Table Of ContentLecture Notes ni
lortnoC dna
noitamrofnI Sciences
Edited by .V.A and Balakrishnan .M Thoma
13
Berc Rustem
Projection Methods
ni Constrained Optimisation
and Applications
to Optimal Policy Decisions
galreV-regnirpS
Berlin Heidelberg New kroY 1891
Series Editors
A.V. Balakrishnan • M. Thoma
Advisory Board
L D. Davisson • A. G. .J MacFarlane • H. Kwakernaak
.J L Massey • Ya. Z. Tsypkin - A. .J Viterbi
Author
Dr. Berc Rustem
Control Section
Dept. of Electrical Engineering
Imperial College of Science and Technology
Exhibition Road
London. SW? 2BT - England
ISBN 3-540-10646-4 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-10646-4 Springer-Verlag NewYork Heidelberg Berlin
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© Springer-Verlag Berlin Heidelberg 1891
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TCARTSBA
This work is denrecnoc with projection sdohtem in
constrained optimisation dna the application of projection techniques
to policy optimisation problems. ehT constrained optimisation melborp
of minimising a nonlinear function of n variables subject to equality
dna inequality constraints is also nwonk sa the nonlinear gnimmargorp
problem. Projection sdohtem for constrained minimisation involve
projections of descent directions. ehT basic idea underlying these
sdohtem is the principle of projections which yam eb considered to eb
the generalisation of the fact that in n-dimensional Euclidean ecaps
the shortest vector from a point to a ecapsbus is orthogonal to the
bus p s .eca
In effect, this work consists of owt parts, ehT first part,
sretpahC l - 4, is denrecnoc with the tnempoleved of projection
techniques for different aspects of constrained optimisation.
retpahC l provides a unified hcaorppa to the derivation of projection
techniques dna reviews existing methods. ehT application of projection
techniques to the computation of a feasible point of a linearly
constrained region is discussed in retpahC 2 . In retpahC 3 a
projection dohtem is discussed for the quadratic gnimmargorp melborp
of minimising a quadratic objective function subject to inequality
constraints. This dohtem requires na initial feasible point. In
retpahC 4 a method is discussed for solving the nonlinear
gnimmargorp problem. This dohtem requires solutions to quadratic
minimisation .smelborpbus
V)
ehT dnoces part of the work is denrecnoc with eht
application of projection techniques to lanoitatupmoc smelborp in
policy optimisation. A latnemadnuf melborp in eht optimisation
of policy decisions is eht specification of a suitable objective
function. In retpahC 5 na iterative dohtem is given for specifying
objective functions. In retpahC 6 policy optimisation algorithms,
desab no extensions of projection sdohtem in retpahC l , era .dessucsid
esehT algorithms quadratic minimise objective functions subject to
large nonlinear sledom treated sa equality constraints.
XI
retpahC 3 NA MHTIROGLA ROF EVITISOP ETINIFED
CITARDAUQ GNIMMARGORP
1.3 Introduction 67
3.1.1 ehT deniartsnocnU muminiM 77
3.1.2 ehT Quadratic gnimmargorP
melborP 78
2.3 Projecti no srotarepO 97
97
3.2.1 Preliminaries
80
3.2.2 Projections in n E
3.3 Motivation for a Positive Definite
Quadratic gnimmargorP 1A gori mht 28
3.3.1 ehT Quadratic Objective
noitcnuF 28
3.3.2 Inequality Constraints dna
eht deniartsnocnU mumitpO 09
4.3 ehT Algorithm 6g
3.4.1 A Positive Definite Quadratic
gnimmargorP 1A gori mht 79
3.4.2 ehT isoP ti ev lleD-imeS ni te
dna Indefinite sesaC 001
3.4.3 snoisnetxE to Nonlinear
Constraints 101
STNEMEGDELWONKCA
I would like to thank .rD Heather Liddell for
her valuable guidance dna great patience.
I ma also grateful to Dr. Alan Frieze for helpful discussions
dna valuable stnemmoc dna to Drs. ymawsaramuK Velupillai, naeS Holly
for providing the scimonoce dnuorgkcab of Chapter .5
Finally, I wish to thank Miss Claudia regrebmahcS for patiently
typing the manuscript.
STNETNOC
egaP
snoitnevnoC dna slobmyS XIII
retpahC l YLRAENILNON DENIARTSNOC NOITASIMITPO
SEUQINHCET DESAB SNNOOITCEJORP
l.l Introduction
l.l.l latnemadnuF stpecnoC dna
seR u Its 3
l.l.2 Projection sdohteM for
snoC trai den Mini im s ati no 7
l.l.3 sdohteM desaB no gnitupmoC
sesaB for 61
l.l.4 sdohteM desaB no gnitupmoC
sesaB for %I o 32
2.1 snoisnetxE of Projection smhtiroglA
for solving eht Linear Equality
deniartsnoC melborP 92
1.2.1 snoisnetxE for raenilnoN
snoC trai stn 03
2.2.1 Active teS Strategies 83
3.1 weiveR dna Original Contributions 04
Vlll
retpahC 2 NOITCEJORP SDOHTEM ROF GNITUPMOC
ELBISAEF STNIOP FO YLRAENIL
DENIARTSNOC SNOIGER
1.2 Introduction 34
2.l.l ehT Feasible noigeR 44
2.1.2 ehT Linear ecnednepeD of
Constrai nt slamroN 84
2.1.3 A Projection Algorithm 25
2.2 Projection Algorithms, ycnadnudeR
dna ycarenegeD 16
2.2.l ,ycarenegeD tnadnudeR
Constraints, Infeasibil ity
dna Computational
snoC i derati sno 16
2.2.2 Projection Algorithms for
gnitupmoC Feasible Points
of a Linearly Constrained
noigeR 76
2.3 Concluding skrameR 47
X
5.3 ecnarruceR snoitaleR II0
3.5.1 gnitadpU *N dna detaleR
srotarepO III
3.5.2 gnitadpU l')mNH~N( 021
3.5.3 Linear ecnednepeD 321
6.3 ecnegrevnoC 521
3.7 gnidulcnoC skrameR 031
:xidneppA laciremuN stluseR 131
retpahC 4 NOITASIMITPO HTIW RAENIL DNA
RAENILNON STNIARTSNOC
1.4 Introduction 531
4.1.1 Projection Algorithms
for Nonlinear
gnimmargorP 731
4.1.2 Approximations to the
Inverse Hessian dna the
noitatupmoC of deniartsnoC
tnecseD Directions 441
4.2 Convergence of the Algorithm 651
4.2.1 Stepsize Strategies 651
lX
4.2.2 ecnegrevnoC : Proofs
Exact dnoceS Derivatives 761
4.2.3 ecnegrevnoC : Proofs
ixorppA etam dnoceS
Deri vati sev 081
4.3 Concluding skrameR 812
retpahC 5 EHT EVITARETI NOITACIFICEPS FO
EVITCEJBO SNOITCNUF NI CIMONOCE
YCILOP :NOITASIMITPO
NA NOITACILPPA FO NOITCEJORP
SDOHTEM
1.5 Introduction 912
5.2 Preliminaries 022
5.3 ehT lautpecnoC Algorithm 822
5.4 lanoitatupmoC Considerations
for the Implementation of the
Algorithm 332
5.5 snoisulcnoC 832
:xidneppA A laciremuN elpmaxE 042
A.l ehT Structure of the
Objective Functi no 342
2.A First icepseR ficati no 542
