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, first 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 theory, we provide different kinds of dual problems to it and examine the relations between the optimal objective values of the duals. Given some additional assumptions, 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 first 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 different location problems are made. We finish our investigations with the study of composed multiobjective optimization problems. In doing like this, first we scalarize this problem and study the scalarized one by using the conjugacy approach developed before. The optimality 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 optimization 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. / In dieser Arbeit wird, anhand der sogenannten konjugierten Dualitätstheorie, ein allgemeines Dualitätsverfahren für die Untersuchung verschiedener Optimierungsaufgaben dargestellt. Um dieses Ziel zu erreichen wird zuerst eine allgemeine Optimierungsaufgabe betrachtet, wobei sowohl die Zielfunktion als auch die Nebenbedingungen zusammengesetzte Funktionen sind. Mit Hilfe der konjugierten Dualitätstheorie, die auf der sogenannten Störungstheorie basiert, werden für die primale Aufgabe drei verschiedene duale Aufgaben konstruiert und weiterhin die Beziehungen zwischen deren optimalen Zielfunktionswerten untersucht. Unter geeigneten Konvexitäts- und Monotonievoraussetzungen wird die Gleichheit dieser optimalen Zielfunktionswerte und zusätzlich die Existenz der starken Dualität zwischen der primalen und den entsprechenden dualen Aufgaben bewiesen. In Zusammenhang mit der starken Dualität werden Optimalitätsbedingungen hergeleitet. Die Ergebnisse werden abgerundet durch die Betrachtung zweier Spezialfälle, nämlich die klassische restringierte bzw. unrestringierte Optimierungsaufgabe, für welche sich die aus der Literatur bekannten Dualitätsergebnisse ergeben. Der zweite Teil der Arbeit ist der Dualität bei Standortproblemen gewidmet. Dazu wird ein sehr allgemeines Standortproblem mit konvexer zusammengesetzter Zielfunktion in Form eines Gauges formuliert, für das die entsprechenden Dualitätsaussagen abgeleitet werden. Als Spezialfälle werden Optimierungsaufgaben mit monotonen Normen betrachtet. Insbesondere lassen sich Dualitätsaussagen und Optimalitätsbedingungen für das klassische Weber und Minmax Standortproblem mit Gauges als Zielfunktion herleiten. Das letzte Kapitel verallgemeinert die Dualitätsaussagen, die im zweiten Kapitel erhalten wurden, auf multikriterielle Optimierungsprobleme. Mit Hilfe geeigneter Skalarisierungen betrachten wir zuerst ein zu der multikriteriellen Optimierungsaufgabe zugeordnetes skalares Problem. Anhand der in diesem Fall erhaltenen Optimalitätsbedingungen formulieren wir das multikriterielle Dualproblem. Weiterhin beweisen wir die schwache und, unter bestimmten Annahmen, die starke Dualität. Durch Spezialisierung der Zielfunktionen bzw. Nebenbedingungen resultieren die klassischen konvexen Mehrzielprobleme mit Ungleichungs- und Mengenrestriktionen. Als weitere Anwendungen werden vektorielle Standortprobleme betrachtet, zu denen wir entsprechende duale Aufgaben formulieren.

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, first 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 theory, we provide different kinds of dual problems to it and examine the relations between the optimal objective values of the duals. Given some additional assumptions, 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 first 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 different location problems are made. We finish our investigations with the study of composed multiobjective optimization problems. In doing like this, first we scalarize this problem and study the scalarized one by using the conjugacy approach developed before. The optimality 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 optimization 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. / In dieser Arbeit wird, anhand der sogenannten konjugierten Dualitätstheorie, ein allgemeines Dualitätsverfahren für die Untersuchung verschiedener Optimierungsaufgaben dargestellt. Um dieses Ziel zu erreichen wird zuerst eine allgemeine Optimierungsaufgabe betrachtet, wobei sowohl die Zielfunktion als auch die Nebenbedingungen zusammengesetzte Funktionen sind. Mit Hilfe der konjugierten Dualitätstheorie, die auf der sogenannten Störungstheorie basiert, werden für die primale Aufgabe drei verschiedene duale Aufgaben konstruiert und weiterhin die Beziehungen zwischen deren optimalen Zielfunktionswerten untersucht. Unter geeigneten Konvexitäts- und Monotonievoraussetzungen wird die Gleichheit dieser optimalen Zielfunktionswerte und zusätzlich die Existenz der starken Dualität zwischen der primalen und den entsprechenden dualen Aufgaben bewiesen. In Zusammenhang mit der starken Dualität werden Optimalitätsbedingungen hergeleitet. Die Ergebnisse werden abgerundet durch die Betrachtung zweier Spezialfälle, nämlich die klassische restringierte bzw. unrestringierte Optimierungsaufgabe, für welche sich die aus der Literatur bekannten Dualitätsergebnisse ergeben. Der zweite Teil der Arbeit ist der Dualität bei Standortproblemen gewidmet. Dazu wird ein sehr allgemeines Standortproblem mit konvexer zusammengesetzter Zielfunktion in Form eines Gauges formuliert, für das die entsprechenden Dualitätsaussagen abgeleitet werden. Als Spezialfälle werden Optimierungsaufgaben mit monotonen Normen betrachtet. Insbesondere lassen sich Dualitätsaussagen und Optimalitätsbedingungen für das klassische Weber und Minmax Standortproblem mit Gauges als Zielfunktion herleiten. Das letzte Kapitel verallgemeinert die Dualitätsaussagen, die im zweiten Kapitel erhalten wurden, auf multikriterielle Optimierungsprobleme. Mit Hilfe geeigneter Skalarisierungen betrachten wir zuerst ein zu der multikriteriellen Optimierungsaufgabe zugeordnetes skalares Problem. Anhand der in diesem Fall erhaltenen Optimalitätsbedingungen formulieren wir das multikriterielle Dualproblem. Weiterhin beweisen wir die schwache und, unter bestimmten Annahmen, die starke Dualität. Durch Spezialisierung der Zielfunktionen bzw. Nebenbedingungen resultieren die klassischen konvexen Mehrzielprobleme mit Ungleichungs- und Mengenrestriktionen. Als weitere Anwendungen werden vektorielle Standortprobleme betrachtet, zu denen wir entsprechende duale Aufgaben formulieren.

