Block-Bench: A Framework for Controllable and Transparent Discrete Optimization Benchmarking

Optimization benchmarks play a fundamental role in assessing algorithm performance; however, existing artificial benchmarks often fail to capture the diversity and irregularity of real-world problem structures, while benchmarks derived from real-world problems are costly and difficult to construct. To address these challenges, we propose an evolutionary automatic benchmark generation framework that leverages a large language model (LLM) as a generative operator, termed the LLM-driven evolutionary benchmark generator (LLM-EBG). In this framework, the LLM serves as an evolutionary operator that generates and evolves benchmark problems within a flexible, expressive representation space. As a case study, we generate unconstrained single-objective continuous minimization problems represented as mathematical expressions designed to induce significant performance differences between a genetic algorithm (GA) and differential evolution (DE). Experimental results show that LLM-EBG successfully produces benchmark problems in which the designated target algorithm consistently outperforms the comparative algorithm in more than 80% of trials. Furthermore, exploratory landscape analysis reveals that benchmarks favoring GA are highly sensitive to variable scaling, demonstrating that the proposed framework can generate problems with distinct geometric characteristics that reflect the intrinsic search behaviors of different optimization algorithms.

Many real-world optimization tasks are governed not only by what should be optimized, but also by what is allowed: a solution must first satisfy physical, operational, safety, budgetary, or structural requirements before its objective value becomes meaningful. Yet benchmark design for constrained continuous optimization often treats these restrictions as fixed properties of predefined test instances. Existing generators and benchmark suites provide essential foundations for reproducible comparison, but they either focus primarily on configurable objective landscapes or expose constraint properties within limited, predefined regimes, making it difficult to ask targeted questions about constraint-induced difficulty: whether an optimizer struggles because feasible regions are small, boundaries are restrictive, variables are tightly coupled, or constraint information is poorly scaled. We introduce GNBG-C, a Generalized Numerical Benchmark Generator for Constrained Continuous Optimization. The central idea is to treat the feasible region itself as a configurable object rather than as an inseparable attachment to a fixed test function. By generating controlled constraint regimes around the known GNBG global minimizer, the generator enables systematic studies in which feasibility pressure can be varied while preserving that point as a global minimizer of the constrained problem. This allows researchers to examine how optimizers discover feasibility, handle constraint pressure, and improve objective quality under diverse and controllable feasibility conditions. GNBG-C therefore provides a flexible and auditable framework for probing the strengths and weaknesses of constrained continuous optimizers under targeted forms of feasibility-side difficulty.

The MOEA/D is one of the most successful multi-objective evolutionary algorithms. It operates by decomposing the optimization problem into single-objective subproblems, which are then solved in a co-evolutionary manner. The performance of the MOEA/D depends critically on this decomposition. In this work, we propose a novel way to adjust this decomposition on the fly, for the case of discrete bi-objective optimization. Different from all previous works in this direction, we support our new algorithm with a proven performance guarantee. This is the first work that rigorously proves the runtime of a dynamic multi-objective evolutionary algorithm. For monotonic distortions of the simple LOTZ benchmark, it is known that classic decompositions leave large holes in the Pareto front, making it very hard for the MOEA/D to cover the entire Pareto front. In contrast, we prove that our self-adjusting MOEA/D solves all these LOTZ problems in an expected number of $O(n^4)$ function evaluations. This relatively low runtime as well as the details of our mathematical proof show that the dynamic decomposition strategy indeed evolves a decomposition adjusted to each of these problem instances.