Analysis of Surrogate-Assisted Information-Geometric Optimization Algorithms

A competitive variable-fidelity surrogate-assisted CMA-ES algorithm using data mining techniques

This paper presents an extended variant of the distance-weighted exponential natural evolution strategy (DXNES) that works well in uncertain environments. Since we often face objective functions with uncertain parameters in real-world problems, function optimization in uncertain environments is an important problem. The covariance matrix adaptation evolution strategy (CMA-ES) and DX-NES have been proposed as promising methods for function optimization in deterministic environments. The performance of these methods, however, deteriorates in uncertain environments. The uncertain handling CMA-ES (TIH-CMA-ES) has been proposed as an extended variant of CMA-ES for uncertain environments and has shown relatively good performance on problems with uncertain parameters. In this paper, we propose an extended variant of DX-NES named DX-NES for uncertain environments (DX-NES-TIE). DX-NES-TIE approximates the objective function by a quadratic function. DXNES-TIE uses approximation function values for updating the mutation distribution if the noise is strong; otherwise it uses observed objective function values. The strength of the noise is quantified by using the approximation function and the evolution path. Through numerical experiments on 20-dimensional uncertain benchmark problems, we demonstrate that DX-NES-TIE can find ten to 2,000 times as accurate solutions as TIH-CMA-ES can. We also apply DX-NES-TIE to 80-dimensional problems and confirm that DX-NES-TIE is scalable with respect to problem dimensionality.

Randomized direct search algorithms for continuous domains, such as evolution strategies, are basic tools in machine learning. They are especially needed when the gradient of an objective function (e.g., loss, energy, or reward function) cannot be computed or estimated efficiently. Application areas include supervised and reinforcement learning as well as model selection. These randomized search strategies often rely on normally distributed additive variations of candidate solutions. In order to efficiently search in non-separable and ill-conditioned landscapes the covariance matrix of the normal distribution must be adapted, amounting to a variable metric method. Consequently, covariance matrix adaptation (CMA) is considered state-of-the-art in evolution strategies. In order to sample the normal distribution, the adapted covariance matrix needs to be decomposed, requiring in general Θ(n 3) operations, where n is the search space dimension. We propose a new update mechanism which can replace a rank-one covariance matrix update and the computationally expensive decomposition of the covariance matrix. The newly developed update rule reduces the computational complexity of the rank-one covariance matrix adaptation to Θ(n 2) without resorting to outdated distributions. We derive new versions of the elitist covariance matrix adaptation evolution strategy (CMA-ES) and the multi-objective CMA-ES. These algorithms are equivalent to the original procedures except that the update step for the variable metric distribution scales better in the problem dimension. We also introduce a simplified variant of the non-elitist CMA-ES with the incremental covariance matrix update and investigate its performance. Apart from the reduced time-complexity of the distribution update, the algebraic computations involved in all new algorithms are simpler compared to the original versions. The new update rule improves the performance of the CMA-ES for large scale machine learning problems in which the objective function can be evaluated fast.

Comparing various metaheuristics based on an equal number of objective function evaluations has become standard practice. Many contemporary publications use a specific number of objective function evaluations by the benchmarking sets definitions. Furthermore, many publications deal with the recurrent theme of late stagnation, which may lead to the impression that continuing the optimization process could be a waste of computational capabilities. But is it? Recently, many challenges, issues, and questions have been raised regarding fair comparisons and recommendations towards good practices for benchmarking metaheuristic algorithms. The aim of this work is not to compare the performance of several well-known algorithms but to investigate the issues that can appear in benchmarking and comparisons of metaheuristics performance (no matter what the problem is). This article studies the impact of a higher evaluation number on a selection of metaheuristic algorithms. We examine the effect of a raised evaluation budget on overall performance, mean convergence, and population diversity of selected swarm algorithms and IEEE CEC competition winners. Even though the final impact varies based on current algorithm selection, it may significantly affect the final verdict of metaheuristics comparison. This work has picked an important benchmarking issue and made extensive analysis, resulting in conclusions and possible recommendations for users working with real engineering optimization problems or researching the metaheuristics algorithms. Especially nowadays, when metaheuristic algorithms are used for increasingly complex optimization problems, and meet machine learning in AutoML frameworks, we conclude that the objective function evaluation budget should be considered another vital optimization input variable.

Particle swarm optimization is a population-based stochastic algorithm designed to solve difficult optimization problems, such as the flexible job shop scheduling problem. This problem consists of scheduling a set of operations on a set of machines while minimizing a certain objective function. This paper presents a two-level particle swarm optimization algorithm for the flexible job shop scheduling problem. The upper level handles the operations-to-machines mapping, while the lower level handles the ordering of operations on machines. A lower bound-checking strategy on the optimal objective function value is used to reduce the number of visited solutions and the number of objective function evaluations. The algorithm is benchmarked against existing state-of-the-art algorithms for the flexible job shop scheduling problem on a significant number of diverse benchmark problems.

