Marlon Alexander Braun

Indicator Based Search in Variable Orderings: Theory and Algorithms

Various real world problems, especially in financial applications, medical engineering, and game theory, involve solving a multi-objective optimization problem with a variable ordering structure. This means that the ordering relation at a point in the (multi-)objective space depends on the point. This is a striking difference from usual multi-objective optimization problems, where the ordering is induced by the Pareto-cone and remains constant throughout the objective space. In addition to variability, in many applications (like portfolio optimization) the ordering is induced by a non-convex set instead of a cone. The main purpose of this paper is to provide theoretical and algorithmic advances for general set-based variable orderings. A hypervolume based indicator measure is also proposed for the first time for such optimization tasks. Theoretical results are derived and properties of this indicator are studied. Moreover, the theory is also used to develop three indicator based algorithms for approximating the set of optimal solutions. Computational results show the niche of population based algorithms for solving multi-objective problems with variable orderings.

Theory and Algorithms for Finding Knees

A multi-objective optimization problem involves multiple and conflicting objectives. These conflicting objectives give rise to a set of Pareto-optimal solutions. However, not all the members of the Pareto-optimal set have equally nice properties. The classical concept of proper Pareto-optimality is a way of characterizing good Pareto-optimal solutions. In this paper, we metrize this concept to induce an ordering on the Pareto-optimal set. The use of this metric allows us to define a proper knee region, which contains solutions below a user-specified threshold metric. We theoretically analyze past definitions of knee points, and in particular, reformulate a commonly used nonlinear program, to achieve convergence results. Additionally, mathematical properties of the proper knee region are investigated. We also develop two multi-objective evolutionary algorithms towards finding proper knees and present simulation results on a number of test problems.

Preference ranking schemes in multi-objective evolutionary algorithms

In recent years, multi-objective evolutionary algorithms have diversified their goal from finding an approximation of the complete efficient front of a multi-objective optimization problem, to integration of user preferences. These user preferences can be used to focus on a preferred region of the efficient front. Many such user preferences come from so called proper Pareto-optimality notions. Although, starting with the seminal work of Kuhn and Tucker in 1951, proper Pareto-optimal solutions have been around in the multi-criteria decision making literature, there are (surprisingly) very few studies in the evolutionary domain on this. In this paper, we introduce new ranking schemes of various state-of-the-art multi-objective evolutionary algorithms to focus on a preferred region corresponding to proper Pareto-optimal solutions. The algorithms based on these new ranking schemes are successfully tested on extensive benchmark test problems of varying complexity, with the aim to find the preferred region of the efficient front. This comprehensive study adequately demonstrates the efficiency of the developed multi-objective evolutionary algorithms in finding the complete preferred region for a large class of complex problems.

Comparison of Multi-objective Evolutionary Optimization in Smart Building Scenarios

The optimization of operating times and operation modes of devices and systems that consume or generate electricity in buildings by building energy management systems promises to alleviate problems arising in today’s electricity grids. Conflicting objectives may have to be pursued in this context, giving rise to a multi-objective optimization problem. This paper presents the optimization of appliances as well as heating and air-conditioning devices in two distinct settings of smart buildings, a residential and a commercial building, with respect to the minimization of energy costs, CO2 emissions, discomfort, and technical wearout. We propose new encodings for appliances that are based on a combined categorization of devices respecting both, the optimization of operating times as well as operation modes, e.g., of hybrid devices. To identify an evolutionary algorithm that promises to lead to good optimization results of the devices in our real-world lab environments, we compare four state-of-the-art algorithms in realistic simulations: ESPEA, NSGA-II, NSGA-III, and SPEA2. The results show that ESPEA and NSGA-II significantly outperform the other two algorithms in our scenario.

A neuro-genetic approach for modeling and optimizing a complex cogeneration process

Graphical abstractDisplay Omitted HighlightsA strategy for modeling and optimizing a cogeneration process of a industrial plant is presented.A multi-objective optimization approach is chosen.A computational study reveals that the ESPEA algorithm performs best.ESPEA approximates the Pareto front and puts an emphasis on regions that maximize efficiency. Cogeneration is the simultaneous generation of electricity and useful heat with the aim of exploiting more efficiently the energy stored in the fuel. Cogeneration is, however, a complex process that encompasses a great amount of sub-systems and variables. This fact makes it very difficult to obtain an analytical model of the whole plant, and therefore providing a mechanism or a methodology able to optimize the global behavior. This paper proposes a neuro-genetic strategy for modeling and optimizing a cogeneration process of a real industrial plant. Firstly, the modeling of the process is carried out by means of several interconnected neural networks where, each neural network deals with a particular sub-system of the plant. Next, the obtained models are used by a genetic algorithm, which solves a multiobjective optimization problem of the plant, where the goal is to minimize the fuel consumption and maximize both the generated electricity and the use of the heat. The proposed approach is evaluated with data of a real cogeneration plant collected over a one-year period. Obtained results show not only that the modeling of the plant is correct but also that the optimization increases significantly the efficiency of the cogeneration plant.

Obtaining Optimal Pareto Front Approximations using Scalarized Preference Information

Scalarization techniques are a popular method for articulating preferences in solving multi-objective optimization problems. These techniques, however, have so far proven to be ill-suited in finding a preference-driven approximation that still captures the Pareto front in its entirety. Therefore, we propose a new concept that defines an optimal distribution of points on the front given a specific scalarization function. It is proven that such an approximation exists for every real-valued problem irrespective of the shape of the corresponding front under some very mild conditions. We also show that our approach works well in obtaining an equidistant approximation of the Pareto front if no specific preference is articulated. Our analysis is complemented by the presentation of a new algorithm that implements the aforementioned concept. We provide in-depth simulation results to demonstrate the performance of our algorithm. The analysis also reveals that our algorithm is able to outperform current state-of-the-art algorithms on many popular benchmark problems.

