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A Dynamic Aggregation Strategy Enhanced Efficient Global Optimization Algorithm for Solving High-Dimensional Turbomachinery Design Problems

Qineng Wang, Zhendong Guo, Yun Chen, Guangjian Ma, Liming Song, Jun Li

arXiv:2609.16067v1cs.LGcs.CE

TL;DR

High-dimensional, expensive black-box optimization makes surrogate modeling and search difficult within limited evaluation budgets. DA-EGO iteratively decomposes the design space and aggregates subspace knowledge to update interactions and search ranges. It performs well on separable and partially separable benchmarks and shows practical value in expensive engineering cases, while GSGA performs better on shifted Rosenbrock at 60 and 90 dimensions.

  • Problem

    High-dimensional, expensive black-box problems are difficult to optimize because evaluations are limited and conventional methods struggle with dimensionality and interaction detection.

    Source: §1 · p. 2

  • Method

    DA-EGO iteratively decomposes the design space into low-dimensional subproblems, aggregates subspace knowledge, and updates decomposition and search ranges using interaction analyses.

    Source: §1 · pp. 3–4, §3.2.1 · p. 11, §3.3.2 · p. 14

  • Results

    DA-EGO performs well on separable and partially separable benchmarks and engineering cases, while GSGA is better on shifted Rosenbrock at 60 and 90 dimensions.

    Source: §6 · p. 33

  • Takeaways & Limitations

    The engineering cases support DA-EGO’s practical value when objective evaluations are expensive.

    Source: §6 · p. 33

    Surrogate construction and interaction analysis add computational cost, reducing DA-EGO’s advantage when objective evaluations are inexpensive.

    Source: §6 · p. 33

Abstract

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In order to solve the high-dimensional ($d \geq 30$) expensive black-box problems within budget, an efficient global optimization (EGO) algorithm with a dynamic aggregation strategy is proposed, labeled as DA-EGO. Specifically, the DA-EGO decomposes the original high-dimensional design space into a set of low-dimensional subspaces for efficient surrogate-based optimization search, and the optimal solutions of subspaces are combined as an elite point for the global search. Most importantly, the subspaces are not fixed. Instead, the subspace variables are updated in each iteration, according to the variable interaction analyses in the sub- and full-spaces. The perturbation method and the analysis of variance are used to detect variable interactions. To further accelerate the optimization progress, the searching ranges of subspaces are also adaptively adjusted according to the analyses of subspace optimization results of the previous iteration. Tests on 21 benchmark instances, comprising seven functions at 30, 60, and 90 dimensions, show that DA-EGO is effective on separable and partially separable problems under a budget of 1500 function evaluations. Its advantage is case-dependent: on the non-separable shifted Rosenbrock function, GSGA performs better at 60 and 90 dimensions, while the 30-dimensional results are statistically comparable to IKAEA and GSGA. Moreover, the advantage of DA-EGO is also seen in the aerodynamic optimization of a transonic rotor blade with 28 variables as well as the compressor stage optimization with 60 variables. With the above, the effectiveness of the proposed DA-EGO has been well demonstrated.

1. Introduction

High-dimensional, expensive engineering optimization strains available evaluations, while existing decomposition and surrogate methods struggle with variable interactions or dimensionality. DA-EGO combines iterative decomposition, subspace knowledge mining, and adaptive search ranges to address these challenges.

1. Introduction

  • High-dimensional engineering problems expand the design space and sample needs exponentially, while high-fidelity CFD evaluations can take hours or days.The paper follows a definition of high dimensionality as 30–100 variables and expensive problems as those limited to thousands of samples.Source: §1 · p. 2
  • Prior VID methods need many samples to detect interactions, while common surrogates lose accuracy in high dimensions and are efficient only below 15 dimensions.Earlier combined approaches retained fixed manual decomposition or relied on high-dimensional surrogates, limiting their applicability.Source: §1 · p. 3
  • DA-EGO iteratively changes its decomposition, mines knowledge from subspace optimization to inform later subtasks, and adapts search ranges using design-space knowledge.Unlike fixed VID decomposition, its decomposition and subtasks develop from rough to detailed across cycles.Source: §1 · pp. 3–4
  • The paper tests DA-EGO on 18 numerical benchmark functions and applies it to a 28-dimensional rotor cascade and a 60-dimensional multistage compressor.The introduction identifies these evaluations as the paper’s validation of the method.Source: §1 · p. 4

2. Related Work

The related work reviews surrogate-based EGO and decomposition strategies, then introduces polynomial chaos expansion as a source of sensitivity information for interaction analysis.

