Adaptive Rotation for iSOMA: Geometry, Benchmarking, and Noise Robustness in Variational Quantum Objectives
Authors: Vojtěch Novák, Ivan Zelinka
Organizations: Department of Computer Science, Faculty of Electrical Engineering and Computer Science, VSB-Technical University of Ostrava, Ostrava, Czech Republic. · IT4Innovations National Supercomputing Center, VSB-Technical University of Ostrava, Ostrava, 708 00, Czech Republic. · Department of Informatics and Statistics, Marine Research Institute, Klaipeda University, Klaipeda, Lithuania.
We study whether the coordinate dependence of the improved Self-Organizing Migrating Algorithm (iSOMA) can be reduced while retaining its inexpensive leader-directed migration mechanism. We introduce iSOMA-AR, which learns a basis from successful migration displacements and selectively applies the standard perturbation mask in that basis. On the complete noiseless BBOB suite, iSOMA- AR significantly outperformed baseline iSOMA across matched conditions, with the largest gains on geometrically difficult landscapes. A targeted ablation shows that the learned orientation is beneficial on a rotated ill-conditioned landscape and that moderate changes of the gate threshold and rotation cap preserve the qualitative result. On CEC 2011 Real World Optimization Problems, iSOMA-AR outperformed iL-SHADE on most problems, although its advantage over baseline iSOMA was not statistically significant. A canonical-jSO rerun is reported as a post-hoc sensitivity check alongside the original jSO-derived comparator. On frustrated-spin variational quantum objectives, adaptive rotation improved most transverse-field conditions, while gains on the diagonal and anisotropic models were absent or selective. Under strong effective sampling noise, the SOMA variants were the most robust population-based methods in the comparison, but iSOMA-AR was not significantly better than baseline iSOMA. Repairing all-zero PRT masks greatly reduced repeated-point evaluations without changing endpoint quality significantly, making this implementation detail unlikely to explain the noise result. Overall, adaptive rotation is most useful on coordinate-sensitive deterministic problems, while the observed noise robustness appears to arise mainly from the underlying SOMA migration mechanism.
Figures & tables
Figure 1: Illustrative proposal geometry on a rotated Rosenbrock valley. With the same partially active perturbation mask, standard iSOMA applies masking in the original coordinate system, whereas iSOMA-AR applies it in the learned basis Qs . The rotated proposals follow the valley geometry more directly and make faster progress toward the leader. Discrete points denote evaluated path locations and diamonds denote accepted replacements.
BBOB
CEC 2011 real-world
Problems
24 noiseless functions
22 real-world problems
Dimensions
D∈{5,10,20}
problem-specific ( 1 – 240 )
Instances
1 – 5
fixed problem definitions
FE budgets
500D,2000D
50,000
Stochastic runs
10 / condition
10 / problem
Primary optimizers
iSOMA, iSOMA-AR, jSO-derived, iL-SHADE, CMA-ES
Table 1: Benchmark protocols used in the final study.
Algorithm
Mean rank
Top-1
Target-hit frac.
Success 10−8
Med. FE/D to 10−8
s/1000 FE
jSO-derived
1.86
49.9%
65.4%
48.9%
1073
0.0279
CMA-ES
2.49
51.0%
55.1%
44.4%
417
0.0676
iL-SHADE
3.13
16.9%
35.8%
15.0%
552
0.0334
iSOMA-AR
3.46
10.6%
36.4%
15.5%
1212
0.0204
iSOMA
4.05
8.3%
29.5%
12.2%
1154
0.0182
Table 2: Overall BBOB results across 720 matched function–dimension–instance–budget conditions. Target-hit fraction is the fraction of all run–target pairs attained among the 51 fixed targets Δf∈[102,10−8] at the 2000D budget. Lower rank and runtime are better; higher shares are better.
Group
Most frequent condition winner
AR/base W/T/L
AR/base ratio
AR/iL-SHADE W/T/L
AR/iL-SHADE ratio
G1 separable
jSO-derived 115/150
84/39/27
0.93
71/33/46
1.00
G2 low/moderate conditioning
CMA-ES 90/120
99/0/21
0.55
45/0/75
1.32
G3 high-conditioning unimodal
CMA-ES 131/150
137/0/13
0.45
45/0/105
2.53
G4 multimodal, adequate structure
jSO-derived 80/150
100/0/50
0.90
21/0/129
2.44
G5 multimodal, weak structure
jSO-derived 67/150
87/3/60
0.99
69/3/78
1.01
Table 3: BBOB official-group pairwise final-error outcomes. W/T/L is the number of matched conditions won/tied/lost by the first method. The ratio is the median error ratio; values below one favor iSOMA-AR.