Car classification, using different machine learning models with optimization frameworks, is done for evaluation in this work. We used different models, namely, Extra Trees, XGBoost, Gaussian Naive Bayes, K-Nearest Neighbors, Histogram-based Gradient Boosting (Hist Gradient Boosting), and Linear Discriminant Analysis, for classification. The car samples are classified as “very good,” “good,” “acceptable,” and “unacceptable.” Among these, Hist Gradient Boosting has the highest value for precision, accuracy, recall, and F1 score. We further tuned this model using Evolutionary Strategies, Evolutionary Programming, Covariance Matrix Adaptation Evolution Strategy, and the Flower Pollination Algorithm. Our outcomes indicate that the Covariance Matrix Adaptation Evolution Strategy and Flower Pollination Algorithm significantly enhanced the performance of the model and outperformed Evolutionary Programming. This work investigates the potential of integrating advanced machine learning models with sophisticated optimization strategies to deliver an effective car evaluation classification process that will be useful in this industry and, perhaps, in many other classification tasks.

A robust atlas-to-subject registration using a statistical deformation model (SDM) is presented. The SDM uses statistics of voxel-wise displacement learned from pre-computed deformation vectors of a training dataset. This allows an atlas instance to be directly translated into an intensity volume and compared with a patient’s intensity volume. Rigid and nonrigid transformation parameters were simultaneously optimized via the Covariance Matrix Adaptation – Evolutionary Strategy (CMA-ES), with image similarity used as the objective function. The algorithm was tested on CT volumes of the pelvis from 55 female subjects. A performance comparison of the CMA-ES and Nelder-Mead downhill simplex optimization algorithms with the mutual information and normalized cross correlation similarity metrics was conducted. Simulation studies using synthetic subjects were performed, as well as leave-one-out cross validation studies. Both studies suggested that mutual information and CMA-ES achieved the best performance. The leave-one-out test demonstrated 4.13 mm error with respect to the true displacement field, and 26,102 function evaluations in 180 seconds, on average.

This work studies the dual formulation of a penalized maximum likelihood reconstruction problem in x-ray CT. The primal objective function is a Poisson log-likelihood combined with a weighted cross-entropy penalty term. The dual formulation of the primal optimization problem is then derived and the optimization procedure outlined. The dual formulation better exploits the structure of the problem, which translates to faster convergence of iterative reconstruction algorithms. A gradient descent algorithm is implemented for solving the dual problem and its performance is compared with the filtered back-projection algorithm, and with the primal formulation optimized by using surrogate functions. The 3D XCAT phantom and an analytical x-ray CT simulator are used to generate noise-free and noisy CT projection data set with monochromatic and polychromatic x-ray spectrums. The reconstructed images from the dual formulation delineate the internal structures at early iterations better than the primal formulation using surrogate functions. However the body contour is slower to converge in the dual than in the primal formulation. The dual formulation demonstrate better noise-resolution tradeoff near the internal organs than the primal formulation. Since the surrogate functions in general can provide a diagonal approximation of the Hessian matrix of the objective function, further convergence speed up may be achieved by deriving the surrogate function of the dual objective function.

In this study, we investigate the problem of min-max continuous optimization in a black-box setting $\min_{x} \max_{y}f(x,y)$. A popular approach updates $x$ and $y$ simultaneously or alternatingly. However, two major limitations have been reported in existing approaches. (I) As the influence of the interaction term between $x$ and $y$ (e.g., $x^\mathrm{T} B y$) on the Lipschitz smooth and strongly convex-concave function $f$ increases, the approaches converge to an optimal solution at a slower rate. (II) The approaches fail to converge if $f$ is not Lipschitz smooth and strongly convex-concave around the optimal solution. To address these difficulties, we propose minimizing the worst-case objective function $F(x)=\max_{y}f(x,y)$ directly using the covariance matrix adaptation evolution strategy, in which the rankings of solution candidates are approximated by our proposed worst-case ranking approximation (WRA) mechanism. Compared with existing approaches, numerical experiments show two important findings regarding our proposed method. (1) The proposed approach is efficient in terms of $f$-calls on a Lipschitz smooth and strongly convex-concave function with a large interaction term. (2) The proposed approach can converge on functions that are not Lipschitz smooth and strongly convex-concave around the optimal solution, whereas existing approaches fail.

Automatic software generation based on some specification is known as program synthesis. Most existing approaches formulate program synthesis as a search problem with discrete parameters. In this paper, we present a novel formulation of program synthesis as a continuous optimization problem using an evolutionary approach, known as Covariance Matrix Adaptation Evolution Strategy. We then propose several mapping schemes to convert the continuous formulation into actual programs and propose different restart policies for the evolutionary approach. This is the first work that demonstrates the feasibility of continuous approach in synthesizing complex programs, not just simple toy programs. We compare our system, Genesys , to several recent program synthesis techniques (in both discrete and continuous domains). We find that Genesys synthesizes more programs within a fixed time budget than those existing schemes. For example, for programs of length 10, Genesys synthesizes 28% more programs than those existing schemes within the same time budget.

