RoTO Benchmark for Constrained Optimization

• RoTO Benchmark is a suite of linear constrained optimization problems that apply rotation and translation to the classical Klee–Minty formulation, eliminating coordinate biases.
• It challenges randomized search and evolutionary algorithms by introducing coupling among decision variables, ensuring robust testing of invariance and constraint handling.
• Empirical results show that advanced methods like ϵMAg-ES outperform others in efficiency and accuracy, highlighting the benchmark’s significance in evolutionary algorithm research.

The RoTO Benchmark is a suite of linear constrained optimization problems specifically constructed to evaluate and stress-test the robustness, scalability, and invariance properties of randomized search algorithms, particularly evolutionary algorithms (EAs), in the domain of constrained optimization. Its foundation is a geometric transformation of the classical Klee–Minty problem, designed to eliminate coordinate-alignment biases and introduce coupling among decision variables, thereby creating challenging, reproducible scenarios for black-box optimization methods (Hellwig et al., 2018).

1. Foundational Problem: The Klee–Minty Linear Program

The original Klee–Minty LP is a perturbed $n$-dimensional unit-cube problem with an objective and constraint structure notorious for exposing the exponential worst-case complexity of the simplex algorithm. The problem is formalized as: $$\min_{x\in\mathbb{R}^n}\;c^T x \quad\text{s.t.}\quad A\,x\le b,$$ where $c \in\mathbb{R}^n$ is $(0,\dots,0,1)^T$, matrix $A\in\mathbb{R}^{2n\times n}$ combines identity and lower-triangular parts with perturbation parameter $\varepsilon=0.1$, and $b\in\mathbb{R}^{2n}$ encodes upper and lower bounds for each variable via $A_1$ (upper) and $A_2$ (lower) blocks. The unique optimum lies at $x^*=(0, \ldots, 0)^T$. This construction yields a highly axis-aligned, separable feasible region (Hellwig et al., 2018).

2. The RoTO Transformation: Rotation and Translation

To eradicate coordinate-system biases and disallow simple coordinate-wise search strategies from exploiting separability, the RoTO benchmark applies a rigid isometry (rotation plus translation) to the Klee–Minty polytope. The transformation is defined as: $$\min_{x\in\mathbb{R}^n}\;c^T x \quad\text{s.t.}\quad A\,x\le b,$$0 where $$\min_{x\in\mathbb{R}^n}\;c^T x \quad\text{s.t.}\quad A\,x\le b,$$1 is an orthogonal rotation matrix (in the 2-plane spanned by $$\min_{x\in\mathbb{R}^n}\;c^T x \quad\text{s.t.}\quad A\,x\le b,$$2 and $$\min_{x\in\mathbb{R}^n}\;c^T x \quad\text{s.t.}\quad A\,x\le b,$$3, with angle $$\min_{x\in\mathbb{R}^n}\;c^T x \quad\text{s.t.}\quad A\,x\le b,$$4), and $$\min_{x\in\mathbb{R}^n}\;c^T x \quad\text{s.t.}\quad A\,x\le b,$$5 translates the optimum deep into the interior. The resulting rotated problem formulation is: $$\min_{x\in\mathbb{R}^n}\;c^T x \quad\text{s.t.}\quad A\,x\le b,$$6 with the unique minimum at $$\min_{x\in\mathbb{R}^n}\;c^T x \quad\text{s.t.}\quad A\,x\le b,$$7 (Hellwig et al., 2018).

3. Benchmark Generation and Scalability

The RoTO benchmark suite is scalable in $$\min_{x\in\mathbb{R}^n}\;c^T x \quad\text{s.t.}\quad A\,x\le b,$$8 and parametrized by rotation matrix $$\min_{x\in\mathbb{R}^n}\;c^T x \quad\text{s.t.}\quad A\,x\le b,$$9 and translation $c \in\mathbb{R}^n$0. Instances are generated by choosing $c \in\mathbb{R}^n$1 from a standard set (e.g., $c \in\mathbb{R}^n$2), fixing $c \in\mathbb{R}^n$3 and $c \in\mathbb{R}^n$4, and applying the specified rotation. Additional diversity is achieved by altering $c \in\mathbb{R}^n$5 or randomizing $c \in\mathbb{R}^n$6, while preserving hardness through the inherited worst-case properties of the Klee–Minty form. For each dimension, at least 15 independent problem instances are typically sampled for robust comparison (Hellwig et al., 2018).