Algorithm
Mean rank
Top-1
Target-hit frac.
Success 10−4
Success 10−6
s/1000 FE
jSO-derived
1.50
77.3%
63.1%
59.5%
29.1%
0.0202
CMA-ES
2.77
18.2%
48.6%
45.5%
7.7%
0.0834
iSOMA-AR
3.27
13.6%
42.0%
33.2%
11.8%
0.0193
iSOMA
3.41
9.1%
43.2%
34.1%
12.3%
0.0110
iL-SHADE
4.05
9.1%
18.3%
11.4%
6.8%
0.0224
Table 4: Overall CEC 2011 results across 22 problems at 50,000 evaluations. The normalized target-hit fraction is diagnostic: it is the fraction of run–target pairs attained for six relative gap reductions 10−1 – 10−6 reconstructed from saved checkpoints.
Group
Most frequent problem winner
AR/base W/T/L
AR/base ratio
AR/iL-SHADE W/T/L
AR/iL-SHADE ratio
A1 control / parameter estimation
jSO-derived 3/3
0/2/1
1
1/2/0
1
A2 molecular / material potentials
jSO-derived 3/3
0/0/3
1.3
1/0/2
1.9
A3 signal / antenna design
CMA-ES 1/2, iL-SHADE 1/2
2/0/0
0.97
0/0/2
4.9
A4 power systems
jSO-derived 10/12
8/1/3
0.87
10/0/2
0.0002
A5 spacecraft trajectory
iSOMA-AR 1/2, jSO-derived 1/2
1/0/1
0.88
2/0/0
0.038
Table 5: CEC 2011 diagnostic application-group outcomes. The groups are interpretive and outside the official CEC 2011 taxonomy. W/T/L compares median final regret over the problems in each group; values below one favor iSOMA-AR.
Study
Comparison
W/T/L
Median ratio
pHolm
BBOB
iSOMA-AR / iSOMA
507/42/171
0.814
1.34e-50
BBOB
iSOMA-AR / iL-SHADE
251/36/433
1.35
8.44e-12
BBOB
iSOMA-AR / CMA-ES
198/46/476
2.78
2.34e-45
BBOB
iSOMA-AR / jSO-derived
58/54/608
22
1.91e-101
BBOB
jSO-derived / CMA-ES
310/115/295
1
0.99
CEC2011
iSOMA-AR / iSOMA
11/3/8
0.99
0.338
Table 6: Selected paired comparisons. Ratios are median error/regret ratios of the first method over the second; Holm-adjusted Wilcoxon p -values are descriptive because benchmark conditions are not fully independent experimental units.
Model
Condition
iSOMA
iSOMA-AR
AR/base
Best core method
Q1
N=12 , 10k
1.50\times10−05
1.46\times10−04
9.708
jSO-derived
Q1
N=12 , 30k
0
1.17\times10−13
–
iSOMA
Q1
N=14 , 10k
6.93\times10−08
5.10\times10−07
7.360
jSO-derived
Q1
N=14 , 30k
0
0
–
iSOMA
Q1
N=16 , 10k
0.458
0.458
1.000
iSOMA
Q1
N=16 , 30k
0.458
0.458
1.000
jSO-derived
Table 7: Exact-objective VQE extension using the original 25-run Q1–Q3 benchmark protocol. Values are median final errors. For Q1 and Q2 the error is relative to the exact physical ground energy; for Q3 it is relative to the unchanged saved best-known variational reference. AR/base is the ratio of iSOMA-AR to iSOMA median error; values below one favor adaptive rotation.
Noise
Model
Budget
iSOMA
iSOMA-AR
jSO-derived
iL-SHADE
CMA-ES
SPSA
Low
Q1
10k
0.066
0.087
0.115
0.177
1.712
2.022
Low
Q1
30k
0.062
0.080
0.083
0.158
1.755
2.017
Low
Q2
10k
0.317
0.308
0.374
0.446
0.342
0.587
Low
Q2
30k
0.319
0.316
0.260
0.272
0.322
0.203
Low
Q3
10k
1.928
1.927
3.689
1.555
1.382
0.356
Low
Q3
30k
1.907
1.910
1.960
1.244
1.216
0.166
Table 8: Sampling-noise robustness at N=12 over 25 runs. Entries are the median exact errors of the parameter vector selected using the noisy objective. Bold marks the smallest median in each condition. Low and high noise correspond to 8192 and 128 effective shots, respectively.
