Duality for convex composed programming problems [Elektronische Ressource] / vorgelegt von Emese Tünde Vargyas
102 Pages
English

Duality for convex composed programming problems [Elektronische Ressource] / vorgelegt von Emese Tünde Vargyas

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Duality for convex composedprogramming problemsVon derFakult at fur Mathematik der Technischen Universit at ChemnitzgenehmigteD i s s e r t a t i o nzur Erlangung des akademischen GradesDoctor rerum naturalium(Dr. rer. nat.)vorgelegtvonDipl.-Math. Emese Tunde Vargyasgeboren am 21.02.1975 in Reghin (Rum anien)eingereicht am: 05.07.2004Gutachter: Prof. Dr. Gert WankaProf. Dr. Kathrin KlamrothConf. Dr. G abor KassayTag der Verteidigung: 25.11.2004Bibliographical descriptionEmese Tunde VargyasDuality for convex composed programming problemsDissertation, 102 pages, Chemnitz University of Technology, Faculty of Math-ematics, 2004ReportThe theory of duality represents an important research area in optimization.The goal of this work is to present a conjugate duality treatment of composedprogramming as well as to give an overview of some recent developments in bothscalar and multiobjective optimization.In order to do this, rst we study a single-objective optimization problem, inwhich the objective function as well as the constraints are given by composedfunctions. By means of the conjugacy approach based on the perturbation the-ory, we provide di eren t kinds of dual problems to it and examine the relationsbetween the optimal objective values of the duals. Given some additional as-sumptions, we verify the equality between the optimal objective values of theduals and strong duality between the primal and the dual problems, respectively.

