DynamicEn Benchmark Framework

• DynamicEn benchmark is a framework for creating realistic dynamic multi-objective optimization environments by evolving Pareto-optimal sets and fronts over time.
• It employs hypersurface-based Pareto dynamics, controlled variable imbalances, rotational non-separability, and irregular perturbations to mimic complex real-world optimization challenges.
• The framework integrates time-linkage and history-dependent mechanisms, offering reproducible and tunable instances for assessing optimization algorithms under dynamic conditions.

Searching arXiv for DynamicEn benchmark and closely related dynamic-environment optimization benchmarks. DynamicEn is a benchmarking framework for dynamic multi-objective optimization (DMOO) that constructs configurable dynamic environments in which objective functions, Pareto-optimal sets, and Pareto fronts evolve over time. In the formulation associated with the framework, a dynamic multi-objective optimization problem is written as

$$\min_{\mathbf{x}(t) \in \mathcal{X}(t)} \;\mathbf{f}(\mathbf{x}(t), t) \;=\; \big(f_1(\mathbf{x}(t), t),\dots,f_m(\mathbf{x}(t), t)\big),$$

subject to time-dependent constraints, with the benchmark designed to make these dynamics more realistic through changing Pareto-optimal sets on hypersurfaces, controlled variable contribution imbalances, time-varying non-separability via rotations, irregular perturbations, and generalized time-linkage (Shao et al., 4 Jan 2026). In this sense, DynamicEn is not a single test function but a generalized benchmark generation framework whose purpose is to provide fair, reproducible, and discriminative evaluation of DMOO algorithms under practical dynamic conditions (Shao et al., 4 Jan 2026).

1. Origins and Benchmarking Context

DynamicEn emerges from the broader problem that dynamic optimization benchmarks have often emphasized relatively simple change mechanisms, such as regular shifts in objective landscapes or Pareto structures, while underrepresenting heterogeneous variable sensitivities, irregular environmental changes, and history-dependent effects. The DynamicEn framework is explicitly presented as a response to this limitation, with the stated goal of constructing “highly realistic and challenging DMOO benchmarks” that go beyond conventional formulations (Shao et al., 4 Jan 2026).

This positioning matters because dynamic optimization has long relied on benchmark families in which either multimodality or temporal change is present, but not always in combinations that reflect real-world complexity. The CEC 2022 benchmark for seeking multiple optima in dynamic environments, for example, focuses on dynamic multimodal optimization problems with 24 problems built from 8 multimodal functions and 8 change modes, including modes that alter the number of global optima over time (Luo et al., 2022). That benchmark is centered on tracking multiple optima in changing landscapes, whereas DynamicEn is centered on dynamic multi-objective optimization, especially the evolution of Pareto-optimal sets and fronts, heterogeneous conditioning, and non-separability (Shao et al., 4 Jan 2026). This suggests that DynamicEn occupies a distinct place in the benchmark landscape: it is aimed less at dynamic single-objective or multimodal tracking and more at the structural and geometric challenges unique to time-varying multi-objective search.

The framework also differs from dynamic discrete optimization benchmarks such as the Dynamic Binary Value problem integrated into IOHprofiler, which preserves monotonicity while changing weights over time to sharply discriminate among genetic algorithms (Vermetten et al., 2024). DynamicEn instead targets continuous multi-objective settings and places particular emphasis on dynamic Pareto geometry, temporal perturbation, and time-linkage (Shao et al., 4 Jan 2026).

2. Formal Problem Structure and Core Construction

At the core of DynamicEn is a factorized objective formulation

$$f_i(\mathbf{x}, t) = g(\mathbf{x}, t)\,\mu_i(\mathbf{x}, t), \quad i=1,\ldots,m,$$

where $g(\cdot)$ encodes distance to the dynamic Pareto-optimal set and $\mu_i(\cdot)$ controls Pareto-front geometry such as convexity, concavity, degeneracy, or disconnectedness (Shao et al., 4 Jan 2026). This factorization preserves the interpretability of classic benchmark constructions while allowing richer temporal variation.

