KT-EGO: A Knowledge Transfer Assisted Efficient Global Optimization Algorithm for Solving High-Dimensional Expensive Black-Box Problems

1. Introduction

• The paper defines HEB problems as having more than 20 dimensions and only a few hundred or thousand affordable evaluations.The authors note that definitions of HEB can vary by field.Source: §1 · pp. 1–2
• Nash-EGO divides a high-dimensional design space into low-dimensional subsets and uses EGO to optimize them before combining their solutions.EGO is described as sample-efficient below 15 dimensions.Source: §1 · p. 2
• Random decomposition addresses variable interactions, while adaptive-range kriging search strengthens local exploitation later in optimization.The authors note that interactions in non-separable problems make correct one-shot decomposition nearly impossible.Source: §1 · pp. 2–3
• KT-EGO fuses previously evaluated samples through surrogate models to transfer information into subset-space searches.This addresses Nash-EGO's failure to reuse samples from earlier optimization cycles.Source: §1 · p. 2
• Tests on 12 benchmark problems and an engineering compressor-blade problem demonstrate KT-EGO's effectiveness for high-dimensional expensive black-box problems.The introduction reports the evaluation scope but gives no comparative scores.Source: §1 · p. 3

2.1 The Nash-EGO algorithm

• Nash-EGO anchors each cycle at an elite point, optimizes low-dimensional subsets with EGO, and combines their optima; KT-EGO reuses prior-cycle samples to aid later searches.The elite point is the current best solution, and is random in the first cycle.Source: §2.1 · pp. 3–4

2.2 Transfer Optimization and Multi-Form Optimization

• Multi-form optimization jointly searches alternate formulations of one target task with continuous knowledge transfer; KT-EGO treats subset-space optimizations as formulations and uses data fusion for transfer.The framework's quality measure Qc quantifies solutions obtained at cycle c.Source: §2.2 · p. 4

2.3 Polynomial Chaos Expansion

• Polynomial chaos expansion represents random variables with polynomial functions and is used as a surrogate for data fusion, with Legendre polynomials chosen for uniformly distributed variables.The paper selects PCE for its approximation accuracy in high-dimensional spaces.Source: §2.3 · pp. 4–5

2.4.1 Kriging

• Kriging predicts an unknown-site response using a trend plus a zero-mean Gaussian process that models local features around sampled points.The paper also introduces prediction and mean squared error at an unknown point.Source: §2.4.1 · p. 5

2.4.2 Hierarchical Kriging

• Hierarchical kriging combines high- and low-fidelity samples by modeling the high-fidelity function as a scaled low-fidelity function plus a Gaussian process.Its scaling factor models correlation between the two fidelity levels.Source: §2.4.2 · pp. 5–6

3. Proposed Method

• KT-EGO combines random decomposition, surrogate-based data fusion, and adaptive local exploitation to address high-dimensional optimization.The framework uses a global PCE surrogate and optimizes the resulting subset spaces.Source: §3 · p. 6

3.1 Decomposition of subset design spaces

• Variable indices are randomly reordered at each cycle to form subset spaces, rather than partitioning variables in their original sequence.The number of variables assigned to each subset can vary.Source: §3.1 · p. 8

3.2.1 DoE in subset design spaces

• Subset-space DoE uses LHS while fixing variables outside the current subset at the elite point; subsets are optimized in parallel.After optimizing all subsets, their variable-wise optima are concatenated into a sample for evaluation.Source: §3.2.1 · p. 8, §3.2.2 · p. 10
• Previously evaluated samples inform each new subset search through a surrogate-based data-fusion strategy, unlike Nash-EGO’s from-scratch subset optimizations.A PCE model captures the high-dimensional trend, and its predictions augment subset-space samples in a multi-fidelity surrogate.Source: §3.2.2 · pp. 9–10

3.2.3 Local exploitation with adaptive variable range

• When EImax ≤ ∆, KT-EGO switches from HK-guided search to kriging and reduces the subset’s search range for local exploitation.The threshold ∆ is set to 10^-3.Source: §3.2.3 · pp. 10–12

• KT-EGO is tested against state-of-the-art algorithms on numerical benchmark problems and an engineering problem.Source: §4 · p. 12

• The comparison includes GA, DE, GSK, IKAEA, GSGA, and Nash-EGO alongside KT-EGO.GA, DE, and GSK use population size 50; IKAEA and GSGA follow their papers’ default settings.Source: §4.1 · p. 12
• Benchmark tests are repeated 20 times with the same initial sample distribution across algorithms in each run.Source: §4.1 · p. 13
• Each benchmark run stops at 3000 function evaluations; KT-EGO sets tmax to six times the subset dimension.Source: §4.1 · p. 13