Angle-Based Preference Models in Multi-objective Optimization

Solutions that provide a balance between different objective values in multi-objective optimization can be identified by assessing the curvature of the Pareto front. We analyze how methods based on angles have been utilized in the past for this task and propose a new angle-based measure--angle utility--that ranks points of the Pareto front irrespective of its shape or the number of objectives. An algorithm for finding angle utility optima is presented and a computational study shows that this algorithm is successful in identifying angle utility optima.

On the interrelationships between knees and aggregate objective functions

Optimizing several objectives that are often at odds with each other provides difficult challenges that are not encountered if having only one goal at hand. One intuitive way to solve a multi-objective problem is to aggregate the objectives and reformulate it as an optimization problem having just a single goal. This goal can be a designer specific aggregation of the objectives or a characterization of knees, trade-offs, utilities, stronger optimality concepts or preferences. This paper examines the theoretical relationships between two knee concepts and aggregate objective functions methods. The changes in the fitness landscape by utilizing different aggregations is also discussed.

Scalarized Preferences in Multi-objective Optimization

Multikriterielle Optimierungsprobleme verfugen uber keine Losung, die optimal in jeder Zielfunktion ist. Die Schwierigkeit solcher Probleme liegt darin eine Kompromisslosung zu finden, die den Praferenzen des Entscheiders genugen, der den Kompromiss implementiert. Skalarisierung – die Abbildung des Vektors der Zielfunktionswerte auf eine reelle Zahl – identifiziert eine einzige Losung als globales Praferenzenoptimum um diese Probleme zu losen. Allerdings generieren Skalarisierungsmethoden keine zusatzlichen Informationen uber andere Kompromisslosungen, die die Praferenzen des Entscheiders bezuglich des globalen Optimums verandern konnten. Um dieses Problem anzugehen stellt diese Dissertation eine theoretische und algorithmische Analyse skalarisierter Praferenzen bereit. Die theoretische Analyse besteht aus der Entwicklung eines Ordnungsrahmens, der Praferenzen als Problemtransformationen charakterisiert, die praferierte Untermengen der Paretofront definieren. Skalarisierung wird als Transformation der Zielmenge in diesem Ordnungsrahmen dargestellt. Des Weiteren werden Axiome vorgeschlagen, die wunschenswerte Eigenschaften von Skalarisierungsfunktionen darstellen. Es wird gezeigt unter welchen Bedingungen existierende Skalarisierungsfunktionen diese Axiome erfullen. Die algorithmische Analyse kennzeichnet Praferenzen anhand des Resultats, das ein Optimierungsalgorithmus generiert. Zwei neue Paradigmen werden innerhalb dieser Analyse identifiziert. Fur beide Paradigmen werden Algorithmen entworfen, die skalarisierte Praferenzeninformationen verwenden: Praferenzen-verzerrte Paretofrontapproximationen verteilen Punkte uber die gesamte Paretofront, fokussieren aber mehr Punkte in Regionen mit besseren Skalarisierungswerten; multimodale Praferenzenoptima sind Punkte, die lokale Skalarisierungsoptima im Zielraum darstellen. Ein Drei-Stufen-Algorith\-mus wird entwickelt, der lokale Skalarisierungsoptima approximiert und verschiedene Methoden werden fur die unterschiedlichen Stufen evaluiert. Zwei Realweltprobleme werden vorgestellt, die die Nutzlichkeit der beiden Algorithmen illustrieren. Das erste Problem besteht darin Fahrplane fur ein Blockheizkraftwerk zu finden, die die erzeugte Elektrizitat und Warme maximieren und den Kraftstoffverbrauch minimiert. Praferenzen-verzerrte Approximationen generieren mehr Energie-effiziente Losungen, unter denen der Entscheider seine favorisierte Losung auswahlen kann, indem er die Konflikte zwischen den drei Zielen abwagt. Das zweite Problem beschaftigt sich mit der Erstellung von Fahrplanen fur Gerate in einem Wohngebaude, so dass Energiekosten, Kohlenstoffdioxidemissionen und thermisches Unbehagen minimiert werden. Es wird gezeigt, dass lokale Skalarisierungsoptima Fahrplane darstellen, die eine gute Balance zwischen den drei Zielen bieten. Die Analyse und die Experimente, die in dieser Arbeit vorgestellt werden, ermoglichen es Entscheidern bessere Entscheidungen zu treffen indem Methoden angewendet werden, die mehr Optionen generieren, die mit den Praferenzen der Entscheider ubereinstimmen.

Multimodal scalarized preferences in multi-objective optimization

Scalarization functions represent preferences in multi-objective optimization by mapping the vector of objectives to a single real value. Optimization techniques using scalarized preferences mainly focus on obtaining only a single global preference optimum. Instead, we propose considering all local and global scalarization optima on the global Pareto front. These points represent the best choice in their immediate neighborhood. Additionally, they are usually sufficiently far apart in the objective space to present themselves as true alternatives if the scalarization function cannot capture every detail of the decision makers true preference. We propose an algorithmic framework for obtaining all scalarization optima of a multi-objective optimization problem. In said framework, an approximation of the global Pareto front is obtained, from which neighborhoods of local optima are identified. Local optimization algorithms are then applied to identify the optimum of every neighborhood. In this way, we have an optima-based approximation of the global Pareto front based on the underlying scalarization function. A computational study reveals that local optimization algorithms must be carefully configured for being able to find all optima.