The Expected R2-Indicator Improvement for Multi-objective Bayesian Optimization

We propose a surrogate-assisted reference vector adaptation (SRVA) method to\nsolve expensive multi- and many-objective optimization problems with various\nPareto front shapes. SRVA is coupled with a multi-objective Bayesian\noptimization (MBO) algorithm using reference vectors for scalarization of\nobjective functions. The Kriging surrogate models for MBO is used to estimate\nthe Pareto front shape and generate adaptive reference vectors uniformly\ndistributed on the estimated Pareto front. We combine SRVA with expected\nimprovement of penalty-based boundary intersection as an infill criterion for\nMBO. The proposed algorithm is compared with two other MBO algorithms by\napplying them to benchmark problems with various Pareto front shapes.\nExperimental results show that the proposed algorithm outperforms the other two\nin the problems whose objective functions are reasonably approximated by the\nKriging models. SRVA improves diversity of non-dominated solutions for these\nproblems with continuous, discontinuous, and degenerated Pareto fronts.\nBesides, the proposed algorithm obtains much better solutions from early stages\nof optimization especially in many-objective problems.\n

Pressure swing adsorption (PSA) is an energy-efficient technology for gas separation, while the multi-objective optimization of PSA is a challenging task. To tackle this, we propose a hybrid optimization framework, which integrates three steps. In the first step, we establish surrogate models for the constraints using Gaussian processes (GPs) and employ multi-objective Bayesian optimization to search for feasible points that satisfy the constraints. In the second step, we establish surrogate models for the objective function and constraints using GPs and utilize constrained multi-objective Bayesian optimization to search for an approximate Pareto front. In the third step, we perform a local search based on the approximate Pareto front. By employing the trust region filter method, we construct quadratic models for each constraint and objective function and refine the Pareto front to achieve local optimality. This framework demonstrates the efficiency of Bayesian optimization and the local optimality of the trust region method. A comparison with the popular evolutionary algorithm, Nondominated Sorting Genetic Algorithm II (NSGA-II), showed that this framework had a higher hypervolume of the Pareto front while halving the runtime and reducing the number of simulations by a factor of 20.

Multi-Objective Bayesian Global Optimization using expected hypervolume improvement gradient

Due to the adversarial limitations between the output solutions in the multi-objective optimization stage, the energy efficiency optimization of large pumping stations has the disadvantage of difficult coordination between power consumption and efficiency. Therefore, a multi-objective Bayesian hyperparameter optimization algorithm design for energy efficiency optimization of large pumping stations has been carried out. A multi-objective optimization model for energy efficiency of large pumping stations was constructed by combining the input power of the main motor, transmission energy loss, transformer loss, and energy consumption of auxiliary equipment, with efficiency as the constraint. We extract sequence features that affect pump water demand, and use feature labels as hyperparameters for the multi-objective optimization model of energy efficiency in large pumping stations. We transform the multi-objective optimization problem into a hyperparameter optimization problem, avoiding adversarial output solutions. We iterate the query points collected by EI, so the output model reaches a convergent optimal solution. The test results show that the designed algorithm can improve the pumping efficiency of the pump station while controlling the power consumption of the pump station, achieving collaborative optimization of efficiency.

Bayesian optimization (BO) methods solve problems with several black-box objectives and constraints.Each black-box is expensive to evaluate and lacks a closed-form.They use a model of each black-box to guide the search for the problem's solution.Sometimes, however, the black-boxes may be evaluated at different fidelity levels.A lower fidelity is simply a cheap proxy of the corresponding black-box.Thus, lower fidelities that correlate with the actual black-box can be used to reduce the optimization cost.We propose Joint Entropy Search for Multi-Fidelity and Multi-objective Bayesian Optimization with Constraints (MF-JESMOC), a BO method for solving the aforementioned problems.It chooses the next point and fidelity level at which to evaluate the black-boxes as the one that is expected to reduce the most the joint entropy of the Pareto set and the Pareto front, normalized by the fidelity's cost.Deep Gaussian processes are used to model each black-box and dependencies between fidelities.In our experiments, MF-JESMOC outperforms other state-of-the-art methods for multi-objective BO with constraints and different fidelity levels.