• The benchmark functions are selected from the CEC 2010 special session; Table 1 summarizes their characteristics.Source: §4.2 · p. 13, Table 1 · p. 13
• Six separable and six non-separable functions test cases with independent versus highly interacting design variables.Source: §4.2 · p. 13

4.3.1 Comparison result of 30-D benchmark functions

• In 30-D tests, KT-EGO was significantly better in most test cases and had better convergence and final solutions on non-separable F4–F6, especially F6.The convergence curves are shown in Figure 5.Source: §4.3.1 · pp. 13–14, Figure 5 · p. 14
• On separable 30-D functions F1–F3, decomposition-based methods had advantages; KT-EGO also excelled early on unimodal F1, while GSGA performed well on multimodal F2 and F3.The paper attributes F1's early result to accurate PCE fitting with few samples.Source: §4.3.1 · p. 14

4.3.2 Comparison results of 60-D benchmark functions

• In 60-D tests, KT-EGO was significantly better or comparable in all cases and usually reached better final solutions, despite IKAEA's sometimes faster early convergence.The convergence curves are shown in Figure 6.Source: §4.3.2 · pp. 14–15, Figure 6 · p. 15

4.4.1 Effectiveness of the KT-EGO algorithm components

• RD-EGO was significantly better than other compared algorithms on most benchmark functions, but random decomposition alone showed no clear acceleration at 1000 NFE.RD-EGO is the variant for the random decomposition strategy.Source: §4.4.1 · p. 15
• KT-EGO's greater efficiency than TF-EGO indicates that random grouping improves global PCE accuracy through a more uniform sample distribution.TF-EGO corresponds to the knowledge transfer strategy, while RD-EGO corresponds to random decomposition.Source: §4.4.1 · pp. 15–16
• TF-EGO tracked KT-EGO early but fell behind later; the paper attributes knowledge transfer's contribution mainly to faster global exploration.Without random grouping, interactive variables are more difficult to place in the same group.Source: §4.4.1 · p. 16
• The variant tests are reported to demonstrate the effectiveness of both proposed components.RD-EGO and TF-EGO are the component variants.Source: §4.4.1 · pp. 15–16

4.5.1 Problem description

• Rotor 37 is a 28-variable blade design parameterized by five spanwise sections, five suction-side control points per section, and three stacking-line translation parameters.Source: §4.5.1 · pp. 16–17, Figure 8 · p. 17

4.5.2 Numerical simulation model

• About 600 s per sample, the CFD simulation uses NUMECA FINE/TURBO with Spalart–Allmaras and a mesh of about 4×10^6 nodes.The outlet static pressure is fixed at 115000 Pa, corresponding to a relative mass flow rate of 99% for the Rotor 37 blade.Source: §4.5.2 · p. 18, Figure 9 · p. 18

4.5.3 Results Analysis

• 1.65 percentage points (0.8539 to 0.8704) improved Rotor 37 total efficiency with KT-EGO after 1000 CFD calculations, the best result among five algorithms.Across five repeats, the result variance for KT-EGO, GA, and DE was very small, indicating less sensitivity to the initial sample distribution.Source: Table 4 · p. 19, §4.5.3 · p. 19
• Across the whole operating range, KT-EGO's optimized design is more efficient than the reference and maintains compressor stability; above 96% relative flow, its pressure ratio is slightly higher.Source: §4.5.3 · pp. 19–20, Figure 11 · p. 20
• For the median optimal designs, the optimized separation line moves downstream, the separation area decreases, and tip back-flow intensity is reduced relative to the baseline.The baseline shows suction-surface shock-wave/boundary-layer interference and serious separation near the trailing edge.Source: §4.5.3 · p. 20

5. Conclusions

• KT-EGO uses surrogate-based data fusion to draw information from previously evaluated samples and accelerate optimization in subset design spaces.Source: §5 · p. 20
• Random decomposition addresses joint effects among design variables, while adaptive variable ranges support local exploitation later in optimization.Source: §5 · p. 20
• Tests on 12 benchmark functions from 30 to 60 dimensions and a compressor-blade engineering problem demonstrate KT-EGO's effectiveness within those evaluated settings.Source: §5 · p. 20

Data availability statement

• Data supporting the study findings are available from the corresponding author upon reasonable request.Source: Data availability statement · p. 21
• The published article identifies the KT-EGO code repository.Source: Data availability statement · pp. 21–22
• The statement does not assert that the repository is currently available.Source: Data availability statement · p. 22