Multi-Objective Bayesian Global Optimization using expected hypervolume improvement gradient

The Expected Hypervolume Improvement (EHVI) is a frequently used infill criterion in Multi-Objective Bayesian Global Optimization (MOBGO), due to its good ability to lead the exploration. Recently, the computational complexity of EHVI calculation is reduced to O(n log n) for both 2-D and 3-D cases. However, the optimizer in MOBGO still requires a significant amount of time, because the calculation of EHVI is carried out in each iteration and usually tens of thousands of the EHVI calculations are required. This paper derives a formula for the Expected Hypervolume Improvement Gradient (EHVIG) and proposes an efficient algorithm to calculate EHVIG. The new criterion (EHVIG) is utilized by two different strategies to improve the efficiency of the optimizer discussed in this paper. Firstly, it enables gradient ascent methods to be used in MOBGO. Moreover, since the EHVIG of an optimal solution should be a zero vector, it can be regarded as a stopping criterion in global optimization, e.g., in Evolution Strategies. Empirical experiments are performed on seven benchmark problems. The experimental results show that the second proposed strategy, using EHVIG as a stopping criterion for local search, can outperform the normal MOBGO on problems where the optimal solutions are located in the interior of the search space. For the ZDT series test problems, EHVIG still can perform better when gradient projection is applied.Algorithms and the Foundations of Software technolog

Generalized decomposition and cross entropy methods for many-objective optimization

Decomposition-based algorithms for multi-objective optimization problems have increased in popularity in the past decade. Although their convergence to the Pareto optimal front (PF) is in several instances superior to that of Pareto-based algorithms, the problem of selecting a way to distribute or guide these solutions in a high-dimensional space has not been explored. In this work, we introduce a novel concept which we call generalized decomposition. Generalized decomposition provides a framework with which the decision maker (DM) can guide the underlying evolutionary algorithm toward specific regions of interest or the entire Pareto front with the desired distribution of Pareto optimal solutions. Additionally, it is shown that generalized decomposition simplifies many-objective problems by unifying the three performance objectives of multi-objective evolutionary algorithms – convergence to the PF, evenly distributed Pareto optimal solutions and coverage of the entire front – to only one, that of convergence. A framework, established on generalized decomposition, and an estimation of distribution algorithm (EDA) based on low-order statistics, namely the cross-entropy method (CE), is created to illustrate the benefits of the proposed concept for many objective problems. This choice of EDA also enables the test of the hypothesis that low-order statistics based EDAs can have comparable performance to more elaborate EDAs

Dimension reduction of decision variables for multireservoir operation: A spectral optimization model

Optimizing the operation of a multireservoir system is challenging due to the high dimension of the decision variables that lead to a large and complex search space. A spectral optimization model (SOM), which transforms the decision variables from time domain to frequency domain, is proposed to reduce the dimensionality. The SOM couples a spectral dimensionality-reduction method called Karhunen-Loeve (KL) expansion within the routine of Nondominated Sorting Genetic Algorithm (NSGA-II). The KL expansion is used to represent the decision variables as a series of terms that are deterministic orthogonal functions with undetermined coefficients. The KL expansion can be truncated into fewer significant terms, and consequently, fewer coefficients by a predetermined number. During optimization, operators of the NSGA-II (e.g., crossover) are conducted only on the coefficients of the KL expansion rather than the large number of decision variables, significantly reducing the search space. The SOM is applied to the short-term operation of a 10-reservoir system in the Columbia River of the United States. Two scenarios are considered herein, the first with 140 decision variables and the second with 3360 decision variables. The hypervolume index is used to evaluate the optimization performance in terms of convergence and diversity. The evaluation of optimization performance is conducted for both conventional optimization model (i.e., NSGA-II without KL) and the SOM with different number of KL terms. The results show that the number of decision variables can be greatly reduced in the SOM to achieve a similar or better performance compared to the conventional optimization model. For the scenario with 140 decision variables, the optimal performance of the SOM model is found with six KL terms. For the scenario with 3360 decision variables, the optimal performance of the SOM model is obtained with 11 KL terms