The decision vector is partitioned into $x_I$, $x_{II,1}$, and $x_{II,2}$, where $x_I$ controls the position and shape of the Pareto-optimal set hypersurfaces and the $x_{II,\cdot}$ blocks contribute to the distance term $g$ (Shao et al., 4 Jan 2026). The benchmark’s generalized hypersurface formulation is

$$f_i(\mathbf{x}, t) = g(\mathbf{x}, t)\,\mu_i(\mathbf{x}, t), \quad i=1,\ldots,m,$$0

with $$f_i(\mathbf{x}, t) = g(\mathbf{x}, t)\,\mu_i(\mathbf{x}, t), \quad i=1,\ldots,m,$$1, and the Pareto-optimal set is characterized by $$f_i(\mathbf{x}, t) = g(\mathbf{x}, t)\,\mu_i(\mathbf{x}, t), \quad i=1,\ldots,m,$$2, implying

$$f_i(\mathbf{x}, t) = g(\mathbf{x}, t)\,\mu_i(\mathbf{x}, t), \quad i=1,\ldots,m,$$3

This construction generalizes older hyperplane-based Pareto-set motion by allowing hypersurface-defined Pareto-optimal sets (Shao et al., 4 Jan 2026).

The time parameterization can be discrete or continuous. For generation-based evolutionary runs, DynamicEn uses

$$f_i(\mathbf{x}, t) = g(\mathbf{x}, t)\,\mu_i(\mathbf{x}, t), \quad i=1,\ldots,m,$$4

where $$f_i(\mathbf{x}, t) = g(\mathbf{x}, t)\,\mu_i(\mathbf{x}, t), \quad i=1,\ldots,m,$$5 controls change frequency and $$f_i(\mathbf{x}, t) = g(\mathbf{x}, t)\,\mu_i(\mathbf{x}, t), \quad i=1,\ldots,m,$$6 controls severity, with smaller $$f_i(\mathbf{x}, t) = g(\mathbf{x}, t)\,\mu_i(\mathbf{x}, t), \quad i=1,\ldots,m,$$7 producing larger jumps per change (Shao et al., 4 Jan 2026). This explicit separation of frequency and severity is standard in dynamic optimization, but DynamicEn extends it with additional perturbation and time-linkage mechanisms.

3. Dynamic Mechanisms: Hypersurfaces, Imbalance, Rotations, and Perturbations

A defining feature of DynamicEn is that dynamic change is not limited to simple parameter drift. The framework introduces four major mechanisms that collectively increase realism and difficulty (Shao et al., 4 Jan 2026).

First, it allows Pareto-optimal sets to move on hypersurfaces rather than hyperplanes. The functions $$f_i(\mathbf{x}, t) = g(\mathbf{x}, t)\,\mu_i(\mathbf{x}, t), \quad i=1,\ldots,m,$$8 and $$f_i(\mathbf{x}, t) = g(\mathbf{x}, t)\,\mu_i(\mathbf{x}, t), \quad i=1,\ldots,m,$$9 embed time-varying hypersurfaces in the decision space, so the optimizer must track a dynamically moving, potentially curved optimal manifold rather than a simpler affine structure (Shao et al., 4 Jan 2026).

Second, DynamicEn introduces controlled variable contribution imbalances through weighted additive components: $g(\cdot)$0 where $g(\cdot)$1 sets the time-varying contribution of variable $g(\cdot)$2 to objective $g(\cdot)$3, and imbalance ratios such as

$g(\cdot)$4

control conditioning heterogeneity (Shao et al., 4 Jan 2026). The component functions $g(\cdot)$5 can induce multimodality or ruggedness; examples given in the framework include Rastrigin-like and Schaffer-like structures (Shao et al., 4 Jan 2026). This means that not all variables matter equally at a given time, and their relative importance can drift.

Third, DynamicEn uses time-varying rotations to induce non-separability. It defines orthogonal rotations via

$g(\cdot)$6

or equivalently through Givens factorizations, and transforms variables as

$g(\cdot)$7

with a drifting center $g(\cdot)$8 (Shao et al., 4 Jan 2026). By replacing $g(\cdot)$9 with $\mu_i(\cdot)$0 inside $\mu_i(\cdot)$1, $\mu_i(\cdot)$2, or $\mu_i(\cdot)$3, the framework produces time-varying correlations among variables. This is important because many dynamic benchmark suites remain at least partially separable; DynamicEn explicitly aims to invalidate divide-and-conquer shortcuts through dynamic interactions (Shao et al., 4 Jan 2026).