The performance of evolutionary algorithms can be heavily undermined when\nconstraints limit the feasible areas of the search space. For instance, while\nCovariance Matrix Adaptation Evolution Strategy is one of the most efficient\nalgorithms for unconstrained optimization problems, it cannot be readily\napplied to constrained ones. Here, we used concepts from Memetic Computing,\ni.e. the harmonious combination of multiple units of algorithmic information,\nand Viability Evolution, an alternative abstraction of artificial evolution, to\ndevise a novel approach for solving optimization problems with inequality\nconstraints. Viability Evolution emphasizes elimination of solutions not\nsatisfying viability criteria, defined as boundaries on objectives and\nconstraints. These boundaries are adapted during the search to drive a\npopulation of local search units, based on Covariance Matrix Adaptation\nEvolution Strategy, towards feasible regions. These units can be recombined by\nmeans of Differential Evolution operators. Of crucial importance for the\nperformance of our method, an adaptive scheduler toggles between exploitation\nand exploration by selecting to advance one of the local search units and/or\nrecombine them. The proposed algorithm can outperform several state-of-the-art\nmethods on a diverse set of benchmark and engineering problems, both for\nquality of solutions and computational resources needed.\n

Evolutionary algorithms generally require a large number of objective function evaluations which can be costly in practice. These evaluations can be replaced by evaluations of a cheaper meta-model of the objective functions. In this paper we describe a multiobjective memetic algorithm utilizing local distance based meta-models. This algorithm is evaluated and compared to standard multiobjective evolutionary algorithms as well as a similar algorithm with a global meta-model. The number of objective function evaluations is considered, and also the conditions under which the algorithm actually helps to reduce the time needed to find a solution are analyzed.KeywordsMultiobjective optimizationmeta-modelevolutionary algorithm

The high-frequency nature of impedance-based structural health monitoring makes the utilization of impedance signature for model-based damage characterization a challenging problem. In this study, a novel damage characterization approach that utilizes impedance signature measured with a single piezoelectric wafer is developed. Length-varying spectral elements are introduced to minimize the total number of elements required to describe the system, along with the number of damage characterization parameters. Several objective function definitions are studied and their behaviour with respect to each damage parameter is investigated. It has been found that an objective function definition based on the frequency shift in impedance peaks is the most effective definition compared to root mean square deviation and correlation-based objective functions. A novel damage localization method, referred to as sine-fit localization method, is developed based on the underlying periodic behaviour of impedance peak shifts as a function of damage location. The sine-fit localization method is integrated with gradient descend method in a two-stage optimization algorithm for damage characterization. The developed algorithm is capable of solving the ill-posed problem of damage characterization with few iterations and small number of objective function evaluations, which makes it computationally very efficient.

Optimum implementation of non-conventional wells allows us to increase considerably hydrocarbon recovery. By considering the high drilling cost and the potential improvement in well productivity, well placement decision is an important issue in field development. Considering complex reservoir geology and high reservoir heterogeneities, stochastic optimization methods are the most suitable approaches for optimum well placement. This paper proposes an optimization methodology to determine optimal well location and trajectory based upon the Covariance Matrix Adaptation-Evolution Strategy (CMA-ES) which is a variant of Evolution Strategies recognized as one of the most powerful derivative-free optimizers for continuous optimization. To improve the optimization procedure, two new techniques are investigated: (1). Adaptive penalization with rejection is developed to handle well placement constraints. (2). A meta-model, based on locally weighted regression, is incorporated into CMA-ES using an approximate ranking procedure. Therefore, we can reduce the number of reservoir simulations, which are computationally expensive. Several examples are presented. Our new approach is compared with a Genetic Algorithm incorporating the Genocop III technique. It is shown that our approach outperforms the genetic algorithm: it leads in general to both a higher NPV and a significant reduction of the number of reservoir simulations. ECMOR XII – 12 th European Conference on the Mathematics of Oil Recovery

This study addresses the two-dimensional weakly homogeneous Bin Packing Problem (2D-BPP) in the context of robotic packing, where items must be arranged in a manner feasible for robotic manipulation. Traditional heuristics for this NP-hard problem often lack adaptability across diverse datasets, while metaheuristics typically suffer from slow convergence. To overcome these limitations, we propose a novel neural-driven constructive heuristic. The method employs a population of simple feed-forward neural networks, which are trained using black-box optimization via the Covariance Matrix Adaptation Evolution Strategy (CMA-ES). The resulting neural network dynamically scores candidate placements within the constructive heuristic. Unlike conventional heuristics, the approach adapts to instance-specific characteristics without relying on predefined rules. Evaluated on datasets generated by 2DCPackGen and real-world logistic scenarios, the proposed method consistently outperforms benchmark heuristics such as MaxRects and Skyline, reducing the average number of bins required across various item types and demand ranges. The most significant improvements occur in complex instances, with up to 86% of 2DCPackGen cases yielding superior results. This heuristic offers a flexible and extremely fast, data-driven solution to the algorithm selection problem, demonstrating robustness and potential for broader application in combinatorial optimization while avoiding the scalability issues of reinforcement learning-based methods.