4. Performance Metrics and Evaluation Protocol

Evaluation under the RoTO benchmark adheres to protocols emphasizing black-box optimization and fairness. One function evaluation comprises the computation of both objective $c \in\mathbb{R}^n$7 and the full constraint vector $c \in\mathbb{R}^n$8. The standard evaluation budget is $c \in\mathbb{R}^n$9 function evaluations per run.

Termination conditions are:

• $(0,\dots,0,1)^T$0
• No improvement for $(0,\dots,0,1)^T$1 evaluations
• Exhaustion of budget

Constraint violation is quantified as: $(0,\dots,0,1)^T$2 and solutions are lexicographically ordered: $(0,\dots,0,1)^T$3 Summary metrics include: best-ever objective $(0,\dots,0,1)^T$4, median objective and violation $(0,\dots,0,1)^T$5, $(0,\dots,0,1)^T$6, feasibility rate ($(0,\dots,0,1)^T$7), mean objective-space distance to $(0,\dots,0,1)^T$8, and mean evaluations to termination. Results are customarily visualized via ECDF plots over achievement of a suite of target values, extending analytic transparency (Hellwig et al., 2018).

5. Empirical Results and Algorithmic Insights

The RoTO benchmark differentiates optimizers based on their invariance to rotation, constraint-handling, and global search capacity. In comparative studies, $(0,\dots,0,1)^T$9MAg-ES and LSHADE44 (state-of-the-art EAs) achieved $A\in\mathbb{R}^{2n\times n}$0 to within $A\in\mathbb{R}^{2n\times n}$1 for $A\in\mathbb{R}^{2n\times n}$2 up to 40 with $A\in\mathbb{R}^{2n\times n}$3. $A\in\mathbb{R}^{2n\times n}$4MAg-ES required fewer evaluations (e.g., $A\in\mathbb{R}^{2n\times n}$5 for $A\in\mathbb{R}^{2n\times n}$6) than LSHADE44 ($A\in\mathbb{R}^{2n\times n}$7), while random search was consistently ineffective. LP solvers (e.g., glpk) reached similar optimal values, but with larger numerical errors in $A\in\mathbb{R}^{2n\times n}$8. ECDFs demonstrated that $A\in\mathbb{R}^{2n\times n}$9MAg-ES exhibits a superior early-time performance profile, increasing with problem dimension (Hellwig et al., 2018).

6. Algorithmic Implications and Best Practices

By eliminating axis-alignment and introducing enforced coupling, the RoTO benchmark invalidates coordinate-based and separable search heuristics, challenging algorithms to adapt via rotationally-invariant sampling or adaptive covariance. Practical recommendations include initializing uniformly on $\varepsilon=0.1$0, employing lexicographic tie-breakers, and reporting the full complement of metrics and ECDFs. The RoTO construction enables modular instance generation for large-scale benchmarking, offering a reproducible platform for testing global search and rotational invariance across linear constraint landscapes (Hellwig et al., 2018).

7. Significance in Evolutionary Algorithm Research

The RoTO benchmark substantially addresses a critical gap in constrained optimization benchmarking by presenting stringent, scalable, and reproducible test cases that reflect real-world geometric ambiguity and variable coupling. Its design ensures avoidance of the “coordinate-bias trap,” allowing robust comparative evaluation of stochastic optimizers under rotation and translation. RoTO thus serves as a focal environment for developing and calibrating covariance-adapting, invariance-respecting, and constraint-resilient search methodologies, and remains a recommended baseline for experimental studies in evolutionary computation (Hellwig et al., 2018).