Evolutionary Algorithms for Static and Dynamic Multiobjective Optimization

Many real-world optimization problems consist of a number of conflicting objectives that have to be optimized simultaneously. Due to the presence of multiple conflicting ob- jectives, there is no single solution that can optimize all the objectives. Therefore, the resulting multiobjective optimization problems (MOPs) resort to a set of trade-off op- timal solutions, called the Pareto set in the decision space and the Pareto front in the objective space. Traditional optimization methods can at best find one solution in a sin- gle run, thereby making them inefficient to solve MOPs. In contrast, evolutionary algo- rithms (EAs) are able to approximate multiple optimal solutions in a single run. This strength makes EAs good candidates for solving MOPs. Over the past several decades, there have been increasing research interests in developing EAs or improving their perfor- mance, resulting in a large number of contributions towards the applicability of EAs for MOPs. However, the performance of EAs depends largely on the properties of the MOPs in question, e.g., static/dynamic optimization environments, simple/complex Pareto front characteristics, and low/high dimensionality. Different problem properties may pose dis- tinct optimization difficulties to EAs. For example, dynamic (time-varying) MOPs are generally more challenging than static ones to EAs. Therefore, it is not trivial to further study EAs in order to make them widely applicable to MOPs with various optimization scenarios or problem properties. This thesis is devoted to exploring EAs’ ability to solve a variety of MOPs with dif- ferent problem characteristics, attempting to widen EAs’ applicability and enhance their general performance. To start with, decomposition-based EAs are enhanced by incorpo- rating two-phase search and niche-guided solution selection strategies so as to make them suitable for solving MOPs with complex Pareto fronts. Second, new scalarizing functions are proposed and their impacts on evolutionary multiobjective optimization are exten- sively studied. On the basis of the new scalarizing functions, an efficient decomposition- based EA is introduced to deal with a class of hard MOPs. Third, a diversity-first- and-convergence-second sorting method is suggested to handle possible drawbacks of convergence-first based sorting methods. The new sorting method is then combined with strength based fitness assignment, with the aid of reference directions, to optimize MOPs with an increase of objective dimensionality. After that, we study the field of dynamic multiobjective optimization where objective functions and constraints can change over time. A new set of test problems consisting of a wide range of dynamic characteristics is introduced at an attempt to standardize test environments in dynamic multiobjective optimization, thereby aiding fair algorithm comparison and deep performance analysis. Finally, a dynamic EA is developed to tackle dynamic MOPs by exploiting the advan- tages of both generational and steady-state algorithms. All the proposed approaches have been extensively examined against existing state-of-the-art methods, showing fairly good performance in a variety of test scenarios. The research work presented in the thesis is the output of initiative and novel attempts to tackle some challenging issues in evolutionary multiobjective optimization. This re- search has not only extended the applicability of some of the existing approaches, such as decomposition-based or Pareto-based algorithms, for complex or hard MOPs, but also contributed to moving forward research in the field of dynamic multiobjective optimiza- tion with novel ideas including new test suites and novel algorithm design

System Architecture Design Using Multi-Criteria Optimization

System architecture is defined as the description of a complex system in terms of its functional requirements, physical elements and their interrelationships. Designing a complex system architecture can be a difficult task involving multi-faceted trade-off decisions. The system architecture designs often have many project-specific goals involving mix of quantitative and qualitative criteria and a large design trade space. Several tools and methods have been developed to support the system architecture design process in the last few decades. However, many conventional problem solving techniques face difficulties in dealing with complex system design problems having many goals. In this research work, an interactive multi-criteria design optimization framework is proposed for solving many-objective system architecture design problems and generating a well distributed set of Pareto optimal solutions for these problems. System architecture design using multi-criteria optimization is demonstrated using a real-world application of an aero engine health management (EHM) system. A design process is presented for the optimal deployment of the EHM system functional operations over physical architecture subsystems. The EHM system architecture design problem is formulated as a multi-criteria optimization problem. The proposed methodology successfully generates a well distributed family of Pareto optimal architecture solutions for the EHM system, which provides valuable insights into the design trade-offs. Uncertainty analysis is implemented using an efficient polynomial chaos approach and robust architecture solutions are obtained for the EHM system architecture design. Performance assessment through evaluation of benchmark test metrics demonstrates the superior performance of the proposed methodology

Improved max-value entropy search for multi-objective bayesian optimization with constraints

We present MESMOC+, an improved version of Max-value Entropy search for Multi-Objective Bayesian optimization with Constraints (MESMOC). MESMOC+ can be used to solve constrained multi-objective problems when the objectives and the constraints are expensive to evaluate. It is based on minimizing the entropy of the solution of the optimization problem in function space (i.e., the Pareto front) to guide the search for the optimum. The cost of MESMOC+ is linear in the number of objectives and constraints. Furthermore, it is often significantly smaller than the cost of alternative methods based on minimizing the entropy of the Pareto set. The reason for this is that it is easier to approximate the required computations in MESMOC+. Moreover, MESMOC+’s acquisition function is expressed as the sum of one acquisition per each black-box (objective or constraint). Therefore, it can be used in a decoupled evaluation setting in which it is chosen not only the next input location to evaluate, but also which black-box to evaluate there. We compare MESMOC+ with related methods in synthetic, benchmark and real optimization problems. These experiments show that MESMOC+ has similar performance to that of state-of-the-art acquisitions based on entropy search, but it is faster to execute and simpler to implement. Moreover, our experiments also show that MESMOC+ is more robust with respect to the number of samples of the Pareto frontThe authors acknowledge financial support from Spanish Plan Nacional I + D+i, grant PID2019-106827 GB-I00/ AEI/ 10.13039/50110001103

Non-dominated sorting Harris’s hawk multi-objective optimizer based on reference point approach

