Gap between the Lower and Upper Bounds for the Iteration Complexity of Interior-Point Methods

ISBN 978-979-95093-9-0

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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

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

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

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 instit utions 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 ou r 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

Table of Content

Page

Board of Editors

111

Foreword

i

\I

Table of Content

v

A. Sustainability and Science Based Agr iculture

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

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

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 a nd 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 Ra ndom 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

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

55

56

D. Biosciences and Bio r eso urces

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., .. ｾＮ＾＠

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

Gap between the Lower and Upper Bounds for the Iteration

Co mplexity 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 Ｐｐｔｬｾｬｬｚａｔｉｏｎ＠

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

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 Ｒｾ＠ - I iterations. Their example uses Dantzig's classic most-negative-reduce-cost pivot rule.

Then-dimensional Klee-M inty (KM) problem is given by :

minyn

subject to py._₁ ｾＮｖｴ＠ ｾ＠ 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

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 bit-length of the input-data. Then in his ｮ･ｾｴ＠ 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

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 o f 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 path-following methods. Small-update path-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

ｮｾ ｯ＠

l

(27)

where p0

> 0 denotes the initial value of the so-called barrier parameter.

Ill. GAP DEnHEN T ll E LOWER AND UPPER

BOUNDS

Figure I shows the central path of the 2-dimensional Klee-Minty (KM) problem for p

=

l / 3.

,_ .... _ ' --·-.... -....

__

·-... ._.

____

---....

--Ｍｾ＠

---1

-

---

---IU M o.a Q..I t y

1

Figure 1. Central path of the 2-dimensional KM problem, p

=

11 3. [image:13.626.73.514.159.794.2]

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

µ0 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.

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 _{ｏＨｮｾ＠ 2°n)}

ｮＨｾＩ＠

[211

PY1 -1 -Y1 S d _{O(n2'n) .}

n ,_._

ＨｾＩ＠

In :\' [22)

PYi-1-Yt Sd _O(n'21n)

ｮＨｾＩ＠

[23]

- y₁ Sd₁ ｏＨｮＲｾｮｽ＠

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 ｾ＠ 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

ｮＨｾＩ＠

iterations, where r is such that N ｾｲ｡ｮ､＠

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

ｾ＠

2n

ｾ＠

(!!_);;.

[image:14.614.289.522.42.313.2]

From N 2:

:r

we derive. by raking the 2-logarithm at both sides. that log: /\' 2: m. whence

n ｾ＠ 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 p0 _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] REFERENCES

G. 8. Oantzig, linear Programming and Extensions, Princeton N.J. : Princeton University Press, 1963.

J. Oongarra and F. Sullivan, "Guest editors : introduction to the top 10 algorithm,"

Computing in Science and Engineering, vol. 2, no. I, pp. 22-23, 2000.

V. Klee and G. Minty, "How good is the simplex algorithm?," in Inequalities, Ill (Proc. Third Sympos., Univ.California, Los Angeles, Calif, 1969; dedicated to the memory of Theodore S. Motzkin, New York, Academic Press, 1972, pp. 159-175.

D. Luenberger and Y. Ye, linear and Nonlinear Programming. Third ed. , New York: Springer, 2008.

0. Iudin and A. Nemirovskii, "Informational Complexity and Effective Methods of Solution for Convex Extremal Problems," Ekonomika i Matematicheskie Metody (Translated into English in Matekon, 13: 2 5-46, 1977), vol. 12, pp. 357-369, 1976.

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