2.1. Efficient global optimization

  • EGO combines a kriging surrogate with an expected-improvement acquisition function to balance global exploitation and local exploration.The reviewed EGO methods are described as typically limited to problems with up to 15 dimensions.Source: §2.1 · p. 4, §2.1.2 · p. 5

2.1.1. Kriging surrogate

  • Kriging predicts responses using a trend function plus a zero-mean random process that captures local features around evaluated samples.The trend is commonly constant, linear, or quadratic, with a constant most widely used.Source: §2.1.1 · p. 4

2.1.2. Maximum expectation improvement

  • Expected improvement guides the search toward locations with the maximum EI value, using the normal cumulative distribution and density functions.The reviewed EGO framework uses EI as its acquisition function.Source: §2.1 · p. 4, §2.1.2 · p. 5

2.2. Decomposition-based optimization algorithms

  • Decomposition groups interacting variables into subtasks and combines their solutions around an elite point, but detecting interactions can require costly sampling.Manual grouping ignores interactions and is limited for nonseparable problems; VID groups variables by their interaction relationships.Source: §2.2 · pp. 5–7

2.3. Polynomial chaos expansion based sensitivity analysis

  • Polynomial chaos expansion represents responses with polynomial functions, and its coefficients support sensitivity analysis; this study uses Legendre polynomials for uniformly distributed variables.The coefficients are calculated with Orthogonal Matching Pursuit, and the PCE-based analysis uses UQLab.Source: §2.3 · pp. 7–8

3. Proposed Method

DA-EGO alternates between decomposition and subproblem optimization with knowledge mining. Subspace outcomes feed back into subsequent decomposition, replacing one-shot interaction detection with low-dimensional mining and aggregation.

3. Proposed Method

  • DA-EGO cycles between generating sub-tasks from whole-design-space knowledge and optimizing them while mining subspace samples, interactions, and search boundaries.The feedback cycle continues until the stopping condition is met.Source: §3 · p. 8
  • DA-EGO adds a space-reduction mechanism that uses design-space knowledge to guide subspace optimization.Its framework treats initialization, decomposition, and subproblem optimization with knowledge mining as three parts.Source: §3 · p. 8
  • Subspace outcomes feed back into the decomposition strategy, so later subtasks can use verified or newly mined problem information.The method aggregates low-dimensional knowledge rather than detecting all variable interactions at once.Source: §3 · p. 8

3.1. Initialization

Initialization establishes the initial samples, interaction assumptions, and search boundaries for DA-EGO.

3.1. Initialization

  • Initialization evaluates 50 Latin hypercube samples, assumes no variable interactions initially, and sets each variable’s search range to [0, 1].The initial search scope is the entire design space.Source: §3.1 · p. 8

3.2. High-dimensional problem decomposition

DA-EGO aggregates feedback from sub-tasks to update the decomposition scheme and search scope, then builds new low-dimensional tasks around the current best sample.

3.2. High-dimensional problem decomposition

  • Each cycle aggregates sub-task feedback and uses accumulated knowledge to update the decomposition plan and create new sub-tasks.Feedback includes samples, variable interactions, and subspace search ranges; global knowledge includes interactions, optimal points, and search ranges.Source: §3.2 · p. 10

3.2.3. Generate new sub-tasks

  • New sub-tasks vary only their assigned variables while fixing all others at the elite-point, selected as the best sample evaluated so far.The resulting tasks use the updated decomposition and search scope.Source: §3.2 · p. 11, §3.2.3 · p. 13

3.2.1. Decomposition scheme aggregation

  • Confirmed variable pairs and ranked suspected pairs guide grouping, with suspected interactions checked by low-dimensional surrogate-based perturbation analysis.PCE-based Sobol sensitivity estimates rank suspected pairs, but high-dimensional PCE accuracy is limited by the curse of dimensionality.Source: §3.2.1 · p. 11

3.2.2. Searching scope aggregation

  • For unchanged variable groups, search ranges reuse prior sub-task boundaries; groups with changed variables are explored over the full normalized range [0, 1].Prior boundaries are recorded in the boundary table.Source: §3.2.2 · pp. 11, 13

3.3. Sub-task optimization and knowledge mining

DA-EGO optimizes low-dimensional subproblems around an elite point, then mines their surrogate models and samples for interaction and search-boundary information that informs later cycles.