A non-dominated sorting Harris’s hawk multi- objective optimizer (NDSHHMO) algorithm is presented in this paper. The algorithm is able to improve the population diversity, convergence of non-dominated solutions toward the Pareto front, and prevent the population from trapping into local optimal. This was achieved by integrating fast non-dominated sorting with the original Harris’s hawk multi-objective optimizer (HHMO). Non-dominated sorting divides the objective space into levels based on fitness values and then selects non-dominated solutions to produce the next generation of hawks. A set of well-known multi-objective optimization problems has been used to evaluate the performance of the proposed NDSHHMO algorithm. The results of the NDSHHMO algorithm were verified against the results of an HHMO algorithm. Experimental results demonstrate the efficiency of the proposed NDSHHMO algorithm in terms of enhancing the ability of convergence toward the Pareto front and significantly improve the search ability of the HHMO

Antecipação na tomada de decisão com múltiplos critérios sob incerteza

Orientador: Fernando José Von ZubenTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: A presença de incerteza em resultados futuros pode levar a indecisões em processos de escolha, especialmente ao elicitar as importâncias relativas de múltiplos critérios de decisão e de desempenhos de curto vs. longo prazo. Algumas decisões, no entanto, devem ser tomadas sob informação incompleta, o que pode resultar em ações precipitadas com consequências imprevisíveis. Quando uma solução deve ser selecionada sob vários pontos de vista conflitantes para operar em ambientes ruidosos e variantes no tempo, implementar alternativas provisórias flexíveis pode ser fundamental para contornar a falta de informação completa, mantendo opções futuras em aberto. A engenharia antecipatória pode então ser considerada como a estratégia de conceber soluções flexíveis as quais permitem aos tomadores de decisão responder de forma robusta a cenários imprevisíveis. Essa estratégia pode, assim, mitigar os riscos de, sem intenção, se comprometer fortemente a alternativas incertas, ao mesmo tempo em que aumenta a adaptabilidade às mudanças futuras. Nesta tese, os papéis da antecipação e da flexibilidade na automação de processos de tomada de decisão sequencial com múltiplos critérios sob incerteza é investigado. O dilema de atribuir importâncias relativas aos critérios de decisão e a recompensas imediatas sob informação incompleta é então tratado pela antecipação autônoma de decisões flexíveis capazes de preservar ao máximo a diversidade de escolhas futuras. Uma metodologia de aprendizagem antecipatória on-line é então proposta para melhorar a variedade e qualidade dos conjuntos futuros de soluções de trade-off. Esse objetivo é alcançado por meio da previsão de conjuntos de máximo hipervolume esperado, para a qual as capacidades de antecipação de metaheurísticas multi-objetivo são incrementadas com rastreamento bayesiano em ambos os espaços de busca e dos objetivos. A metodologia foi aplicada para a obtenção de decisões de investimento, as quais levaram a melhoras significativas do hipervolume futuro de conjuntos de carteiras financeiras de trade-off avaliadas com dados de ações fora da amostra de treino, quando comparada a uma estratégia míope. Além disso, a tomada de decisões flexíveis para o rebalanceamento de carteiras foi confirmada como uma estratégia significativamente melhor do que a de escolher aleatoriamente uma decisão de investimento a partir da fronteira estocástica eficiente evoluída, em todos os mercados artificiais e reais testados. Finalmente, os resultados sugerem que a antecipação de opções flexíveis levou a composições de carteiras que se mostraram significativamente correlacionadas com as melhorias observadas no hipervolume futuro esperado, avaliado com dados fora das amostras de treinoAbstract: The presence of uncertainty in future outcomes can lead to indecision in choice processes, especially when eliciting the relative importances of multiple decision criteria and of long-term vs. near-term performance. Some decisions, however, must be taken under incomplete information, what may result in precipitated actions with unforeseen consequences. When a solution must be selected under multiple conflicting views for operating in time-varying and noisy environments, implementing flexible provisional alternatives can be critical to circumvent the lack of complete information by keeping future options open. Anticipatory engineering can be then regarded as the strategy of designing flexible solutions that enable decision makers to respond robustly to unpredictable scenarios. This strategy can thus mitigate the risks of strong unintended commitments to uncertain alternatives, while increasing adaptability to future changes. In this thesis, the roles of anticipation and of flexibility on automating sequential multiple criteria decision-making processes under uncertainty are investigated. The dilemma of assigning relative importances to decision criteria and to immediate rewards under incomplete information is then handled by autonomously anticipating flexible decisions predicted to maximally preserve diversity of future choices. An online anticipatory learning methodology is then proposed for improving the range and quality of future trade-off solution sets. This goal is achieved by predicting maximal expected hypervolume sets, for which the anticipation capabilities of multi-objective metaheuristics are augmented with Bayesian tracking in both the objective and search spaces. The methodology has been applied for obtaining investment decisions that are shown to significantly improve the future hypervolume of trade-off financial portfolios for out-of-sample stock data, when compared to a myopic strategy. Moreover, implementing flexible portfolio rebalancing decisions was confirmed as a significantly better strategy than to randomly choosing an investment decision from the evolved stochastic efficient frontier in all tested artificial and real-world markets. Finally, the results suggest that anticipating flexible choices has lead to portfolio compositions that are significantly correlated with the observed improvements in out-of-sample future expected hypervolumeDoutoradoEngenharia de ComputaçãoDoutor em Engenharia Elétric