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Published 01 January 2004
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Duality for convex composed
programming problems
Von der
Fakult at fur Mathematik der Technischen Universit at Chemnitz
genehmigte
D i s s e r t a t i o n
zur Erlangung des akademischen Grades
Doctor rerum naturalium
(Dr. rer. nat.)
vorgelegt
von
Dipl.-Math. Emese Tunde Vargyas
geboren am 21.02.1975 in Reghin (Rum anien)
eingereicht am: 05.07.2004
Gutachter: Prof. Dr. Gert Wanka
Prof. Dr. Kathrin Klamroth
Conf. Dr. G abor Kassay
Tag der Verteidigung: 25.11.2004Bibliographical description
Emese Tunde Vargyas
Duality for convex composed programming problems
Dissertation, 102 pages, Chemnitz University of Technology, Faculty of Math-
ematics, 2004
Report
The theory of duality represents an important research area in optimization.
The goal of this work is to present a conjugate duality treatment of composed
programming as well as to give an overview of some recent developments in both
scalar and multiobjective optimization.
In order to do this, rst we study a single-objective optimization problem, in
which the objective function as well as the constraints are given by composed
functions. By means of the conjugacy approach based on the perturbation the-
ory, we provide di eren t kinds of dual problems to it and examine the relations
between the optimal objective values of the duals. Given some additional as-
sumptions, we verify the equality between the optimal objective values of the
duals and strong duality between the primal and the dual problems, respectively.
Having proved the strong duality, we derive the optimality conditions for each of
these duals. As special cases of the original problem, we study the duality for the
classical optimization problem with inequality constraints and the optimization
problem without constraints.
The second part of this work is devoted to location analysis. Considering rst
the location model with monotonic gauges, it turns out that the same conjugate
duality principle can be used also for solving this kind of problems. Taking in the
objective function instead of the monotonic gauges several norms, investigations
concerning duality for di eren t location problems are made.
We nish our investigations with the study of composed multiobjective opti-
mization problems. In doing like this, rst we scalarize this problem and study the
scalarized one by using the conjugacy approach developed before. The optimal-
ity conditions which we obtain in this case allow us to construct a multiobjective
dual problem to the primal one. Additionally the weak and strong duality are
proved. In conclusion, some special cases of the composed multiobjective opti-
mization problem are considered. Once the general problem has been treated,
particularizing the results, we construct a multiobjective dual for each of them
and verify the weak and strong dualities.Keywords
composed functions, convex programming, perturbation theory, conjugate du-
ality, optimality conditions, duality in multiobjective optimization, Pareto e -
cient and properly e cien t solutions, gauges, norms, location problems, Weber
problems, minmax problemsContents
1 Introduction 9
1.1 Convex composed programming: A survey of the literature . . . . 9
1.2 A description of the contents . . . . . . . . . . . . . . . . . . . . . 11
2 Duality for a single-objective composed optimization problem 15
2.1 The composed optimization problem and its conjugate duals . . . 15
2.1.1 General notations and formulation . . . . . . . . . 16
2.1.2 The Lagrange dual problem . . . . . . . . . . . . . . . . . 19
2.1.3 The Fenchel dual . . . . . . . . . . . . . . . . . . 20
2.1.4 The Fenchel-Lagrange dual problem . . . . . . . . . . . . . 21
2.2 The relations between the optimal objective values of the dual
problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 The general case . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2 The equivalence of the dual problems (D ) and (D ) . . 27L FL
2.2.3 The equiv of the dual (D ) and (D ) . . 28F FL
2.3 Strong duality and optimality conditions . . . . . . . . . . . . . . 30
2.3.1 Strong duality for (D ), (D ) and (D ) . . . . . . . . . . 30L F FL
2.3.2 Optimality conditions . . . . . . . . . . . . . . . . . . . . 32
2.4 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.1 The classical optimization problem with inequality con-
straints and its dual problems . . . . . . . . . . . . . . . . 35
2.4.2 The optimization problem without constraints . . . . . . . 38
3 Location problems 41
3.1 Duality for location problems . . . . . . . . . . . . . . . . . . . . 42
3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.2 Notations and preliminaries . . . . . . . . . . . . . . . . . 42
3.1.3 The composed problem with monotonic gauges . . . . . . . 44
3.1.4 The case of monotonic norms . . . . . . . . . . . . . . . . 46
3.1.5 The location model with unbounded unit balls . . . . . . . 47
3.1.6 The Weber problem with gauges of closed convex sets . . . 51
3.1.7 The minmax with gauges of closed convex sets . . 53
74 Multiobjective optimization problems 57
4.1 Duality in multiobjective optimization . . . . . . . . . . . . . . . 58
4.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . 59
4.1.3 Duality for the scalarized problem . . . . . . . . . . . . . . 60
4.1.4 The multiobjective dual . . . . . . . . . . . . . . 63
4.1.5 Duality for the classical multiobjective optimization prob-
lem with inequality constraints . . . . . . . . . . . . . . . 66
4.1.6 Duality for the multiobjective optimization problem with-
out constraints . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.1 The case of monotonic norms . . . . . . . . . . . . . . . . 75
4.2.2 The multiobjective location model involving sets as existing
facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.3 The biobjective Weber-minmax problem with in mal dis-
tances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.4 The multiobjective Weber problem with in mal distances . 83
4.2.5 The mjective minmax with in mal 84
Theses 87
Index of notation 93
Bibliography 95
Lebenslauf 101
Selbststandigkeitserklarung 102Chapter 1
Introduction
1.1 Convex composed programming: A survey
of the literature
In the last years convex composed programming (CCP) has received consider-
able attention since it o ers a uni ed framework for solving di eren t types of
optimization problems. By (CCP) we mean a class of optimization problems
in which the objective function as well as the constraints are convex composed
functions. Problems of this form occur, for instance, when nding a feasible
point of the system of inequalities F (x) 0; i = 1;:::;m, by minimizing thei
T n mnorm kF(x)k, where F = (F ;:::;F ) :R !R is a vector function. Similar1 m
problems arise when solving the Weber problem with in mal distances by min-
mP
imizing w d(x;A ), where d(x;A ) = inf (x a ), A = fA ;:::;A g is ai i i i i 1 m
a 2Ai ii=1
family of convex sets, are the gauges of the sets A and w , i = 1;:::;m, arei i i
positive weights. All these examples can be cast within the structure of a convex
composed optimization problem.
There are many papers on composed optimization problems both in nite
and in nite dimensions. Among the many contributors to the study of these
problems we mention A. D. IOFFE, who provided in 1979 (see [29], [30], [31])
the theoretical foundation for the composed problem
c(P ) min f(F(x));
nx2R
n m mwhere F :R !R is a di eren tiable function and f :R !R is a sublinear
function. In [7], J. V. BURKE extended this theory to the case where f is
convex. Later, V. JEYAKUMAR and X. Q. YANG provide in [38] rst-order
cLagrangian conditions and second-order optimality conditions for (P ), in case
when f is a lower semicontinuous convex function and F is a locally Lipschitzian
and (G^ ateaux) di eren tiable function. Further optimality conditions under twice
continuously di eren tiability hypotheses can be found in [31] and [7].
910 CHAPTER 1. INTRODUCTION
Recently, G. WANKA, R. I. BOT and E. VARGYAS treated in [73] the composed
problem with inequality constraints
c(P ) inf f(F(x));i
x2A
where ( )
A = x2 X : g(G(x)) 5 0 ;
k
R
+
n T m T lX R , F = (F ;:::;F ) : X ! R , G = (G ;:::;G ) : X ! R , f :1 m 1 l
m T l kR !R and g = (g ;:::;g ) :R !R . The authors showed the existence of a1 k
solution to this problem via conjugate duality. Under some convexity assumptions
and requiring a quite general constraint quali cation they proved several duality
results and derived the corresponding optimality conditions.
Extended real-valued composed problems of the form
c(P ) min f(F(x));e nx2R ;
F(x)2dom(f)
n m mwhere F : R ! R is a di eren tiable function and f : R ! R[f+1g is a
convex function, have been studied by J. V. BURKE and R. A. POLIQUIN in [8].
The authors derived optimality conditions for these problems by reducing them
to real-valued minimization problems and requiring a constraint quali cation.
Similar problems have also been studied by R. T. ROCKAFELLAR in [54] and
[55], in case when F is twice continuously di eren tiable and f is piecewise linear
quadratic function.
Multiobjective composed problems arise in many applications, subsuming
most of the problem models used in mathematical programming. Problems of
the form
c(P ) v-min f(F(x));v
x2A
where ( )
A = x2 X : g(G(x)) 5 0 ;
k
R+
n T m TX is a convex subset of R , F = (F ;:::;F ) : X ! R , G = (G ;:::;G ) :1 m 1 k
k T m m T k kX ! R , f = (f ;:::;f ) : R ! R , g = (g ;:::;g ) : R ! R , f ; g are1 m 1 k i j
real-valued convex functions and F , G are locally Lipschitz and di eren tiablei j
functions, were studied by V. JEYAKUMAR and X. Q. YANG in [39], [75] and
by C. J. GOH and X. Q. YANG in [22], respectively. In [39], using the Clarke
subdi eren tial, the authors gave rst-order optimality conditions and duality
results for them. In [22] and [75], second-ordery are given
cfor a special case of the problem (P ).v
In what follows we brie y outline the contents of this work.