Surrogate-based method can dramatically reduce the number of expensive function evaluations in real-world multi- objective optimization problems (MOP). When the number of objectives is small, using surrogate models combined with expected hypervolume improvement (EHVI) infill sampling criterion (ISC) has been proved to be efficient to provide a set of solutions with good diversity and good proximity to the Pareto front (PF) in aerodynamic shape optimization. However, traditional hypervolume-based infilling strategies use only one kind of ISC to generate one or multiple sample points, the advantages of various kinds of ISC cannot be comprehensively utilized and the parallelization is not easy to implement. This paper proposes a combined multi-point infilling strategy based on Kriging models and develops an efficient global multi-objective constrained optimization method (EGMOCO) to solve multi- objective aerodynamic shape optimization with complex constraints. Multiple sample points are generated by using four ISC considering hypervolume at each iteration and then evaluated in parallel. Firstly, the performance of EGMOCO is compared with that of single criterion EHVI strategy on six benchmarks within the same computational budget to prove its effectiveness, and then EGMOCO is implemented in an aerodynamic shape optimization problem with complex constraints. The result shows that EGMOCO has good performance in balancing local exploitation and global exploration with faster convergence rate and high robustness, the whole PF can be fully explored in limited evaluations and the constraint handling is effective especially for real-world problems with complex and nonlinear constraints, the comprehensive aerodynamic performance of the airfoil is greatly improved. It can be confirmed that Kriging-based multi-objective optimization method combined with multi-point infilling strategy performs better than single infilling criterion EHVI, since different sample infilling criteria can complement with each other, both local exploitation and global exploration can be considered and well balanced.

The physical models governing surfactant-polymer (SP) flooding process are subject to parametric uncertainties, accurate quantification of which is crucial for improved decision making. Moreover, history matching of SP flooding is an ill-posed problem, typically characterized by a multimodal posterior distribution of these model parameters. This paper presents a systematic approach for Bayesian history matching and uncertainty quantification in the model calibration stage of SP flooding using coreflood experimental data. The approach is as follows. First, we construct a surrogate of the computationally expensive physics-based model using a polynomial chaos expansion (PCE-proxy). Second, we formulate a Bayesian calibration problem for inferring the model parameters from a single coreflood experiment that measures pressure drop and oilcut profiles. Third, we solve the Bayesian calibration problem by sampling directly from the posterior using Markov chain Monte Carlo (MCMC). We validate the calibrated parameters by successfully predicting the result of two other coreflood experiments. Then, we extend this framework to stochastic multiobjective optimization of injection slug size design under uncertainties in model parameters (captured by the posterior of Bayesian calibration problem) and oil price (modeled as a geometric random walk with constant drift and volatility). To identify the Pareto frontier of the stochastic multiobjective optimization problem, we employ a variant of Bayesian global optimization (BGO), a class of algorithms capable of optimizing black-box, gradient-free, computationally expensive functions. In particular, we use the extended expected improvement over the dominated hypervolume to sequentially select simulations that seek to reveal the Pareto frontier. An addendum of the implemented BGO is that it quantifies the epistemic uncertainty about the Pareto frontier as induced by the limited number of simulations used to construct it.

We report a human-in-the-loop implementation of the multi-objective experimental design via a Bayesian optimization platform (EDBO+) towards the optimization of butylpyridinium bromide synthesis under continuous flow conditions. The algorithm simultaneously optimized reaction yield and production rate (or space-time yield) and generated a well defined Pareto front. The versatility of EDBO+ was demonstrated by expanding the reaction space mid-campaign by increasing the upper temperature limit. Incorporation of continuous flow techniques enabled improved control over reaction parameters compared to common batch chemistry processes, while providing a route towards future automated syntheses and improved scalability. To that end, we applied the open-source Python module, nmrglue, for semi-automated nuclear magnetic resonance (NMR) spectroscopy analysis, and compared the acquired outputs against those obtained through manual processing methods from spectra collected on both low-field (60 MHz) and high-field (400 MHz) NMR spectrometers. The EDBO+ based model was retrained with these four different datasets and the resulting Pareto front predictions provided insight into the effect of data analysis on model predictions. Finally, quaternization of poly(4-vinylpyridine) with bromobutane illustrated the extension of continuous flow chemistry to synthesize functional materials.

Rather than optimizing expensive objective functions such as complex engineering simulations directly, Bayesian optimization methodologies fit a surrogate model (typically Kriging or a Gaussian Process) on evaluations of the objective function(s). To determine the next evaluation, an acquisition function is optimized (also referred to as infill criterion or sampling policy) which incorporates the model prediction and uncertainty and balances exploration and exploitation. Therefore, Bayesian optimization methodologies replace a single optimization of the objective function by a sequence of optimization problems: this makes sense as the acquisition function is cheap-to-evaluate whereas the objective is not. Depending on the goal different acquisition functions are available: multi-objective acquisition functions are relatively new and this chapter gives a state-of-the-art overview and illustrates some approaches based on hypervolume improvement. It is shown that the quality of the model is crucial for the performance of Bayesian optimization and illustrate this by using the more flexible Student-t processes as surrogate models.

Optimizing multiple conflicting black-box objectives simultaneously is a prevalent occurrence in many real-world applications, such as neural architecture search, and machine learning. These problems are known as expensive multi-objective optimization problems (EMOPs) when the function evaluations are computationally or financially costly. Multi-objective Bayesian optimization (MOBO) offers an efficient approach to discovering a set of Pareto optimal solutions. However, the data deficiency issue caused by limited function evaluations has posed a great challenge to current optimization methods. Moreover, most current methods tend to prioritize the quality of candidate solutions, while ignoring the quantity of promising samples. In order to tackle these issues, our paper proposes a novel multi-objective Bayesian optimization algorithm with a data augmentation strategy that provides ample high-quality samples for Pareto set learning (PSL). Specifically, it utilizes Generative Adversarial Networks (GANs) to enrich data and a dominance prediction model to screen out high-quality samples, mitigating the predicament of limited function evaluations in EMOPs. Additionally, we adopt the regularity model to expensive multi-objective Bayesian optimization for PSL. Experimental results on both synthetic and real-world problems demonstrate that our algorithm outperforms several state-of-the-art and classical algorithms.