Appendix figures & tables13 assets
Supplementary material from the paper’s appendix.
Appendix
Method
Role / implementation
Settings used
Notes
iSOMA
Baseline SOMA used in all three benchmark families.
Pop 50 ; Njump=10 ; Step =0.3 ; m=10 , n=5 , k=15 ; masks regenerated every proposal; PRT=0.1+0.9NFE/B ; path t=3.0,2.7,…,0.3 ; first non-worsening proposal accepted; box clipping; 10% pop reinitialization after >50PopSize failed attempts without global best improvement.
FE budget is effective stopping rule. BBOB/CEC driver uses nonbinding migration cap of 105 ; VQE wrapper uses 104 .
iSOMA-AR
Proposed variant; identical baseline selection, path, repair, and acceptance with occasional rotated masking.
All iSOMA settings above plus Cs(0)=I , success update scale 0.12 (cap 0.35 ), eigenspace update every 5 accepted improving steps, ridge 10−4I , min. successful-step count max(6,⌈1.5D⌉) , off-axis threshold 0.18 , max rotation probability 0.45 .
No extra FEs spent on basis learning. Covariance memory and rotation gate reset at baseline stagnation restart.
jSO-derived
Fixed custom DE used throughout BBOB/CEC and VQE comparison.
N0=max(30,⌊25Dlog10(max(D,2))⌋) , linear reduction to Nmin=4 ; H=5 ; MF=0.5 , MCR=0.8 ; Cauchy F (scale 0.1 ), Gaussian CR ( σ=0.1 ); p decreases linearly 0.25→0.125 ; external archive with pop capacity; binomial crossover; clipping repair; improvement-weighted memory.
Trial mutation: xi+F(xpbest−xi)+F(xr1−xr2) . Reported as jSO-derived ; specific differences are listed below.
jSO-canonical
Post-hoc BBOB/CEC implementation-sensitivity comparator following the published jSO/iL-SHADE mechanisms.
N0=max(4,round(25ln(max(D,2))D)) , linear reduction to Nmin=4 ; H=5 ; MF=0.3 , MCR=0.8 with the special final (0.9,0.9) memory entry; p:0.25→0.125 ; early CR lower bounds and F cap; weighted current-to- p best mutation; external archive; midpoint-to-parent bound repair.
Uses the paper-prose p schedule. Run on the complete BBOB and CEC 2011 grids as a sensitivity comparator; VQE uses jSO-derived.
iL-SHADE
Standardized adaptive-DE reference through PyADE.
pyade.ilshade.get_default_params(D) with pop size 4D , explicit box bounds, supplied replicate seed, and max_evals =B . All other settings are package defaults.
No internal fallback results are included in reported experiments.
Objective-level FE guard enforces VQE budget if internal termination occurs later.
Appendix
Table 9: Optimizer implementations and numerical settings. B denotes the FE budget and D the decision-space dimension.
Algorithm
BBOB s/1000 FE
vs base
CEC s/1000 FE
vs base
iSOMA
0.0182
1.00 ×
0.0110
1.00 ×
iSOMA-AR
0.0204
1.12 ×
0.0193
1.75 ×
jSO-derived
0.0279
1.54 ×
0.0202
1.83 ×
jSO-canonical †
0.0304
1.67 ×
0.0253
2.30 ×
iL-SHADE
0.0334
1.84 ×
0.0224
2.03 ×
CMA-ES
0.0676
3.72 ×
0.0834
7.55 ×
Appendix
Table 10: Median computational overhead normalized per 1000 objective evaluations. Wall-clock values are implementation- and hardware-dependent and are included only as relative overhead indicators.
Problem
Control / default AR
W/T/L
Median ratio
Wilcoxon p
AR rotation frac.