3.3. Sub-task optimization and knowledge mining

  • Each subproblem varies only a subset of the original variables, with its design space determined by the elite-point, variable grouping, and search scope.The mapping G(·) embeds a subspace design into the global space.Source: §3.3 · p. 13

3.3.1. Sub-task optimization

  • EGO searches each subproblem using maximum expected improvement, adding samples to improve the Kriging surrogate and returning the best sample and predictive surrogate.The initial sample count is 3d and the maximum iteration count is 6d, where d is the subspace dimension.Source: §3.3.1 · p. 13

3.3.2. Sub-task knowledge mining

  • Pairwise perturbation analysis of the subspace surrogate confirms selected variable interactions, with pairs meeting Hij ≥ 0.2 recorded for feedback.The method evaluates d(d−1)/2 pairs and restricts analysis to [0.1, 0.9]D to improve precision; NP_A is set to 10000.Source: §3.3.2 · pp. 14–15
  • Sub-task samples, confirmed interactions, and reduced search boundaries feed back into global tables, clarifying interaction structure and enabling smaller search scopes in later cycles.Space reduction focuses search on the surrogate-predicted potential optimal region to avoid calculations in less important regions.Source: §3.3.2 · pp. 14–15

4. Numerical Experimental Study

The numerical study compares DA-EGO with model-free, surrogate-assisted, and surrogate-based approaches on repeated benchmark runs under a shared evaluation budget.

4.1. Experimental settings

  • The comparison includes GA, DE, GSK, IKAEA, GSGA, Nash-EGO, and RG-EGO, a DA-EGO variant using random grouping while holding other settings fixed.Comparing RG-EGO with DA-EGO is intended to reflect the effect of the proposed interaction-analysis strategy.Source: §4.1 · pp. 15–16
  • Algorithms are compared only when their applicable dimensions and sample-size ranges are consistent, excluding methods such as SEE, OMID, and CCVIL.IKAEA and GSGA use their papers’ default settings.Source: §4.1 · p. 16
  • Each benchmark test is repeated 20 times with the same initial sample distribution across algorithms and terminates at 1500 function evaluations.Source: §4.1 · p. 16

4.2. Benchmark functions

The benchmark suite contains seven functions tested across three dimensions, with instances grouped by variable separability.

4.2. Benchmark functions

  • Table 1 summarizes the characteristics of the benchmark functions selected from the CEC 2010 special session.Source: §4.2 · p. 16, Table 1 · p. 16
  • The suite comprises 21 instances: nine separable, nine partially separable, and three non-separable shifted Rosenbrock instances.The seven functions are each tested at 30, 60, and 90 dimensions.Source: §4.2 · p. 16
  • Table 2 presents the benchmark-function equations with index and description columns.Source: Table 2 · p. 17

4.3. Result of benchmark functions

DA-EGO has the lowest reported mean on 17 of 21 benchmark instances and is competitive on several others, with results depending on the function’s interaction structure. Its advantage on separable and partially separable functions does not extend uniformly to shifted Rosenbrock, where GSGA is favored at 60-D and 90-D.

