization Potentials of Harvested Biomass and Tri Kumala Dewi and Agung Heavy Metal I lasan Lukman
17 Protein Content Enhancement ofSpirulina platensis by Irving Noor Arifin, lin 99 Phosphorus Limitation and Nitrogen Addition in Beef Carrie Supaninah Noer and Asri Peni Wastewater Medium Wulandari
18 Development immobilized enzyme of white-rot fungus for Ajeng Arum Sari and Sanro 103 decolorization of RBBR Tachibana
19 Simple and Rapid Screening Method for Early Identification of Sintho Wahyuning Ardie, Nuru l 109 Salt Tolerant Foxtail Millet (Setaria italica L. Beauv) Khumaida, and Amin Nur
20 Synthesis of Silver Nanoparticles by Using Extracellular Suryani, Ridho Pratama, Dimas 113 Metabolites of Lactobacillus delbrueckii subsp. bulgaricus Andrianto
B. Science of Complexity 119
21 Regional Heat Capacity Changes due to Changes of Land Cover Winda Aryani, !dung Risdiyanto 121 Composition Using Landsat-5 TM Data
22 Microbial Cellulolytic Isolation and Identification from Durian Hapsoh, Gusmawartati dan 129 Leather Waste Ujang Al Husnah
23 Predicting Water Surplus and Water Deficit in the Paddy Rice Johanis H. Panelewen, Johannes 135 Production Center in North Sulawesi Using the Water Balance E. X. Rogi and Wiske Model Rotinsulu3
24 Prediction of Oustfall Generation in Ambient Air over an Arief Sabdo Yuwono, Lia 143 lnceptisol Soil Area Amaliah
vi
25 Carboxymethyla1ton of Microfibrillated Cellulose to Improve Fitri Adilla, Lisman 149 Thermal and Mechanical Properties of Polylactic Acid Composites Suryanegara. Suminar S.
Achmadi
26 Esterification of Microfibrillated Cellulose with Various Ajeng Ma\\ angi. Lisman 155 Anhydrides to Improve Thermal and Mechanical Properties of Suryanegara, Suminar S. Polylactic Acid Composite Achmad1
27 Thermal and Mechanical Properties Improvement of Pol) lactic Resty Dwi Andinie, Lisman 161 Acid-Nanocellulose Composites by Acetylation Suryanegara. Suminar S.
Achmadi
C. Mathematics, Sta tistics and Computer Science 167
28 The comparison spatial distribution observed. cstimatated using Rado Yendra, Ari Pani Desvina, 169 Neyman-Scon Rectangular Pulse Method (NSRP}, and simulation Abdul Aziz Jemain for mean of one-hour rain and probability of 24-hour rain
29 Optimal VAR Injection Based on Neural Network Current State Dimas Fajar Uman P, Ontoseno 175 Estimator for 20kV Surabaya Electrical Distribution System Penangsang, Adi Soeprijanto
30 Fire-Fighting Robot Navigat ion System Using Wall Following Karlisa Priandana. Erwin M Y 181 Algorithm and Fuzzy Logic Chriswantoro, Mushthofa
31 Analysis and Solving of Outliers in Longitudinal Data Viani Eminita, lndahwati. 187 Anang Kumia
32 Implementation of Flowers and Ornamental Plants Landscape Meuthia Rachmaniah and 193 Information System using Cloud Computing Technology lswarawati
33 Cluster Information of Non-sampled Area in Small Area Rahma Anisa, Anang Kumia, 199 Estimation with Non-normally Distributed Area Random Effects lndahwati and Auxiliary Variables
34 Study of Overdispersion for Poisson and Zero-Inflated Poisson Lili Puspita Rahayu, Kusman 203 Regression on Some Characteristics of the Data Sadik, lndahwati
35 The Effect of Two-Way and Three-Way Interaction of Perceived Enny Kristiani, Ujang 209 Rewards on the Relationship Quality Sumarwan, Lilik Noor Yulianti
& Asep Saefuddin
36 Implementation of Inverse Kinematics for the Coordination Wulandari, Karlisa Priandana, 213 Control of Six Legged Robot Agus Buono
37 Detection ofC Code Plagiarism by Using K-Means Ahmad Ridha, Abi Panca 219 Gumilang
38 Temporal Entity Tagging for Indonesian Documents Ahmad Ridha, Agus Simamora 223
39 Multidimensional Poverty Measurement Using Counting lndah Soraya. lrwan Susanto, 229 Approach and Dual Cutoff Method in District of Banyumas Mania Roswitha
