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Overcoming the failure of the classical generalized interior-point regularity conditions in convex optimization. Applications of the duality theory to enlargements of maximal monotone operatorsCsetnek, Ernö Robert 14 December 2009 (has links) (PDF)
The aim of this work is to present several new results concerning
duality in scalar convex optimization, the formulation of sequential
optimality conditions and some applications of the duality to the theory
of maximal monotone operators.
After recalling some properties of the classical generalized
interiority notions which exist in the literature, we give some
properties of the quasi interior and quasi-relative interior,
respectively. By means of these notions we introduce several
generalized interior-point regularity conditions which guarantee
Fenchel duality. By using an approach due to Magnanti, we derive
corresponding regularity conditions expressed via the quasi
interior and quasi-relative interior which ensure Lagrange
duality. These conditions have the advantage to be applicable in
situations when other classical regularity conditions fail.
Moreover, we notice that several duality results given in the
literature on this topic have either superfluous or contradictory
assumptions, the investigations we make offering in this sense an
alternative.
Necessary and sufficient sequential optimality conditions for a
general convex optimization problem are established via
perturbation theory. These results are applicable even in the
absence of regularity conditions. In particular, we show that
several results from the literature dealing with sequential
optimality conditions are rediscovered and even improved.
The second part of the thesis is devoted to applications of the
duality theory to enlargements of maximal monotone operators in
Banach spaces. After establishing a necessary and sufficient
condition for a bivariate infimal convolution formula, by
employing it we equivalently characterize the
$\varepsilon$-enlargement of the sum of two maximal monotone
operators. We generalize in this way a classical result
concerning the formula for the $\varepsilon$-subdifferential of
the sum of two proper, convex and lower semicontinuous functions.
A characterization of fully enlargeable monotone operators is also
provided, offering an answer to an open problem stated in the
literature. Further, we give a regularity condition for the
weak$^*$-closedness of the sum of the images of enlargements of
two maximal monotone operators.
The last part of this work deals with enlargements of positive sets in SSD spaces. It is shown that many results from the literature concerning enlargements of maximal monotone operators can be generalized to the setting of Banach SSD spaces.
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Overcoming the failure of the classical generalized interior-point regularity conditions in convex optimization. Applications of the duality theory to enlargements of maximal monotone operatorsCsetnek, Ernö Robert 08 December 2009 (has links)
The aim of this work is to present several new results concerning
duality in scalar convex optimization, the formulation of sequential
optimality conditions and some applications of the duality to the theory
of maximal monotone operators.
After recalling some properties of the classical generalized
interiority notions which exist in the literature, we give some
properties of the quasi interior and quasi-relative interior,
respectively. By means of these notions we introduce several
generalized interior-point regularity conditions which guarantee
Fenchel duality. By using an approach due to Magnanti, we derive
corresponding regularity conditions expressed via the quasi
interior and quasi-relative interior which ensure Lagrange
duality. These conditions have the advantage to be applicable in
situations when other classical regularity conditions fail.
Moreover, we notice that several duality results given in the
literature on this topic have either superfluous or contradictory
assumptions, the investigations we make offering in this sense an
alternative.
Necessary and sufficient sequential optimality conditions for a
general convex optimization problem are established via
perturbation theory. These results are applicable even in the
absence of regularity conditions. In particular, we show that
several results from the literature dealing with sequential
optimality conditions are rediscovered and even improved.
The second part of the thesis is devoted to applications of the
duality theory to enlargements of maximal monotone operators in
Banach spaces. After establishing a necessary and sufficient
condition for a bivariate infimal convolution formula, by
employing it we equivalently characterize the
$\varepsilon$-enlargement of the sum of two maximal monotone
operators. We generalize in this way a classical result
concerning the formula for the $\varepsilon$-subdifferential of
the sum of two proper, convex and lower semicontinuous functions.
A characterization of fully enlargeable monotone operators is also
provided, offering an answer to an open problem stated in the
literature. Further, we give a regularity condition for the
weak$^*$-closedness of the sum of the images of enlargements of
two maximal monotone operators.