4.3.1. Comparison result of 30-D benchmark functions

  • At 30-D, DA-EGO significantly beats seven comparison algorithms on F1–F6; on F7 it is comparable to IKAEA and GSGA, whose reported mean is slightly lower.The results distinguish separable F1–F3, where decomposition methods have advantages, from partially separable F4–F6, where DA-EGO outperforms manual-decomposition methods Nash-EGO and RG-EGO.Source: §4.3.1 · pp. 17–19, Table 3 · p. 18

4.3.2. Comparison result of 60-D benchmark functions

  • At 60-D, DA-EGO has the lowest reported mean on F1–F6, though F3 is comparable to GSGA; on F7, GSGA’s lower mean and Wilcoxon result favor GSGA.DA-EGO converges more slowly early on but faster later, as interaction information accumulates.Source: §4.3.2 · pp. 19–21, Table 4 · p. 20

4.3.3. Comparison result of 90-D benchmark functions

  • At 90-D, DA-EGO remains competitive; on F7, GSGA is significantly better than DA-EGO, while the comparison with IKAEA is statistically comparable.DA-EGO’s comparisons with IKAEA on F3, F4, and F6 are also statistically comparable, and IKAEA has the lower reported mean on F6.Source: Table 5 · p. 21, §4.3.3 · pp. 21–22
  • 17 of 21 instances have the lowest reported mean with DA-EGO, which is competitive on several others; on shifted Rosenbrock, GSGA is favored at 60-D and 90-D.The results support DA-EGO on separable and partially separable functions, but not uniformly on the non-separable shifted Rosenbrock function.Source: §4.3.3 · p. 22

4.3.4. Computational time

  • DA-EGO takes 35, 36, and 40 minutes at 30-D, 60-D, and 90-D, respectively, trading surrogate construction and interaction-analysis overhead for fewer expensive function evaluations.The trade-off is most relevant when physical-objective evaluation dominates optimization-algorithm costs.Source: §4.3.4 · p. 23, Table 6 · p. 24

4.4. Parameter sensitivity analysis

Across the tested titer values, changes have limited effect on DA-EGO’s overall convergence trend. Shorter sub-task runs can make interaction information available to the next decomposition cycle earlier.

4.4. Parameter sensitivity analysis

  • Within titer values 6, 9, and 12, changes have limited effect on DA-EGO’s overall convergence trend across F1, F4, and F7.The sensitivity study is limited to these three functions and parameter values.Source: §4.4 · p. 25, Figure 8 · p. 25
  • Shorter sub-task runs can make interaction information available to the next decomposition cycle earlier.The maximum sub-task iterations are itermax = titerd, with d denoting subspace dimension and titer = 6 as the default.Source: §4.4 · p. 25
  • Figure 8 compares average convergence curves for titer values 6, 9, and 12 on F1, F4, and F7, with GSGA as a comparator.NFE denotes the number of function evaluations.Source: Figure 8 · p. 25, §4.4 · p. 25

5. Engineering Test Case

The engineering evaluation applies DA-EGO to Rotor 37 and a multistage compressor, comparing it with other algorithms. The reported results show efficiency gains in both design tasks.

5.1.1. Problem description

5.1.2. Numerical simulation model

5.1.3. Results Analysis

5.2.1. Problem description

5.2.2. Result Analysis

  • 1.07%: DA-EGO improved multistage-compressor total efficiency and significantly outperformed all four comparators in the 1,000-CFD-evaluation test.Source: Table 11 · p. 31, §5.2.2 · p. 31
  • Efficiency increased across the first-stage span by about 1%; whole-compressor gains were concentrated in the middle span, while some endwall regions declined slightly.Source: §5.2.2 · p. 31, Figure 13 · p. 32
  • Entropy decreased in the three highest-entropy flow regions, especially A and B, as changes to rotor R1 weakened shock waves and improved circumferential pressure uniformity.Source: §5.2.2 · p. 31, Figure 14 · p. 32

6. Conclusions

DA-EGO dynamically decomposes high-dimensional problems into low-dimensional subtasks and aggregates learned interactions to update decomposition. It performs well on several benchmark and engineering cases, while added computational overhead limits its advantage when evaluations are inexpensive.

  • DA-EGO aggregates subproblem knowledge, using perturbation and analysis of variance to identify interactions that guide later decompositions and help improve the best evaluated solution.Source: §6 · p. 33
  • Across 21 benchmarks and two engineering designs, DA-EGO performs well on separable and partially separable cases, while GSGA is better on shifted Rosenbrock at 60 and 90 dimensions.Source: §6 · p. 33
  • Surrogate construction and interaction analysis add computational cost, reducing DA-EGO’s advantage when objective evaluations are inexpensive.Source: §6 · p. 33
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