Optimality and Lagrangian regularity in vector optimization
Ph.D. Thesis in Mathematics
Giancarlo Bigi
Summary
Life involves decision making constantly. Whenever a decision is taken
rationally, the possible alternatives are compared to select the ''most
suitable'' one; such a judgement generally requires the understanding of
what is ''better'' and ''worse'' and thus it can be aided taking into
account a performance criterion. Hence, many real world problems have been
often modeled as optimization problems, whose objective function gives a
numerical measure of the performance of the different alternatives,
according to the considered criterion. Obviously, building an
appropriate and accurate optimization model is a very important and
difficult task; one of the main difficulties to overcome is to
identify the right criterion and sometimes it may be even impossible.
Infact, there is a wide range of real world decision making problems,
which involve a certain number of reasonable performance criteria and
not just a single one. These criteria often happen to be independent
so that selecting just one of them as objective function appears arbitrary
and inadequate; moreover, they are very often also conflicting: there is
no chance to achieve the best for all the criteria with the same choice.
Clearly, decision making problems with these features can not be efficiently
modeled as optimization problems: composing somehow the different criteria
into a single one may seem an unavoidable necessity but in this way the
very nature of the problem often gets lost. To preserve it within an
optimization framework, some idea of optimality with respect to many
criteria is needed and it can be achieved through some kind of compromise
between the criteria. At the beginning of the last century the famous italian
economist Vilfredo Pareto wrote in his Manual of Political Economy:
''We will say that the members of a collectivity enjoy the maximum
ophelimity in a certain position when it is impossible to move even slightly
away from that position without producing benefits or causing disadvantages
to all the members of the collectivity; any minimum change from that position
necessarily benefit some members and trouble others.''
Though it does not explicitly refer to any optimization context, this
sentence is widely recognized as the very root of what nowadays is
known as multiobjective or vector optimization.
Actually, Pareto's interests were mainly about welfare economics and
economic equilibrium and he was not really concerned with optimization.
He didn't even give a precise mathematical formulation of his concept;
however, it follows from his qualitative description, just considering
members as criteria and positions as alternatives: a choice is ''optimal''
if any other choice improving the value of one criterion does worsen
the value of at least another one; in his honor, these ''optimal''
choices are often called Pareto optima. However, these ideas had
very little impact on mathematicians for a long time (a detailed
account on historical developments of vector otimization can be found
in the survey paper by Stadler). Some interest
arose at the beginning of the fifties with Koopmans' paper on a multiobjective
optimization model for the analysis of production activities and the
pioneering paper by Kuhn and Tucker on nonlinear programming, in which the term
vector optimization problem was introduced for the first time and
some optimality conditions were discussed; only starting from the late
sixties an increasing number of reaserchers devoted their efforts to study
vector optimization.
The core of this dissertation is devoted to the discussion of
optimality conditions for nonlinear vector optimization problems. Much
work has been done on this topic, starting from the pioneering papers
of Da Cuna and Polak, Borwein, Lin, Censor, Ben-Israel et al. and many others
followed. The main focus has generally been on first order
optimality conditions both for problems with smooth and nonsmooth
functions or on saddlepoint type criteria; on the contrary, just a few
papers developed a second order analysis. Another issue that seems to have
not been deeply investigated is Lagrangian regularity, that deals
with those conditions which guarantee multipliers rules to hold with
nonzero multipliers corresponding to all or at least one of the
objective functions. In the case of classical nonlinear optimization
such conditions generally involve just the constraint and therefore
they are usually referred to as constraint qualifications; on the contrary,
conditions involving also the objective functions are often needed in the case
of vector optimization.
A large part of Chapter 2 is devoted to the analysis of
regularity conditions both for first and second order multipliers
rules. In particular, not only very weak ones are presented but also
those regularity conditions, which guarantee additional properties of
multipliers such as boundedness and uniquess, are analysed. It is
worth stressing that the focus of that chapter is on first and second
order optimality criteria both for the classical problems, where the
constraint is described by inequalities and equalities, and for those
problems where the constraint is given just as a set. Actually, necessary
optimality criteria are first developed for the latter case, exploiting
suitable approximations of the constraint; these results allow to deduce
multipliers rules and analyse regularity conditions for the former case quite
easily. This procedure does not work for sufficient criteria and the
relationships between the criteria developed for the two cases are
investigated: quite surprisingly, the second order ones are completely
unrelated.
Since the results of Chapter 2 require that functions are
smooth, precisely differentiable or twice differentiable, Chapter 3
is devoted to optimality criteria for problems involving nonsmooth
functions. Two different approaches are considered and compared: the
first one relies on the components of vector functions so that
well-known nonsmooth analysis tools for the real--valued case can be
exploited; the second one is based on set--valued directional
derivatives, which can be introduced considering suitable limits
of vector incremental ratios without relying on components. The
latter allows also to recover tools for vector functions
such as generalized Jacobians and Hessians as particular cases and
furthermore it provides sharper results than the former.
Finally, Chapter 4 deals with optimality criteria based on the
saddlepoints of suitable Lagrangian functions. Though this issue has
been deeply investigated, vector Lagrangians with multipliers associated
to the objective functions have generally been disregarded. On the contrary,
they allow to achieve stronger results and to recover those based on
the saddlepoint of real Lagrangians as particular cases.
The thesis was sent out for evaluation on October, 2002. The referees
have been Prof. Angelo Guerraggio (Università dell'Insubria)
and Prof. Kathrin Klamroth (Universität Erlangen-Nürnberg).
The thesis has been successfully defended on September 19, 2003. The
committee was chaired by Prof. Franco Giannessi (Università di Pisa) and
composed by Prof. Massimo Pappalardo (Thesis advisor), Prof. Fioravante
Patrone (Università di Genova), Proff. Angelo Guerraggio and Kathrin
Klamroth.
Download the .pdf file (1.2M) of the thesis:
Optimality and Lagrangian regularity in vector optimization
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