Ellipsoid (axis-aligned)
random basis / AR default
10/0/15
1.41
0.692
7.4%
Ellipsoid (axis-aligned)
iSOMA / AR default
7/0/18
12.74
0.0451
7.4%
Ellipsoid (rotated)
random basis / AR default
3/0/22
3.16
1.20\times10−04
22.5%
Ellipsoid (rotated)
iSOMA / AR default
6/0/19
4.33
1.20\times10−04
22.5%
Rastrigin (axis-aligned)
random basis / AR default
11/0/14
1.00
0.874
12.4%
Rastrigin (axis-aligned)
iSOMA / AR default
14/0/11
0.90
0.491
12.4%
Appendix
Table 11: Controlled adaptive-rotation mechanism ablation at D=10 and 20,000 FEs (25 paired runs). W/T/L and the median error ratio compare the listed control with default learned AR; ratios above one favor learned AR.
Problem
Variant
W/T/L
Median ratio
Wilcoxon p
Ellipsoid (axis-aligned)
AR pmax=0.25
11/0/14
2.85
0.23
Ellipsoid (axis-aligned)
AR pmax=0.65
13/0/12
0.79
0.916
Ellipsoid (axis-aligned)
AR τ=0.10
15/0/10
0.45
0.474
Ellipsoid (axis-aligned)
AR τ=0.30
9/0/16
8.51
0.21
Ellipsoid (rotated)
AR pmax=0.25
6/0/19
1.98
0.0125
Ellipsoid (rotated)
AR pmax=0.65
12/0/13
1.07
0.23
Appendix
Table 12: One-factor sensitivity of the adaptive-rotation gate. Each variant is paired against the default setting τ=0.18 , pmax=0.45 .
Function
base W/T/L
base/AR
p
random W/T/L
random/AR
p
f2
6/0/14
6.88
0.0973
14/0/6
0.05
0.0637
f3
10/0/10
1.00
0.784
10/0/10
1.00
0.898
f8
4/0/16
3.53
0.00639
8/0/12
1.79
0.245
f9
4/0/16
1.41
0.00365
3/0/17
1.67
7.08\times10−04
f10
4/0/16
1.94
0.00315
4/0/16
3.01
0.00315
f15
6/0/14
1.37
0.114
11/0/9
0.84
0.596
Appendix
Table 13: Selected BBOB corroboration of the AR mechanism at D=10 , instances 1–2, 2000 FE/D (20 paired runs per function after pooling instances). Ratios above one favor default learned AR.
Model
Algorithm
FE
W/T/L
Error ratio
Wilcoxon p
Same-point allowed
Same-point repaired
Q1
iSOMA
10k
14/0/11
0.90
0.833
2.34%
0.50%
Q1
iSOMA
30k
15/0/10
0.88
0.596
2.71%
0.92%
Q2
iSOMA
10k
10/0/15
1.18
0.396
0.49%
0.46%
Q2
iSOMA
30k
9/0/16
1.15
0.252
0.91%
0.40%
Q1
iSOMA-AR
10k
14/0/11
0.82
0.672
2.26%
0.37%
Q1
iSOMA-AR
30k
14/0/11
0.96
0.833
2.56%
0.53%
Appendix
Table 14: All-zero PRT-mask control under high effective-shot Gaussian noise at N=12 (25 paired runs). The nonzero variant repairs an all-zero Bernoulli mask by activating one uniformly random coordinate. Performance ratios below one favor the repair.
Study
Algorithm
Mean rank
Top-1
All targets
Tight target
Tight runtime
BBOB
jSO-derived
2.00
46.8%
65.4%
48.8%
1.08×103
BBOB
jSO-canonical
3.33
18.6%
57.8%
36.8%
1.52×103
Direct derived/canonical: 557/104/59 W/T/L; median error ratio 0.48; Wilcoxon p=6.32\text{\times}{10}^{-79}$$ .
CEC 2011
jSO-derived
1.77
63.6%
62.9%
29.1%
0.504
CEC 2011
jSO-canonical
2.73
27.3%
56.7%
23.6%
0.222
Direct derived/canonical: 15/2/5 W/T/L; median regret ratio 0.47; Wilcoxon p=0.0853 .
Appendix
Table 15: Post-hoc canonical-jSO sensitivity analysis. Ranks and Top-1 shares use the six-method sensitivity panels. The tight target is Δf=10−8 for BBOB and the 10−6 relative-gap target for CEC 2011; tight runtime is FE/D for BBOB and a fraction of the 50,000-FE budget for CEC. Direct rows compare jSO-derived with jSO-canonical, and ratios below one favor jSO-derived.
