Synthetic Pandora’s Box Benchmark

• Synthetic Pandora’s Box Benchmark is a collection of 24 high-dimensional box-constrained optimization problems that assess numerical global optimization methods.
• Each instance is generated using the GNBG methodology, embedding controlled attributes such as modality, ruggedness, symmetry, conditioning, variable interactions, basin linearity, and deceptiveness.
• The benchmark offers reproducible MATLAB implementations with fixed seeds, enabling precise evaluation of algorithm performance across diverse landscape challenges.

The Synthetic Pandora’s Box Benchmark, formally known as the GNBG-Generated Test Suite for Box-Constrained Numerical Global Optimization, is a systematically designed collection of 24 high-dimensional test problems developed for the rigorous assessment and comparison of numerical global optimization algorithms. Using the Generalized Numerical Benchmark Generator (GNBG), each instance introduces carefully controlled attributes—modality, ruggedness, symmetry, conditioning, variable interactions, basin linearity, and deceptiveness—across a spectrum of computational challenges, all within fixed box constraints in $\mathbb{R}^{30}$ and providing explicit, reproducible implementations (Gandomi et al., 2023).

1. GNBG Baseline Formula and Design Parameters

Each instance in the Pandora's Box benchmark is a minimization problem over $x \in [-100,100]^{30}$ (i.e., $d=30$), parameterized by the number of components $o$, with objective function:

$$f(x) = \min_{k=1,\dots,o}\left\{ \sigma_k + \left( T_k(R_k(x - m_k))^\top H_k \, T_k(R_k(x - m_k)) \right)^{\lambda_k} \right\}.$$

• $m_k \in \mathbb{R}^{30}$: Component center, uniformly sampled from $[-80,80]^{30}$ under fixed seed.
• $\sigma_k \leq 0$: Component offset, drawn from prescribed ranges with fixed seed for reproducibility.
• $H_k = \operatorname{diag}(h_{k,1},\dots,h_{k,30})$: Axis scaling matrix, sets the conditioning.
• $R_k \in \mathbb{R}^{30 \times 30}$: Orthonormal rotation, constructed via a Givens-rotation-based procedure encoding variable interactions.
• $\lambda_k > 0$: Exponent controlling basin linearity; super-linear if $>0.5$, linear if $=0.5$, sub-linear if $<0.5.$
• $T_k(\cdot)$: Applied element-wise, injects controlled ruggedness:

$$T_k(a_j) = \begin{cases} \exp\left(\ln a_j + \mu_{k,1} [\sin(\omega_{k,1} \ln a_j) + \sin(\omega_{k,2} \ln a_j)]\right) & a_j > 0 \ 0 & a_j = 0 \ -\exp\left(\ln|a_j| + \mu_{k,2} [\sin(\omega_{k,3} \ln|a_j|) + \sin(\omega_{k,4} \ln|a_j|)]\right) & a_j < 0 \end{cases}$$

with $\mu_{k,i},\,\omega_{k,j}$ controlling ruggedness phase and frequency.

All random draws are implemented with fixed random seeds, ensuring global reproducibility for each instance (Gandomi et al., 2023).

2. The 24-Instance Suite: Structure and Explicit Problem Forms

The test suite comprises:

• Unimodal Instances ($f_1$–$f_6$): Single-basin landscapes ranging from sphere-like, severely ill-conditioned, linearly and nonlinearly transformed, and narrowed/rotated valleys.
• Single-Component Multimodal Instances ($f_7$–$f_{15}$): Functions exhibiting moderate-to-extreme multimodality and ruggedness, basin asymmetry, various separability patterns, block-diagonal or fully random rotations, and highly ill-conditioned cases.
• Multi-Component Multimodal Instances ($f_{16}$–$f_{24}$): Compositions involving $o \geq 2$ heterogeneous basins—ranging from tightly clustered, deceptive, highly rugged, to entirely overlapping, ill-conditioned, asymmetric, and non-separable formulations.

Each $f_i(x)$ is specified with full parameterization: dimensionality, number and placement of components, condition matrices, rotation matrices, ruggedness controls, and explicit domain and randomization settings. The explicit forms, detailed in the source, ensure direct mapping from benchmark specification to MATLAB implementation (Gandomi et al., 2023).

