Traditional simulation-based protein design considers energy minimization of candidate conformations as a singleobjective combinatorial optimization problem. In this paper we consider a challenging protein design problem, producing twelve protein species based on collagen that uniquely assort into four groups of three: a problem defined herein as a 4-level heterotrimer. We formulate a bi-objective combinatorial minimization problem that targets both stability and specificity of the 4-level heterotrimer. In order to approximate its Pareto frontier, we utilize both evolutionary and non-evolutionary approaches, operating in either Pareto or aggregation fashions. Our practical observations suggest that the SMS-EMOA with Evolution Strategies' operators is more effective than standard heuristics deployed in computational protein design, such as Simulated Annealing, Replica Exchange or the Canonical Genetic Algorithm. We investigate the attained Pareto optimal sets using Barrier Tree analysis, aiming to provide insights into the chemical search-space, as well as to explain the observed algorithmic trends. In particular, we identify Replica Exchange as a promising non-evolutionary technique for this problem class, due to its efficient exploration capabilities. Overall, a common high-level protocol for simultaneous landscape analysis of evolutionary and non-evolutionary search methodologies is put forward for the first time.
In this paper, two novel algorithms for features selection are proposed. The first one is a filter method while the second is wrapper method. Both the proposed algorithms use the crowding distance used in the multiobjective optimization as a metric in order to sort the features. The less crowded features have great effects on the target attribute (class). The experimental results have shown the effectiveness and the robustness of the proposed algorithms.
More and more, optimization methods are used to find diverse solution sets. We compare solution diversity in multi-objective optimization, multimodal optimization, and quality diversity in a simple domain. We show that multiobjective optimization does not always produce much diversity, multimodal optimization produces higher fitness solutions, and quality diversity is not sensitive to genetic neutrality and creates the most diverse set of solutions. An autoencoder is used to discover phenotypic features automatically, producing an even more diverse solution set with quality diversity. Finally, we make recommendations about when to use which approach.
Deep reinforcement learning (DRL) has recently shown its success in tackling complex combinatorial optimization problems. When these problems are extended to multiobjective ones, it becomes difficult for the existing DRL approaches to flexibly and efficiently deal with multiple subproblems determined by weight decomposition of objectives. This paper proposes a concise meta-learning-based DRL approach. It first trains a meta-model by meta-learning. The meta-model is fine-tuned with a few update steps to derive submodels for the corresponding subproblems. The Pareto front is then built accordingly. Compared with other learning-based methods, our method can greatly shorten the training time of multiple submodels. Due to the rapid and excellent adaptability of the meta-model, more submodels can be derived so as to increase the quality and diversity of the found solutions. The computational experiments on multiobjective traveling salesman problems and multiobjective vehicle routing problem with time windows demonstrate the superiority of our method over most of learning-based and iteration-based approaches.
Multiobjective optimization problems with heterogeneous objectives are defined as those that possess significantly different types of objective function components (not just incommensurable in units or scale). For example, in a heterogeneous problem the objective function components may differ in formal computational complexity, practical evaluation effort (time, costs, or resources), determinism (stochastic vs deterministic), or some combination of all three. A particularly challenging variety of heterogeneity may occur by the combination of a time-consuming laboratory-based objective with other objectives that are evaluated using faster computer-based calculations. Perhaps more commonly, all objectives may be evaluated computationally, but some may require a lengthy simulation process while others are computed from a relatively simple closed-form calculation. In this chapter, we motivate the need for more work on the topic of heterogeneous objectives (with reference to real-world examples), expand on a basic taxonomy of heterogeneity types, and review the state of the art in tackling these problems. We give special attention to heterogeneity in evaluation time (latency) as this requires sophisticated approaches. We also present original experimental work on estimating the amount of heterogeneity in evaluation time expected in many-objective problems, given reasonable assumptions, and survey related research threads that could contribute to this area in future.
Real-world multiobjective optimization problems usually involve conflicting objectives that change over time, which requires the optimization algorithms to quickly track the Pareto optimal front (POF) when the environment changes. In recent years, evolutionary algorithms based on prediction models have been considered promising. However, most existing approaches only make predictions based on the linear correlation between a finite number of optimal solutions in two or three previous environments. These incomplete information extraction strategies may lead to low prediction accuracy in some instances. In this paper, a novel prediction algorithm based on incremental support vector machine (ISVM) is proposed, called ISVM-DMOEA. We treat the solving of dynamic multiobjective optimization problems (DMOPs) as an online learning process, using the continuously obtained optimal solution to update an incremental support vector machine without discarding the solution information at earlier time. ISVM is then used to filter random solutions and generate an initial population for the next moment. To overcome the obstacle of insufficient training samples, a synthetic minority oversampling strategy is implemented before the training of ISVM. The advantage of this approach is that the nonlinear correlation between solutions can be explored online by ISVM, and the information contained in all historical optimal solutions can be exploited to a greater extent. The experimental results and comparison with chosen state-of-the-art algorithms demonstrate that the proposed algorithm can effectively tackle dynamic multiobjective optimization problems.