A diversity indicator based on reference vectors for many-objective optimization

This paper provides a novel framework for solving multiobjective discrete optimization problems with an arbitrary number of objectives. Our framework represents these problems as network models, in that enumerating the Pareto frontier amounts to solving a multicriteria shortest-path problem in an auxiliary network. We design techniques for exploiting network models in order to accelerate the identification of the Pareto frontier, most notably a number of operations to simplify the network by removing nodes and arcs while preserving the set of nondominated solutions. We show that the proposed framework yields orders-of-magnitude performance improvements over existing state-of-the-art algorithms on five problem classes containing both linear and nonlinear objective functions. Summary of Contribution: Multiobjective optimization has a long history of research with applications in several domains. Our paper provides an alternative modeling and solution approach for multiobjective discrete optimization problems by leveraging graphical structures. Specifically, we encode the decision space of a problem as a layered network and propose graph reduction operators to preserve only solutions whose image are part of the Pareto frontier. The nondominated solutions can then be extracted through shortest-path algorithms on such a network. Numerical results comparing our method with state-of-the-art approaches on several problem classes, including the knapsack, set covering, and the traveling salesperson problem (TSP), suggest orders-of-magnitude runtime speed-ups for exactly enumerating the Pareto frontier, especially when the number of objective functions grows.

Wireless cellular networks have many parameters that are normally tuned upon deployment and re-tuned as the network changes. Many operational parameters affect reference signal received power (RSRP), reference signal received quality (RSRQ), signal-to-interference-plus-noise-ratio (SINR), and, ultimately, throughput. In this paper, we develop and compare two approaches for maximizing coverage and minimizing interference by jointly optimizing the transmit power and downtilt (elevation tilt) settings across sectors. To evaluate different parameter configurations offline, we construct a realistic simulation model that captures geographic correlations. Using this model, we evaluate two optimization methods: deep deterministic policy gradient (DDPG), a reinforcement learning (RL) algorithm, and multi-objective Bayesian optimization (BO). Our simulations show that both approaches significantly outperform random search and converge to comparable Pareto frontiers, but that BO converges with two orders of magnitude fewer evaluations than DDPG. Our results suggest that data-driven techniques can effectively self-optimize coverage and capacity in cellular networks.

Global Sensitivity Analysis (GSA) is the study of the influence of any given input on the outputs of a model. In the context of engineering design, GSA has been widely used to understand both individual and collective contributions of design variables on the design objectives. So far, global sensitivity studies have often been limited to design spaces with only quantitative (numerical) design variables. However, many engineering systems also contain, if not only, qualitative (categorical) design variables in addition to quantitative design variables. In this paper, we integrate Latent Variable Gaussian Process (LVGP) with Sobol’ analysis to develop the first metamodel-based mixed-variable GSA method. Through numerical case studies, we validate and demonstrate the effectiveness of our proposed method for mixed-variable problems. Furthermore, while the proposed GSA method is general enough to benefit various engineering design applications, we integrate it with multi-objective Bayesian optimization (BO) to create a sensitivity-aware design framework in accelerating the Pareto front design exploration for metal-organic framework (MOF) materials with many-level combinatorial design spaces. Although MOFs are constructed only from qualitative variables that are notoriously difficult to design, our method can utilize sensitivity analysis to navigate the optimization in the many-level large combinatorial design space, greatly expediting the exploration of novel MOF candidates.

Global Sensitivity Analysis (GSA) is the study of the influence of any given inputs on the outputs of a model. In the context of engineering design, GSA has been widely used to understand both individual and collective contributions of design variables on the design objectives. So far, global sensitivity studies have often been limited to design spaces with only quantitative (numerical) design variables. However, many engineering systems also contain, if not only, qualitative (categorical) design variables in addition to quantitative design variables. In this paper, we integrate the novel Latent Variable Gaussian Process (LVGP) with Sobol’ analysis to develop the first metamodel-based mixed-variable GSA method. Through two analytical case studies, we first validate and demonstrate the effectiveness of our proposed method for mixed-variable problems. Furthermore, while the new metamodel-based mixed-variable GSA method can benefit various engineering design applications, we employ our method with multi-objective Bayesian optimization (BO) to accelerate the Pareto front design exploration in many-level combinatorial design spaces. Specifically, we implement a sensitivity-aware design framework for metal-organic framework (MOF) materials that are constructed only from qualitative design variables and show the benefits of our method for expediting the exploration of novel MOF candidates from a many-level large combinatorial design space.