vii
40 Minimizing Linear Optimization Model of Basic Reproduc1ion 0 . Chaerani. A. Anisah. N. 235 Number in a Fixed Number of Vaccination Coverage using Anggriani, Firdaniza Interior Poim Method Approach
41 Expert System for Plant Gr0\\1h using Hormones and Exogenous Yaasiin1a Cariens. Karlina Nisa 241 Factors based on Fuzzy Approach
42 The Effect of Divergent Branches on GPU-Based Parallel Program Hendra Rahmawan, Yudi Sa1ria 247 Pcrfom1ance Gondoka1yono
43 Ensemble of Extreme Estimates Based on Modified Aji Hamim Wigena. Anik 253 Champemownc and Generalized Pareto Distributions Djuraidah, Muhammad Hafid
44 Genetic Algorithms Application for Case Study of Septian Rahardiantoro, Oagus 259 Multi-Criteria Decision Analysis (MCDA} Sartono. Totong Martono on the Data Contained Missing Value
45 An Implementation of Parallel AES Algorithm Adit>a Erlangga. Endang 265 for Data Encl') pt ion with GPU Pumama Giri, Karlisa Priandana
46 Constructing Orthogonal Fractional Factorial Split-Plot Designs Bagus Sartono, Yenni Angraini, 269 by Selecting a Subdesign Dependently to Another Subdesign lndahwati
47 Spatial Clustering of Hotspots using DBSCAN and ST-DBSCAN Utsri Yustina Purwanto, Baba 275 Barus.and Hari Agung Adrianto
48 Gap between the Lower and Upper Bounds for the Iteration Bib Paruhum Silalahi 281 Complexity of Interior-Point Methods
49 Black Approximation To Determine Value Of Call Option On Jacob Stevy Seleky, Endar H. 287 Stock In Indonesian Stock Exchange Nugrahani. I Gusti Putu Purnaba
50 Analysis of Portfolio Optimization With and Without Shortselling Kalecm Saleem, Abdul Kohar 291 Basd on Diagonal Model : Evidence from Indonesian Stock Market lrwanto, Endar Hasafah
Nugrahani
51 Community Network Framework as a Support of Successful Rina Trisminingsih, Christine 299 Agricultural Community Suryadi. Husni S. Sastramihardja
52 THE TRANSMISSION MODEL OF DENGUE FEVER Paian Sianturi, Ali Kusnanto, 305 DISEASE: A COMPUTER SIMULA TJON MODEL Fahren Bukhari
53 Improving the Independence of the Components ofa Hari Wijayanto, Bagus Sartono, 311
Decomposition in Time Series Data Casia Nursyifa
54 Modeling and Empirical Mapping of Vehicular Traffic System: Endar H. Nugrahani, Hadi 322 Case Study of Jabodetabek Region Sumamo, Ali Kusnanto
viii
55
56
D. Biosciences and Bioresources
A QuEChERS Based Method for The Detennination of Pesticide Harmoko. Rahmana Emran Residues in Indonesian Green Coffee Beans Using Liquid Chromatography Tandem Mass Spectrometry
Design and Implementation of Roaster Control System Using Image Processing
Kartasasmita. and Astika Tresn;m ati
Mohamad Agung Prawira Negara, Satryo Budi Utomo. Sumardi
3., .. ~.>
325
333
57 Genetic Varia11on of DGA TI Eael Gene of Holstein Friesian m Santiananda A Asmarasari 339 National Dail)' Cattle Stations
58 The Potency of Dahlia Tubers as Prebiotic for Functional Food Ainia Herminiati, Sri Pudjiraharti, Budi Setiawan
59 DNA identification using Markov Chain as feature extraction and Toto Haryanto, Habib Rijzaani, Probabilistic Neural Network as classifier Muhammad Luthfi Fajar
60 Multiple Sequence Alignment with Star Method in Graphical Processing Unit using CU DA
61 Abalone (Haliotis asinina) Wound Detection System Using Histogram and Morphology
62 Local Alignment of DNA Sequence Using Smith-Waterman Algorithm
Muhammad Adi Puspo Sujiwo, Wisnu Ananta Kusuma
Noer Fitria Putra Setyono, Aziz Kustiyo. Dwi Eny Djoko Setyono
Fariz Ashar Himawan,Wisnu Anania Kusuma
345
35 1
359
365
371
63 Agronomic perfonnance and yield potential of 18 com varieties in Anggi Nindita, Willy Bayuardi 377 Indonesia Suwarno, Surjono Hadi Sutjahjo.
