Approximation in Multiobjective Optimization with Applications

Over the last couple of decades, the field of multiobjective optimization has received much attention in solving real-life optimization problems in science, engineering, economics and other fields where optimal decisions need to be made in the presence of trade-offs between two or more conflicting objective functions. The conflicting nature of objective functions implies a solution set for a multiobjective optimization problem. Obtaining this set is difficult for many reasons, and a variety of approaches for approximating it either partially or entirely have been proposed. In response to the growing interest in approximation, this research investigates developing a theory and methodology for representing and approximating solution sets of multiobjective optimization problems. The concept of the tolerance function is proposed as a tool for modeling representation quality. Two types of subsets of the set being represented, covers and approximations, are defined, and their properties are examined. In addition, approximating the solution set of the multiobjective set covering problem (MOSCP), one of the challenging combinatorial optimization problems that has seen limited study, is investigated. Two algorithms are proposed for approximating the solution set of the MOSCP, and their approximation quality is derived. A heuristic algorithm is also proposed to approximate the solution set of the MOSCP. The performance of each algorithm is evaluated using test problems. Since the MOSCP has many real-life applications, and in particular designing reserve systems for ecological species is a common field for its applications, two optimization models are proposed in this dissertation for preserving reserve sites for species and their natural habitats

Precise Characterization and Multiobjective Optimization of Low Noise Amplifiers

Although practically all function blocks of the satellite navigation receivers are realized using the CMOS digital integrated circuits, it is appropriate to create a separate low noise antenna preamplifier based on a low noise pHEMT. Such an RF front end can be strongly optimized to attain a suitable tradeoff between the noise figure and transducer power gain. Further, as all the four principal navigation systems (GPS, GLONASS, Galileo, and COMPASS) work in similar frequency bands (roughly from 1.1 to 1.7 GHz), it is reasonable to create the low noise preamplifier for all of them. In the paper, a sophisticated method of the amplifier design is suggested based on multiobjective optimization. A substantial improvement of a standard optimization method is also outlined to satisfy a uniform coverage of Pareto front. Moreover, for enhancing efficiency of many times repeated solutions of large linear systems during the optimization, a new modification of the Markowitz criterion is suggested compatible with fast modes of the LU factorization. Extraordinary attention was also given to the accuracy of modeling. First, an extraction of pHEMT model parameters was performed including its noise part, and several models were compared. The extraction was carried out by an original identification procedure based on a combination of metaheuristic and direct methods. Second, the equations of the passive elements (including transmission lines and T-splitters) were carefully defined using frequency dispersion of their parameters as Q, ESR, etc. Third, an optimal selection of the operating point and essential passive elements was performed using the improved optimization method. Finally, the s-parameters and noise figure of the amplifier were measured, and stability and third-order intermodulation products were also checked

Evolving dynamic multiple-objective optimization problems with objective replacement

This paper studies the strategies for multi-objective optimization in a dynamic environment. In particular, we focus on problems with objective replacement, where some objectives may be replaced with new objectives during evolution. It is shown that the Pareto-optimal sets before and after the objective replacement share some common members. Based on this observation, we suggest the inheritance strategy. When objective replacement occurs, this strategy selects good chromosomes according to the new objective set from the solutions found before objective replacement, and then continues to optimize them via evolution for the new objective set. The experiment results showed that this strategy can help MOGAs achieve better performance than MOGAs without using the inheritance strategy, where the evolution is restarted when objective replacement occurs. More solutions with better quality are found during the same time span

A Bayesian approach to constrained single- and multi-objective optimization

This article addresses the problem of derivative-free (single- or multi-objective) optimization subject to multiple inequality constraints. Both the objective and constraint functions are assumed to be smooth, non-linear and expensive to evaluate. As a consequence, the number of evaluations that can be used to carry out the optimization is very limited, as in complex industrial design optimization problems. The method we propose to overcome this difficulty has its roots in both the Bayesian and the multi-objective optimization literatures. More specifically, an extended domination rule is used to handle objectives and constraints in a unified way, and a corresponding expected hyper-volume improvement sampling criterion is proposed. This new criterion is naturally adapted to the search of a feasible point when none is available, and reduces to existing Bayesian sampling criteria---the classical Expected Improvement (EI) criterion and some of its constrained/multi-objective extensions---as soon as at least one feasible point is available. The calculation and optimization of the criterion are performed using Sequential Monte Carlo techniques. In particular, an algorithm similar to the subset simulation method, which is well known in the field of structural reliability, is used to estimate the criterion. The method, which we call BMOO (for Bayesian Multi-Objective Optimization), is compared to state-of-the-art algorithms for single- and multi-objective constrained optimization

Multi-objective combinatorial optimization problems in transportation and defense systems

Multi-objective Optimization problems arise in many applications; hence, solving them efficiently is important for decision makers. A common procedure to solve such problems is to generate the exact set of Pareto efficient solutions. However, if the problem is combinatorial, generating the exact set of Pareto efficient solutions can be challenging. This dissertation is dedicated to Multi-objective Combinatorial Optimization problems and their applications in system of systems architecting and railroad track inspection scheduling. In particular, multi-objective system of systems architecting problems with system flexibility and performance improvement funds have been investigated. Efficient solution methods are proposed and evaluated for not only the system of systems architecting problems, but also a generic multi-objective set covering problem. Additionally, a bi-objective track inspection scheduling problem is introduced for an automated ultrasonic inspection vehicle. Exact and approximation methods are discussed for this bi-objective track inspection scheduling problem --Abstract, page iii

Differential Evolution for Multiobjective Portfolio Optimization

Financial portfolio optimization is a challenging problem. First, the problem is multiobjective (i.e.: minimize risk and maximize profit) and the objective functions are often multimodal and non smooth (e.g.: value at risk). Second, managers have often to face real-world constraints, which are typically non-linear. Hence, conventional optimization techniques, such as quadratic programming, cannot be used. Stochastic search heuristic can be an attractive alternative. In this paper, we propose a new multiobjective algorithm for portfolio optimization: DEMPO - Differential Evolution for Multiobjective Portfolio Optimization. The main advantage of this new algorithm is its generality, i.e., the ability to tackle a portfolio optimization task as it is, without simplifications. Our empirical results show the capability of our approach of obtaining highly accurate results in very reasonable runtime, in comparison with quadratic programming and another state-of-art search heuristic, the so-called NSGA II.Portfolio Optimization, Multiobjective, Real-world Constraints, Value at Risk, Expected Shortfall, Differential Evolution