The goal of this thesis is two-fold. On the one hand, it pursues to provide a contribution to the conjugate duality by proposing a new duality concept, which can be understood as an umbrella for different meaningful perturbation methods. On the other hand, this thesis aims to investigate minimax location problems by means of the duality concept introduced in the first part of this work, followed by a numerical approach using epigraphical splitting methods. After summarizing some elements of the convex analysis as well as introducing important results needed later, we consider an optimization problem with geometric and cone constraints, whose objective function is a composition of n+1 functions. For this problem we propose a conjugate dual problem, where the functions involved in the objective function of the primal problem are decomposed. Furthermore, we formulate generalized interior point regularity conditions for strong duality and give necessary and sufficient optimality conditions. As applications of this approach we determine the formulae of the conjugate as well as the biconjugate of the objective function of the primal problem and analyze an optimization problem having as objective function the sum of reciprocals of concave functions. In the second part of this thesis we discuss in the sense of the introduced duality concept three classes of minimax location problems. The first one consists of nonlinear and linear single minimax location problems with geometric constraints, where the maximum of nonlinear or linear functions composed with gauges between pairs of a new and existing points will be minimized. The version of the nonlinear location problem is additionally considered with set-up costs. The second class of minimax location problems deals with multifacility location problems as suggested by Drezner (1991), where for each given point the sum of weighted distances to all facilities plus set-up costs is determined and the maximal value of these sums is to be minimized. As the last and third class the classical multifacility location problem with geometrical constraints is considered in a generalized form where the maximum of gauges between pairs of new facilities and the maximum of gauges between pairs of new and existing facilities will be minimized. To each of these location problems associated dual problems will be formulated as well as corresponding duality statements and necessary and sufficient optimality conditions. To illustrate the results of the duality approach and to give a more detailed characterization of the relations between the location problems and their corresponding duals, we consider examples in the Euclidean space. This thesis ends with a numerical approach for solving minimax location problems by epigraphical splitting methods. In this framework, we give formulae for the projections onto the epigraphs of several sums of powers of weighted norms as well as formulae for the projection onto the epigraphs of gauges. Numerical experiments document the usefulness of our approach for the discussed location problems.

The goal of this thesis is two-fold. On the one hand, it pursues to provide a contribution to the conjugate duality by proposing a new duality concept, which can be understood as an umbrella for different meaningful perturbation methods. On the other hand, this thesis aims to investigate minimax location problems by means of the duality concept introduced in the first part of this work, followed by a numerical approach using epigraphical splitting methods. After summarizing some elements of the convex analysis as well as introducing important results needed later, we consider an optimization problem with geometric and cone constraints, whose objective function is a composition of n+1 functions. For this problem we propose a conjugate dual problem, where the functions involved in the objective function of the primal problem are decomposed. Furthermore, we formulate generalized interior point regularity conditions for strong duality and give necessary and sufficient optimality conditions. As applications of this approach we determine the formulae of the conjugate as well as the biconjugate of the objective function of the primal problem and analyze an optimization problem having as objective function the sum of reciprocals of concave functions. In the second part of this thesis we discuss in the sense of the introduced duality concept three classes of minimax location problems. The first one consists of nonlinear and linear single minimax location problems with geometric constraints, where the maximum of nonlinear or linear functions composed with gauges between pairs of a new and existing points will be minimized. The version of the nonlinear location problem is additionally considered with set-up costs. The second class of minimax location problems deals with multifacility location problems as suggested by Drezner (1991), where for each given point the sum of weighted distances to all facilities plus set-up costs is determined and the maximal value of these sums is to be minimized. As the last and third class the classical multifacility location problem with geometrical constraints is considered in a generalized form where the maximum of gauges between pairs of new facilities and the maximum of gauges between pairs of new and existing facilities will be minimized. To each of these location problems associated dual problems will be formulated as well as corresponding duality statements and necessary and sufficient optimality conditions. To illustrate the results of the duality approach and to give a more detailed characterization of the relations between the location problems and their corresponding duals, we consider examples in the Euclidean space. This thesis ends with a numerical approach for solving minimax location problems by epigraphical splitting methods. In this framework, we give formulae for the projections onto the epigraphs of several sums of powers of weighted norms as well as formulae for the projection onto the epigraphs of gauges. Numerical experiments document the usefulness of our approach for the discussed location problems.