Group
Frequent winner
AR/base W/T/L
AR/base ratio
AR/iLS W/T/L
AR/iLS ratio
C1 ill-conditioned / anisotropic
CMA-ES 137/210
186/2/22
0.49
60/2/148
2.72
C2 valley / ridge path-following
CMA-ES 122/150
109/0/41
0.80
35/0/115
1.76
C3 structured multimodality
jSO-derived 125/210
138/0/72
0.91
80/0/130
1.42
C4 basin selection / weak structure
jSO-derived 60/120
72/3/45
0.99
65/3/52
0.92
C5 plateau / nonsmooth / rugged
jSO-derived 48/120
83/0/37
0.86
24/0/96
1.67
C6 asymmetry / deception
CMA-ES 42/90
63/0/27
0.92
39/0/51
1.37
Appendix
Table 16: Diagnostic BBOB groups used to separate landscape mechanisms. These groups overlap by design.
f
Function
Landscape feature
Frequent winner
AR/base
AR/iLS
jSO-derived/CMA
1
Sphere
smooth, isotropic, unimodal
CMA-ES 30/30
19/9/2
1/10/19
0/20/10
2
Ellipsoidal (sep.)
separable; unimodal; κ≈106
CMA-ES 25/30
27/0/3
4/0/26
5/15/10
3
Rastrigin (sep.)
separable; regular multimodality; ∼10D optima
jSO-derived 23/30
22/0/8
29/0/1
28/0/2
4
Bueche–Rastrigin
asymmetric/deceptive Rastrigin; ∼10D optima
jSO-derived 22/30
16/0/14
30/0/0
29/0/1
5
Linear Slope
linear; optimum at domain boundary
4-way tie (30/30)
0/30/0
7/23/0
0/30/0
6
Attractive Sector
strongly asymmetric unimodal
CMA-ES 30/30
29/0/1
9/0/21
0/10/20
Appendix
Table 17: Function-level BBOB outcomes across 30 matched dimension–instance–budget conditions per function. AR/base and AR/iLS are W/T/L for iSOMA-AR against baseline iSOMA and iL-SHADE.
F
Problem
Application family
D
Frequent winner
AR/base
AR/iLS
jSO-derived/CMA
1
FM sound parameter estimation
A1 Control & param.
6
jSO-derived 1/1
0/0/1
1/0/0
1/0/0
2
Lennard–Jones potential
A2 Materials
30
jSO-derived 1/1
0/0/1
0/0/1
1/0/0
3 †
Bifunctional catalyst opt. control
A1 Control & param.
1
All tied (1/1)
0/1/0
0/1/0
0/1/0
4 †
Stirred-tank reactor opt. control
A1 Control & param.
1
jSO-derived 1/1
0/1/0
0/1/0
1/0/0
5
Tersoff potential Si(B)
A2 Materials
30
jSO-derived 1/1
0/0/1
0/0/1
1/0/0
6
Tersoff potential Si(C)
A2 Materials
30
jSO-derived 1/1
0/0/1
1/0/0
1/0/0
Appendix
Table 18: CEC 2011 problem-level outcomes at 50,000 evaluations. A dagger ( † ) marks problems for which the Minion C++ rewrite has documented discrepancies relative to the Octave/MATLAB reference.
Model
Condition
iSOMA median
AR median
pHolm
Effect
Q1
N=12 , 10k
1.50\times10−05
1.46\times10−04
0.886
-0.136
Q1
N=12 , 30k
0
1.17\times10−13
0.005
-0.485
Q1
N=14 , 10k
6.93\times10−08
5.10\times10−07
0.159
-0.341
Q1
N=14 , 30k
0
0
1.000
-0.110
Q1
N=16 , 10k
0.458
0.458
1.000
0.014
Q1
N=16 , 30k
0.458
0.458
0.030
-0.469
Appendix
Table 19: Direct exact-objective comparison of iSOMA-AR with baseline iSOMA. pHolm is the Holm-adjusted two-sided Mann–Whitney U result within each benchmark condition. The effect column is oriented so that positive values favor iSOMA-AR because lower error is better.
(a) Condition-wise mean rank
(b) Median selection penalty
Optimizer
Low noise
High noise
Low noise
High noise
iSOMA
3.00
2.17
0.034
0.313
iSOMA-AR
3.00
1.83
0.046
0.381
jSO-derived
4.00
3.67
0.069
0.620
iL-SHADE
3.67
3.50
0.073
0.583
CMA-ES
3.83
4.50
0.044
0.410
Appendix
Table 20: Performance metrics in the N=12 noisy VQE extension across low- and high-noise regimes. Lower is better for both metrics.