In this paper, we approach the problem of optimizing blackbox functions over large hybrid search spaces consisting of both combinatorial and continuous parameters. We demonstrate that previous evolutionary algorithms which rely on mutation-based approaches, while flexible over combinatorial spaces, suffer from a curse of dimensionality in high dimensional continuous spaces both theoretically and empirically, which thus limits their scope over hybrid search spaces as well. In order to combat this curse, we propose ES-ENAS, a simple and modular joint optimization procedure combining the class of sample-efficient smoothed gradient techniques, commonly known as Evolutionary Strategies (ES), with combinatorial optimizers in a highly scalable and intuitive way, inspired by the one-shot or supernet paradigm introduced in Efficient Neural Architecture Search (ENAS). By doing so, we achieve significantly more sample efficiency, which we empirically demonstrate over synthetic benchmarks, and are further able to apply ES-ENAS for architecture search over popular RL benchmarks.
Large-scale multiobjective optimization problems (LSMOPs) are characterized as involving hundreds or even thousands of decision variables and multiple conflicting objectives. An excellent algorithm for solving LSMOPs should find Pareto-optimal solutions with diversity and escape from local optima in the large-scale search space. Previous research has shown that these optimal solutions are uniformly distributed on the manifold structure in the low-dimensional space. However, traditional evolutionary algorithms for solving LSMOPs have some deficiencies in dealing with this structural manifold, resulting in poor diversity, local optima, and inefficient searches. In this work, a generative adversarial network (GAN)-based manifold interpolation framework is proposed to learn the manifold and generate high-quality solutions on this manifold, thereby improving the performance of evolutionary algorithms. We compare the proposed algorithm with several state-of-the-art algorithms on large-scale multiobjective benchmark functions. Experimental results have demonstrated the significant improvements achieved by this framework in solving LSMOPs.
Techniques to reduce the energy burden of an industrial ecosystem often require solving a multiobjective optimization problem. However, collecting experimental data can often be either expensive or time-consuming. In such cases, statistical methods can be helpful. This article proposes Pareto Front (PF) and Pareto Set (PS) estimation methods using Bayesian Additive Regression Trees (BART), which is a non-parametric model whose assumptions are typically less restrictive than popular alternatives, such as Gaussian Processes (GPs). These less restrictive assumptions allow BART to handle scenarios (e.g. high-dimensional input spaces, nonsmooth responses, large datasets) that GPs find difficult. The performance of our BART-based method is compared to a GP-based method using analytic test functions, demonstrating convincing advantages. Finally, our BART-based methodology is applied to a motivating engineering problem. Supplementary materials, which include a theorem proof, algorithms, and R code, for this article are available online.
We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the $\ell_1$-norm as a weighted penalty term. In order to gain a better understanding and to allow for an informed model selection, we directly solve the corresponding multiobjective optimization problem (MOP) that arises when we minimize the main objective and the $\ell_1$-norm simultaneously. As this MOP is in general non-convex for nonlinear objectives, the weighting method will fail to provide all optimal compromises. To avoid this issue, we present a continuation method which is specifically tailored to MOPs with two objective functions one of which is the $\ell_1$-norm. Our method can be seen as a generalization of well-known homotopy methods for linear regression problems to the nonlinear case. Several numerical examples - including neural network training - demonstrate our theoretical findings and the additional insight that can be gained by this multiobjective approach.
The theoretical understanding of MOEAs is lagging far behind their success in practice. In particular, previous theory work considers mostly easy problems that are composed of unimodal objectives. As a first step towards a deeper understanding of how evolutionary algorithms solve multimodal multiobjective problems, we propose the OJZJ problem, a bi-objective problem composed of two objectives isomorphic to the classic jump function benchmark. We prove that SEMO with probability one does not compute the full Pareto front, regardless of the runtime. In contrast, for all problem sizes $n$ and all jump sizes ${k \in [4..\frac n2 - 1]}$, the global SEMO (GSEMO) covers the Pareto front in an expected number of $\Theta((n-2k)n^{k})$ iterations. For $k = o(n)$, we also show the tighter bound $\frac 32 e n^{k+1} \pm o(n^{k+1})$, which might be the first runtime bound for an MOEA that is tight apart from lower-order terms. We also combine the GSEMO with two approaches that showed advantages in single-objective multimodal problems. When using the GSEMO with a heavy-tailed mutation operator, the expected runtime improves by a factor of at least $k^{\Omega(k)}$. When adapting the recent stagnation-detection strategy of Rajabi and Witt (2022) to the GSEMO, the expected runtime also improves by a factor of at least $k^{\Omega(k)}$ and surpasses the heavy-tailed GSEMO by a small polynomial factor in $k$. Via an experimental analysis, we show that these asymptotic differences are visible already for small problem sizes: A factor-$5$ speed-up from heavy-tailed mutation and a factor-$10$ speed-up from stagnation detection can be observed already for jump size~$4$ and problem sizes between $10$ and $50$. Overall, our results show that the ideas recently developed to aid single-objective evolutionary algorithms to cope with local optima can be effectively employed also in multiobjective optimization.