Perdinan
64 Characteristic and Phisychochemical Properties of Sweet Potatoes Ai Mahmudatussa'adah 381 (Ipomoea batatas L)
65 Detennination of Harvesting Time of Three Peanut Varieties Heni Pumamawati, Yoga 387 Based on Heat Unit Accumulation Setiawan Santoso, Yudiwanti
Wahyu
66 Respon of Celery (Apium graveolens) Leaves Yield to Plant Karo, B, Marpaung, A. E., 391 Population and Seed Number Per Planting Hole Tarigan, R., Barus, S. and
Khaririyatun, N .
ix
Gap between the Lower and Upper Bounds for the Iteration Complexity of Interior-Point Methods
Bib Parulwm Silalahi'J
11 Lecturer a1 Depanment of Machemacics. Facult\ of Mathematics and Natural Sciences, Bogor Agricultural Universicy. Bogor, Indonesia (bibparuhum l@l)ahoo.com)
Abstract Recenrly, the uu of interior-point methods to solve linear optimi:arion problems, have been becoming great a/fenrion ro the researchers. The most important thing is that the interior point methods have the best comple..tlf}' compared to other methods and also efficient in practice Gon:aga. Afomeiro and Adler presented small-update path-following methods. a varionr of interior-point methods. which is the best known upper bound for the iteration complexity of an interior-point method. Roos. Terlaky am/ Vial presented an interior-point method using primal-dual full-Newton step algorithm and state the upper bound fur the iteration complexity of an interior-point method in difference expression. De:a. Nematollahi. Peyghami and Terlaky showed several worst cases of the interior-point method by using Klee-Minty problem. Us mg their worst cases. we present a lemma. where from this lemma we may obtain the lower bound for the iteration complex11y of an !PM.
KeywQrd:i: interior-point method, upper bound. lower bound
I. INTRODtrCTION Optimization is the branch of applied
mathematics which studies problems where one seeks to minimize (or maximize) a real function of real variables, subject to constraints on the variables. The solution set of the constraints defines 1he feasible region (or 1he domain) of an optimization problem.
II. BRIEF HISTORY Of Ll:'llEAR 0PTl~llZATION
Linear Optimization (LO) is concerned with the minimization or maximization of a linear function, subject to constraints described by linear equations and/or linear inequalities. A. Simplex Methods
LO emerged as a mathematical model after World War II, when Dantzig in 1947 proposed his simplex method for solving "linear programming" (then known as optimization) problems [I).
The feasible region of an LO problem is a polyhedron, the solution set to a system of linear constraints. Simplex methods move along vertices of the polyhedron in order to find an optimal vertex. These methods are designed in such a way that during this process the value of the objective function changes monotonically to its optimal value.
After its discovery, the Dantzig simplex method has inspired much research in mathematics. Simplex methods were placed as the top I 0 algorithms in the twentieth century by the journal Computing in Science and Engineering [2]. There are many variants of a simplex method, distinguished by rules for selecting the next venex (so-called pivot rules). The success of simplex methods have raised some questions such as: whether there exists a pivot rule that requires a
Proceedings ISS 2013
polynomial number of iterations, and whether there are linear optimization problems that require an exponential number of iterations.
The last question was answered by Klee and Minty (3 ] in 1972. They gave an example of an LO problem with 2n inequalities for which the simplex meihod may need as much as 2~ - I iterations. Their example uses Dantzig's classic most-negativereduce-cost pivot rule.