Model
Budget
iSOMA median
AR median
pHolm
Effect
High noise
Q1
10k
1.037
0.958
1.000
0.181
Q1
30k
0.719
0.649
1.000
0.034
Q2
10k
0.593
0.707
0.898
-0.043
Q2
30k
0.433
0.522
0.664
-0.162
Q3
10k
2.318
2.425
0.923
0.018
Appendix
Table 21: Direct noisy-objective comparison of iSOMA-AR with baseline iSOMA. Endpoints are scored by their exact energy after noisy selection. No direct AR–baseline comparison remains significant after Holm correction.
Feedback-based adaptive quantum optimization (FALQON) is a promising approach for solving combinatorial problems on noisy intermediate-scale quantum (NISQ) devices, requiring only single circuit evaluations per layer. However, standard FALQON relies on fixed hyperparameters that severely limit convergence speed, requiring hundreds to thousands of layers for acceptable solutions. This paper proposes Optimal FALQON, an optimization-based formulation that treats the per-layer time step (δk) and scaling factor (Mk) as decision variables optimized via classical methods. We present a comprehensive empirical study on all 94 non-isomorphic 3-regular graphs with 12 vertices, comparing Optimal FALQON with standard FALQON and multiple QAOA variants. Results demonstrate statistically significant improvements in success probability, evaluation efficiency, and depth-normalized cost across the evaluated benchmarks. Furthermore, initializing QAOA with parameters from Optimal FALQON yields superior warm-start performance compared to fixed initialization.
Michael Mancini, Shabnam Sodagari
Computer Engineering & Computer Science, California State University, Long Beach, CA 90840
Barren-plateau results have established exponential gradient suppression as a widely cited obstacle to the scalability of variational quantum algorithms. When and whether these results extend to a given objective has been addressed through loss-specific arguments, but a general structural characterization has remained open. We show that the objective itself admits a fixed-observable representation if and only if the loss is affine in the measured statistics, thereby identifying the exact boundary of the standard concentration-based proof template. Existing transfer results for non-affine losses achieve this reduction under additional assumptions; our characterization implies that such a reduction is not structurally available for a class of non-affine objectives, placing them outside the automatic reach of the existing proof template. Beyond the affine regime, a chain-rule decomposition reveals three governing factors -- model responsivity, loss-side signal, and transmittance -- and induces a loss-class dichotomy: bounded-gradient losses inherit suppression, while amplification-capable losses can in principle counteract it. In the exponentially wide setting, both classes fail, but for different structural reasons. When the interface is instead designed at polynomial width -- exposing coarse-grained statistics rather than individual bitstring probabilities -- the exponential-dimensional obstruction is relaxed and the dichotomy plays a genuine role. In a numerical demonstration on a charge-conserving quantum system, the amplification-capable objective produces resolved gradients several orders of magnitude larger than affine and inheriting baselines at comparable shot budgets. Over the tested interval, its scaling trend is statistically distinguished from the exponential trend of both alternatives. The boundary is affine; what lies beyond it is a representation-design problem.
Gordon Ma, Xiufan Li
Centre for Quantum Technologies, National University of Singapore, Singapore
This paper presents an Optuna-based formulation-level auto-tuning framework for support vector machines (SVMs) implemented on multiple quantum-inspired annealers. In an annealing-based SVM, continuous dual variables are discretized and converted into a quadratic unconstrained binary optimization (QUBO) model. This transformation introduces three coupled classes of parameters: representation parameters-the encoding base B and bit depth K-which determine numerical range, resolution, and QUBO size; the RBF kernel parameter γ, which determines classifier geometry; and the equality-constraint penalty ξ, which controls feasibility and coefficient balance. We formulate their joint selection as a mixed discrete-continuous black-box optimization problem. The framework has two optimization levels: an inner annealer minimizes the generated QUBO, while an outer Optuna loop reconstructs the formulation in every trial and maximizes validation accuracy. The same solver-agnostic procedure is applied to Fixstars Amplify Annealing Engine, Toshiba SQBM+, and Fujitsu Digital Annealer using TPE and Gaussian-process samplers and is compared with conventional grid search. Experiments on linear and nonlinear classification tasks with 0-20% label noise show mean gains over grid search of approximately 0.8 and 2.1 percentage points, respectively. The results demonstrate that formulation quality and backend capability must be evaluated jointly and that task-level feedback can compensate for discretization, penalty imbalance, and backend-dependent approximate optimization.