3. Feature Characterization and Diversity Coverage

Quantitative features for each instance, as codified in the suite's summary table, include:

• Modality: Unimodal, multi-1 (single-component but multimodal), or multi-m (multi-component).
• Number of Basin Local Optima: Distinguishes single versus multi-global/local optima.
• Separability: Fully separable (F), partially separable (P), or nonseparable (N).
• Variable Interaction Complexity: Quantified by nonzero entries in $\Theta_k$ controlling $R_k$.
• Symmetry: Symmetric (S) or asymmetric (A) basin structure.
• Ill-Conditioning: From well-conditioned (51) to severely ill-conditioned (53).
• Basin Linearity: Super-linear (E), linear (Lin), or sub-linear (L) basins via $\lambda_k$.
• Deceptiveness: Explicit rating quantifying the likelihood of misleading attraction to non-global optima.

Instance	Modality	Separability	Ill-Cond.	Ruggedness	Deceptiveness
$f_1$	Unimodal	F	53	S	53
$f_7$	Multi-1	F	53	S	53
$f_{17}$	Multi-m	N	51	A	51
...	...	...	...	...	...

Each problem was constructed to systematically stress optimizers with respect to one or more of these characteristics, offering rigorous and targeted insights into algorithmic strengths and failure modes (Gandomi et al., 2023).

4. GNBG Instance Construction: Algorithmic Procedure

The generation of each benchmark problem follows a reproducible pseudocode procedure:

1. Parameter Sampling: For each component $k$:
   • Sample center $m_k$, offset $\sigma_k$ under prescribed distributions and fixed RNG seed.
   • Select $\lambda_k$, $H_k$ (condition), $\mu_k$, $\omega_k$ (ruggedness) as specified.
   • Build the upper-triangular interaction matrix $\Theta_k$ with sparsity/structure controlled by $p_k$ and nonzero entries via prescribed randomization.
2. Rotation Construction: $R_k$ is computed via a Givens-rotation loop, processing only entries of $\Theta_k$ that are nonzero.
3. Function Definition: Assemble $f(x)$ according to the baseline formula, apply box constraints by assigning $\infty$ or a large penalty for $x$ infeasible.
4. Verification and Reproducibility: All random processes use rng(fixedSeed) to guarantee that each run yields identical instances.

This construction both ensures comprehensive diversity and enables direct inference of the landscape's analytic and geometric properties (Gandomi et al., 2023).

5. Implementation and Usage

All 24 Pandora’s Box benchmark instances can be exactly reproduced in MATLAB:

• GNBG-Generator: Toolbox for generating arbitrary GNBG problems by specifying dimension, number of components, and all relevant parameter arrays (GNBG-Generator GitHub).
• GNBG-Instances: Scripts for the 24 canonical instances ($f_1$–$f_{24}$), setting all random seeds and parameterizations per the reference, callable via

  [f,info] = GNBG_Instance(i);

  where $f$ is a function handle for evaluation and info is a struct with all relevant parameters and feature codes. Full reproducibility is guaranteed by seed initialization for all random elements (Gandomi et al., 2023).

6. Benchmarking Rationale and Research Applications

Systematic variation of the GNBG parameters in Pandora’s Box enables coverage of:

• Unimodal and multimodal cases;
• Both single- and multi-basin landscapes;
• Transition from well- to severely ill-conditioned domains;
• Complete separation to total nonseparability of variables;
• Symmetric to maximally asymmetric basins;
• Rich basin nonlinearity (super-linear to sub-linear);
• User-controllable basin ruggedness and deceptiveness.

This comprehensive space is designed for stress-testing both the exploration and exploitation capabilities of global optimization algorithms under mathematically transparent and user-verifiable conditions, permitting precise attribution of algorithmic performance to underlying landscape attributes. All landscape characteristics are fully analyzable, documented, and reproducible, facilitating comparative studies, ablation analyses, and the development of novel optimization techniques benchmarked to established, interpretable standards (Gandomi et al., 2023).