Then-dimensional Klee-M inty (KM) problem is given by : minyn
subject to py._1 ~.Vt ~ 1- py._1, k = l, ... ,n,
(26) where p is small positive number by which the unit
cube (0, I]" is squashed, and y0 = 0. The domain is
a perturbation of the unit cube in R•. If p = 0 then
the domain is the unit cube and for p e (0, 1/2) it is a perturbation of the unit cube which is contained in the unit cube itself, as can easily be verified. Since the perturbation is small, the domain has the same number ofvenices as the unit cube, i.e. 2". Klee and Minty showed that in their example the simplex method with the Dantzig rule walks along all these venices. Thus it became clear that the computational time needed by the Dantzig simplex method may grow exponentially fast in terms of the number of inequalities. Since then exponential examples have been found for almost every pivot rule. B. The ellipsoid Method
The shortcomings of simplex methods (at least theoretically) stimulated researchers to look for other methods with a running time that grows
201
polynomially fast if the number of inequalities grows. The first polynomial-time algorithm for LO problems is the ellipsoid method. The basic ideas of this method evolved from research done in the 1960s and 1970s in the Soviet Union (as it then was called). The idea of the ellipsoid method is to enclose the region of interest by an ellipsoid and to decrease the volume of the ellipsoid in each iteration (4]. This method was first published in a paper of ludin and Nemirovskii [5]. Independently in 1977, Shor [ 6) also presented the ellipsoid method. Khachiyan modified this method and in 1979 (7) he introduced this method as the first polynomial-time algorithm for LO problems.
He proved that the ellipsoid method solves an LO problem in O(n! l) iterations with the total
number of arithmetic operations O(ns l), where n is
the number of inequalities and L is the total bitlength of the input-data. Then in his ne~t paper (8) he gave a better bound, 0(11. l) . for the total number of arithmetic operations.
Following Khachiyan's work, the ellipsoid method was studied intensively for its theoretical and practical aspects, with the hope that LO problems could be solved faster than by simplex methods. The results were not as expected. In practice, the rate of convergence of the ellipsoid method is rather slow. when compared to simplex methods. The worst-case iteration bound for simplex methods, in any of its several implementations, is an extremely poor indicator of the method's actual perfonnance. On the other hand, the worst-case bound for the ellipsoid method appears to be a good indicator for the practical behavior of the ellipsoid method (9). which makes the method become too slow for practical purposes. C. Interior-point Methods
A really effective breakthrough occurred in 1984, when Kannarkar (I OJ proposed a different polynomial-time method (known as Karmarkar's projective method) for LO problems. Contrary to simplex methods, whose iterates are always on the boundary of the domain, Kannarkar's method passes through the interior of the domain to find an optimal solution.
Jn the worst-case, for a problem with n inequalities and L bits of input data, his method requires O(nl) iterations. Jn each iteration,
Kannarkar's algorithm requires O(nB) arithmetic operations and each arithmetic operation needs a precision of O(l) bits. Jn total, in the worst-case,
Kannarkar's algorithm requires O(nH l) arithmetic
operations on numbers with O(l) bits. The theoretical running time of this algorithm is better than that of the ellipsoid algorithm by a factor of
0( J; ). More exciting, Kannarkar claimed that the algorithm is not only efficient in theory, but also in practice.
282
Karmarkar's paper initiated a revolution in the , field of optimization. It gave rise to so-called interior-point methods (lPMs), fi rst for LO but later also for the more general class of convex problems.
Renegar. in 1988 ( 11 ). improved the number of
Iterations to 0( f;;L ) iterations. Other variants of lPMs, called potential reduction methods, require
also only 0( f;;L) iterations. This was shown by Ye
[12). Freund (13), Todd and Ye (14] and Kojima, Mizuno and Yoshise [ 15]. The main idea of these methods is the usage of a potential function for measuring the progress of the method. Kannarkar's projective method also uses a potential function.
A wide class of IPMs uses the so-called central path, which was introduced by Sonnevend [ 16) and Meggido [ 17], as a guide line to the set of optimal solutions: these methods arc therefore called pathfollowing methods. Small-update path-following methods, a variant of path-folio\\ ing methods, were presented Monteiro and Adler [ 18] and Roos and
Vial ( 19). Their methods require 0( f;;L) iterations. which is the best known upper bound for the iteration complexity of an IPM. Roos, Terlaky and Vial in their book (20] obtained the same upper bound by using an algorithm which is a so-called primal-dual full-Newton step algorithm. Expressing the absolute accuracy of the objective function by &
their upper bound for the number of iterations is
r Ji; In n~o l (27)
where p 0 > 0 denotes the initial value of the socalled barrier parameter.
Ill. GAP DEnHEN TllE LOWER AND UPPER
BOUNDS Figure I shows the central path of the 2-dimensional Klee-Minty (KM) problem for p = l / 3.