The last part of this work deals with enlargements of positive sets in SSD spaces. It is shown that many results from the literature concerning enlargements of maximal monotone operators can be generalized to the setting of Banach SSD spaces.
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New insights into conjugate dualityGrad, Sorin - Mihai 19 July 2006 (has links) (PDF)
With this thesis we bring some new results and improve some
existing ones in conjugate duality and some of the areas it is
applied in.
First we recall the way Lagrange, Fenchel and Fenchel - Lagrange
dual problems to a given primal optimization problem can be
obtained via perturbations and we present some connections between
them. For the Fenchel - Lagrange dual problem we prove strong
duality under more general conditions than known so far, while for
the Fenchel duality we show that the convexity assumptions on the
functions involved can be weakened without altering the
conclusion. In order to prove the latter we prove also that some
formulae concerning conjugate functions given so far only for
convex functions hold also for almost convex, respectively nearly
convex functions.
After proving that the generalized geometric dual problem can be
obtained via perturbations, we show that the geometric duality is
a special case of the Fenchel - Lagrange duality and the strong
duality can be obtained under weaker conditions than stated in the
existing literature. For various problems treated in the
literature via geometric duality we show that Fenchel - Lagrange
duality is easier to apply, bringing moreover strong duality and
optimality conditions under weaker assumptions.
The results presented so far are applied also in convex composite
optimization and entropy optimization. For the composed convex
cone - constrained optimization problem we give strong duality and
the related optimality conditions, then we apply these when
showing that the formula of the conjugate of the precomposition
with a proper convex K - increasing function of a K - convex
function on some n - dimensional non - empty convex set X, where
K is a k - dimensional non - empty closed convex cone, holds under
weaker conditions than known so far. Another field were we apply
these results is vector optimization, where we provide a general
duality framework based on a more general scalarization that
includes as special cases and improves some previous results in
the literature. Concerning entropy optimization, we treat first
via duality a problem having an entropy - like objective function,
from which arise as special cases some problems found in the
literature on entropy optimization. Finally, an application of
entropy optimization into text classification is presented.
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New insights into conjugate dualityGrad, Sorin - Mihai 13 July 2006 (has links)
With this thesis we bring some new results and improve some
existing ones in conjugate duality and some of the areas it is
applied in.
First we recall the way Lagrange, Fenchel and Fenchel - Lagrange
dual problems to a given primal optimization problem can be
obtained via perturbations and we present some connections between
them. For the Fenchel - Lagrange dual problem we prove strong
duality under more general conditions than known so far, while for
the Fenchel duality we show that the convexity assumptions on the
functions involved can be weakened without altering the
conclusion. In order to prove the latter we prove also that some
formulae concerning conjugate functions given so far only for
convex functions hold also for almost convex, respectively nearly
convex functions.
After proving that the generalized geometric dual problem can be
obtained via perturbations, we show that the geometric duality is
a special case of the Fenchel - Lagrange duality and the strong
duality can be obtained under weaker conditions than stated in the
existing literature. For various problems treated in the
literature via geometric duality we show that Fenchel - Lagrange
duality is easier to apply, bringing moreover strong duality and
optimality conditions under weaker assumptions.
The results presented so far are applied also in convex composite
optimization and entropy optimization. For the composed convex
cone - constrained optimization problem we give strong duality and
the related optimality conditions, then we apply these when
showing that the formula of the conjugate of the precomposition
with a proper convex K - increasing function of a K - convex
function on some n - dimensional non - empty convex set X, where
K is a k - dimensional non - empty closed convex cone, holds under
weaker conditions than known so far. Another field were we apply
these results is vector optimization, where we provide a general
duality framework based on a more general scalarization that
includes as special cases and improves some previous results in
the literature. Concerning entropy optimization, we treat first
via duality a problem having an entropy - like objective function,
from which arise as special cases some problems found in the
literature on entropy optimization. Finally, an application of
entropy optimization into text classification is presented.
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