The synthesis of a metasurface exhibiting a specific set of desired scattering properties is a time-consuming and resource-demanding process, which conventionally relies on many cycles of full-wave simulations. It requires an experienced designer to choose the number of the metallic layers, the scatterer shapes and dimensions, and the type and the thickness of the separating substrates. Here, we propose a generative machine learning (ML)-based approach to solve this one-to-many mapping and automate the inverse design of dual- and triple-layer metasurfaces. Using this approach, it is possible to solve multiobjective optimization problems by synthesizing thin structures composed of potentially brand-new scatterer designs, in cases where the inter-layer coupling between the layers is non-negligible and synthesis by traditional methods becomes cumbersome. Various examples to provide specific magnitude and phase responses of $x$- and $y$-polarized scattering coefficients across a frequency range as well as mask-based responses for different metasurface applications are presented to verify the practicality of the proposed method.
Demand response for residential users is essential to the realization of modern smart grids. This paper proposes a multiobjective approach to designing a demand response program that considers the energy costs of residential users and the load factor of the underlying grid. A multiobjective optimization problem (MOP) is formulated and Pareto optimality is adopted. Stochastic search methods of generating feasible values for decision variables are proposed. Theoretical analysis is performed to show that the proposed methods can effectively generate and preserve feasible points during the solution process, which comparable methods can hardly achieve. A multiobjective evolutionary algorithm is developed to solve the MOP, producing a Pareto optimal demand response (PODR) program. Simulations reveal that the proposed method outperforms the comparable methods in terms of energy costs while producing a satisfying load factor. The proposed PODR program is able to systematically balance the needs of the grid and residential users.
Chemical kinetics and reaction engineering consists of the phenomenological framework for the disentanglement of reaction mechanisms, optimization of reaction performance and the rational design of chemical processes. Here, we utilize feed-forward artificial neural networks as basis functions to solve ordinary differential equations (ODEs) constrained by differential algebraic equations (DAEs) that describe microkinetic models (MKMs). We present an algebraic framework for the mathematical description and classification of reaction networks, types of elementary reaction, and chemical species. Under this framework, we demonstrate that the simultaneous training of neural nets and kinetic model parameters in a regularized multi-objective optimization setting leads to the solution of the inverse problem through the estimation of kinetic parameters from synthetic experimental data. We analyze a set of scenarios to establish the extent to which kinetic parameters can be retrieved from transient kinetic data, and assess the robustness of the methodology with respect to statistical noise. This approach to inverse kinetic ODEs can assist in the elucidation of reaction mechanisms based on transient data.
We study the problem of learning the objective functions or constraints of a multiobjective decision making model, based on a set of sequentially arrived decisions. In particular, these decisions might not be exact and possibly carry measurement noise or are generated with the bounded rationality of decision makers. In this paper, we propose a general online learning framework to deal with this learning problem using inverse multiobjective optimization. More precisely, we develop two online learning algorithms with implicit update rules which can handle noisy data. Numerical results show that both algorithms can learn the parameters with great accuracy and are robust to noise.
A multi-task learning (MTL) system aims at solving multiple related tasks at the same time. With a fixed model capacity, the tasks would be conflicted with each other, and the system usually has to make a trade-off among learning all of them together. For many real-world applications where the trade-off has to be made online, multiple models with different preferences over tasks have to be trained and stored. This work proposes a novel controllable Pareto multi-task learning framework, to enable the system to make real-time trade-off control among different tasks with a single model. To be specific, we formulate the MTL as a preference-conditioned multiobjective optimization problem, with a parametric mapping from preferences to the corresponding trade-off solutions. A single hypernetwork-based multi-task neural network is built to learn all tasks with different trade-off preferences among them, where the hypernetwork generates the model parameters conditioned on the preference. For inference, MTL practitioners can easily control the model performance based on different trade-off preferences in real-time. Experiments on different applications demonstrate that the proposed model is efficient for solving various MTL problems.