,_ .... _ ' --·- .... -.... __ ·-... ._. ____ ---.... --
-~ ---1 ------------------------
IU M o.a Q..I t y1
Figure 1. Central path of the 2-dimensional KM problem, p = 113.
The central path is an analytic curve that moves through the interior of the domain to the optimal sel.
Proceedings JSS 2013
Ideally it is a nice smooth curve that goes more or less straightforward to the optimal set. In that case path-following methods are extremely efficient. In practice the curve may possess some sharp turns. We assume that each such tum will require at least one iteration of a path-following IPM. As a consequence we may consider the number of sharp turns in the central path as a lower bound for the number of iteration of any path-following IPM.
Recemly, Deza, Nematollahi. Peyghami and Terlaky [21) showed that when adding abundanily many suitable chosen redundant constraints to the KM cube, the squashed cube which is formed by. the inequalities of KM problem, then one may force the central path to visit small neighborhoods of all the vertices of the KM cube.
Deza et al. concluded that an IPM needs at least 2n - I iterations to solve their problem. Hence the number of iterations may be exponential in the dimension n of the cube. This does not contradict the polynomial-time iteration bound (2) however. Note that the number of inequalities of the /1 -
dimensional KM problem is 2n. If N denotes the number of inequalities in the problem that causes the central path to visit small neighborhoods of all the vertices of the KM cube, then the upper bound
should be o( JN ln(N 1-l I e) ). Assuming that
µ 0 I c = 0(1), one may consider this bound as
o( JN In N). The bound imp I ies that the number
N of inequalities must be exponential in n as well,
because we should have 2n -I= o( JN In N). In [21 ), Deza et al. also argued that such a
redundant KM problem, whose central path visits small neighborhoods of all the vertices of the KM cube, gives rise to a lower bound for the maximal number of iterations in terms of N. Several papers appeared since then, each new paper using less redundant constraints and, as a consequence, yielding a higher lower bound for the iteration complexity of an IPM. The results of these papers are summarized in Table I.
Proceedings ISS 2013
Table I. Results from the literature.
Type of Number Lower Refe redundant of bound for . constraints redundant iteration renc
inequalitie complexity e s
PY1-1-Y1 Sd O(n~ 2°n) n(~) [211
PY1-1 - Y1 S d O(n2'n) . n ,_._ (~) In :\' [22)
PYi-1-Yt Sd O(n'21n) n(~) [23]
- y 1 Sd1 O(n2~n} n(Ji;) In,\' (24)
Column 4 gives the related references. Column I shows the type of constraints used in the corresponding paper, column 2 the order of the number N of inequalities used, and column 3 the resulting lower bound for the number of iterations of
IPMs. In each case one has N ~ 2n. The next lemma explains how the lower
bounds in column 3 can be deduced from the figures in column 2 of Table I.
Lemma I. If the number N of inequalities
describing a redundant KM problem is O(n1'2"")
and the central path enters a small neighborhood of each vertex, then any /PM requires at least
n(~) iterations, where r is such that N ~rand p,q,r > 0.
Proof: It is well known that if an iterate x is on (or close to) the central path, then the search direction at x in any interior-point method is (about) tangent to the central path. The KM path consists of 2" - I line segments. Since for each sharp tum in the central path an IPM requires at least one iteration, when solving the redundant KM problem at least 2n - I iterations are needed. The number N of inequalities being O(n"2""), we have N5.cn,.2"' for some
c > 0. This implies !!..._ < 29" for some c > 0. cnP -
Thus we obtain I
# iterations +I ~ 2n ~ (!!_);;. cnP
283
From N 2: :r we derive. by raking the 2-logarithm at both sides. that log: /\' 2: m. whence
n ~ log: N I r . Substituting this we get I
# iterations +I 2: . ( NrP J;
c(log? N)"
This implies the statement in the lemma. o
As staced before, the best known upper bound for the iteration complexity of an IPM is the bound in (27). For fixed values of p 0 and & we may \Hite
N o o JN In__!:!__ = JN In N +JN In!:!_
e e
= 0( .JN In N).
Comparing this with the highest lower bound in
Table I, which is n( -/N / In N ). we conclude that
there is still a gap between the lower and upper bounds for the iteration complexity of IPMs: the
bounds differ by a factor In 2 N.
I J
2]
3)
4]
5)
6]
284
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