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.

Diss. Stockholm : Stockholms universitet, 2007.

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.

This thesis aims to scrutinize the Visegrad Quadruple Initiative as a device of the Central European countries in the process of involving into the re-negotiations in Europe and in world politics. The thesis argues that the Visegrad group was built in order to respond the demands of changing Europe and Euro-Atlantic structures, and thus to overcome the double process of transition and integration. However that was not the only reason to launch the Visegrad regional cooperation. The group produced an affirmative discourse that its members are distinguished from the other countries in transition, so that they are constantly one step forward to &lsquo / return to Europe&rsquo / . In the aftermath of the eastern enlargements of NATO and the EU alike, the original mission of the group, integration with the West, was achieved. That created a profound discussion about the survival of the group. As it is argued in this thesis, the group, as a prosperous and substantial regional cooperation, should rather continue to work in order to have more words to say in the re-negotiations processes. Another argument of the thesis is that the Visegrad group, taking Benelux group as a model in its continuity, is beneficial to produce a common foreign policy tendency among its members as long as the interests of its members are overlapping, otherwise the group is just being a political platform in which its members can share their views in such areas as regional regulations.

This thesis aims to scrutinize the Visegrad Quadruple Initiative as a device of the Central European countries in the process of involving into the re-negotiations in Europe and in world politics. The thesis argues that the Visegrad group was built in order to respond the demands of changing Europe and Euro-Atlantic structures, and thus to overcome the double process of transition and integration. However that was not the only reason to launch the Visegrad regional cooperation. The group produced an affirmative discourse that its members are distinguished from the other countries in transition, so that they are constantly one step forward to &lsquo / return to Europe&rsquo / . In the aftermath of the eastern enlargements of NATO and the EU alike, the original mission of the group, integration with the West, was achieved. That created a profound discussion about the survival of the group. As it is argued in this thesis, the group, as a prosperous and substantial regional cooperation, should rather continue to work in order to have more words to say in the re-negotiations processes. Another argument of the thesis is that the Visegrad group, taking Benelux group as a model in its continuity, is beneficial to produce a common foreign policy tendency among its members as long as the interests of its members are overlapping, otherwise the group is just being a political platform in which its members can share their views in such areas as regional regulations.