Multi-objective optimization (MOO) problems are prevalent in machine learning. These problems have a set of optimal solutions, called the Pareto front, where each point on the front represents a different trade-off between possibly conflicting objectives. Recent MOO methods can target a specific desired ray in loss space however, most approaches still face two grave limitations: (i) A separate model has to be trained for each point on the front; and (ii) The exact trade-off must be known before the optimization process. Here, we tackle the problem of learning the entire Pareto front, with the capability of selecting a desired operating point on the front after training. We call this new setup Pareto-Front Learning (PFL). We describe an approach to PFL implemented using HyperNetworks, which we term Pareto HyperNetworks (PHNs). PHN learns the entire Pareto front simultaneously using a single hypernetwork, which receives as input a desired preference vector and returns a Pareto-optimal model whose loss vector is in the desired ray. The unified model is runtime efficient compared to training multiple models and generalizes to new operating points not used during training. We evaluate our method on a wide set of problems, from multi-task regression and classification to fairness. PHNs learn the entire Pareto front at roughly the same time as learning a single point on the front and at the same time reach a better solution set. Furthermore, we show that PHNs can scale to generate large models like ResNet18. PFL opens the door to new applications where models are selected based on preferences that are only available at run time.
Inverse multiobjective optimization provides a general framework for the unsupervised learning task of inferring parameters of a multiobjective decision making problem (DMP), based on a set of observed decisions from the human expert. However, the performance of this framework relies critically on the availability of an accurate DMP, sufficient decisions of high quality, and a parameter space that contains enough information about the DMP. To hedge against the uncertainties in the hypothetical DMP, the data, and the parameter space, we investigate in this paper the distributionally robust approach for inverse multiobjective optimization. Specifically, we leverage the Wasserstein metric to construct a ball centered at the empirical distribution of these decisions. We then formulate a Wasserstein distributionally robust inverse multiobjective optimization problem (WRO-IMOP) that minimizes a worst-case expected loss function, where the worst case is taken over all distributions in the Wasserstein ball. We show that the excess risk of the WRO-IMOP estimator has a sub-linear convergence rate. Furthermore, we propose the semi-infinite reformulations of the WRO-IMOP and develop a cutting-plane algorithm that converges to an approximate solution in finite iterations. Finally, we demonstrate the effectiveness of our method on both a synthetic multiobjective quadratic program and a real world portfolio optimization problem.
Overparameterization and overfitting are common concerns when designing and training deep neural networks, that are often counteracted by pruning and regularization strategies. However, these strategies remain secondary to most learning approaches and suffer from time and computational intensive procedures. We suggest a multiobjective perspective on the training of neural networks by treating its prediction accuracy and the network complexity as two individual objective functions in a biobjective optimization problem. As a showcase example, we use the cross entropy as a measure of the prediction accuracy while adopting an l1-penalty function to assess the total cost (or complexity) of the network parameters. The latter is combined with an intra-training pruning approach that reinforces complexity reduction and requires only marginal extra computational cost. From the perspective of multiobjective optimization, this is a truly large-scale optimization problem. We compare two different optimization paradigms: On the one hand, we adopt a scalarization-based approach that transforms the biobjective problem into a series of weighted-sum scalarizations. On the other hand we implement stochastic multi-gradient descent algorithms that generate a single Pareto optimal solution without requiring or using preference information. In the first case, favorable knee solutions are identified by repeated training runs with adaptively selected scalarization parameters. Preliminary numerical results on exemplary convolutional neural networks confirm that large reductions in the complexity of neural networks with neglibile loss of accuracy are possible.
We propose the cone epsilon-dominance approach to improve convergence and diversity in multiobjective evolutionary algorithms (MOEAs). A cone-eps-MOEA is presented and compared with MOEAs based on the standard Pareto relation (NSGA-II, NSGA-II*, SPEA2, and a clustered NSGA-II) and on the epsilon-dominance (eps-MOEA). The comparison is performed both in terms of computational complexity and on four performance indicators selected to quantify the quality of the final results obtained by each algorithm: the convergence, diversity, hypervolume, and coverage of many sets metrics. Sixteen well-known benchmark problems are considered in the experimental section, including the ZDT and the DTLZ families. To evaluate the possible differences amongst the algorithms, a carefully designed experiment is performed for the four performance metrics. The results obtained suggest that the cone-eps-MOEA is capable of presenting an efficient and balanced performance over all the performance metrics considered. These results strongly support the conclusion that the cone-eps-MOEA is a competitive approach for obtaining an efficient balance between convergence and diversity to the Pareto front, and as such represents a useful tool for the solution of multiobjective optimization problems.