Fourth, it supports irregular temporal perturbations. A reproducible irregularity mechanism based on digits of $\mu_i(\cdot)$4 is given by

$\mu_i(\cdot)$5

where $\mu_i(\cdot)$6 is the relevant decimal digit of $\mu_i(\cdot)$7 (Shao et al., 4 Jan 2026). Optional stochastic perturbations include Poisson shocks and AR(1)-style random walks on parameters (Shao et al., 4 Jan 2026). The use of deterministic $\mu_i(\cdot)$8-based jitter is notable because it introduces irregularity while preserving reproducibility.

4. Time-Linkage and History-Dependent Difficulty

One of the most distinctive components of DynamicEn is generalized time-linkage. Instead of treating each environment as independent once the new time index is known, the framework allows past decisions or past approximation quality to influence the current problem instance (Shao et al., 4 Jan 2026).

The general form is

$\mu_i(\cdot)$9

where $x_I$0 is a history penalty, $x_I$1 is memory length, and $x_I$2 controls linkage strength (Shao et al., 4 Jan 2026). The framework provides examples such as squared displacement penalties and delayed-target penalties. It also defines a Pareto-set time-linkage mechanism through a scaling factor $x_I$3: $x_I$4 A practical example is

$x_I$5

so that larger errors in the previous stage inflate current difficulty (Shao et al., 4 Jan 2026).

This mechanism is intended to capture error accumulation and time deception. A plausible implication is that DynamicEn can benchmark not only tracking performance but also resilience to compounding mistakes, which is especially relevant for prediction-based or memory-based DMOO algorithms. That emphasis is comparatively uncommon in earlier dynamic benchmark designs, which often assume that environmental changes are exogenous and independent of prior approximation quality.

5. Benchmark Instances, Generation Procedure, and Evaluation Protocol

DynamicEn is modular rather than monolithic. Users specify the number of objectives $x_I$6, number of variables $x_I$7, variable partitions, Pareto-front geometry, change schedule, rotation speeds, heterogeneity profiles, time-linkage parameters, and perturbation processes (Shao et al., 4 Jan 2026). The framework provides a generation procedure that updates time-dependent parameters, constructs rotations, evaluates transformed variables, and computes objective values over a horizon of changes (Shao et al., 4 Jan 2026).

Three example instances are explicitly described. “DynEn-HypSurf-2” is a two-objective problem with hypersurface Pareto sets, rotation, and irregular perturbation; “DynEn-ZDT1′” is a dynamic variant of ZDT1 with heterogeneity and shocks; and “DynEn-DTLZ2-3” is a three-objective problem with spherical front radius drift and rotation (Shao et al., 4 Jan 2026). These examples are not merely illustrative. They show how the framework can recreate recognizable objective geometries while adding multiple layers of dynamic complexity.

Evaluation follows standard dynamic multi-objective metrics. DynamicEn reports per-time Inverted Generational Distance,

$x_I$8

Hypervolume,

$x_I$9

and Tracking Error,

$x_{II,1}$0

together with temporal aggregates such as $x_{II,1}$1, $x_{II,1}$2, and $x_{II,1}$3 (Shao et al., 4 Jan 2026). The protocol recommends $x_{II,1}$4, $x_{II,1}$5, approximately 50 changes, and at least 20 independent runs with statistical testing such as Friedman ranking and Wilcoxon comparison (Shao et al., 4 Jan 2026).

The benchmark also includes an online responsiveness notion through first hitting time after each environmental change, which is useful for measuring recovery speed rather than only average approximation quality (Shao et al., 4 Jan 2026). This suggests that DynamicEn is designed to support both offline quality assessment and online adaptation assessment.

6. Distinctiveness, Practical Use, and Limitations

DynamicEn’s main distinctive properties are explicitly identified as Pareto-set dynamics on hypersurfaces, heterogeneous landscapes via controlled imbalance, dynamic non-separability through rotations, irregular changes via $x_{II,1}$6-digit jitter and shocks, and time-linkage for error accumulation and deception (Shao et al., 4 Jan 2026). These mechanisms collectively differentiate it from earlier suites based primarily on dynamic versions of ZDT-, DTLZ-, DF-, or FDA-style structures. The authors argue that these additions improve realism and discriminative power, and they report that imbalances and interactions can yield order-of-magnitude changes in dynamic IGD and dynamic HV while runtime remains nearly unchanged (Shao et al., 4 Jan 2026).

For practical use, the framework defines difficulty tiers. Easy instances use diagonal matrices, small rotations, constant weights, and no shocks; medium instances introduce moderate rotations, imbalanced diagonals, $x_{II,1}$7-jitter, and mild time-linkage; hard instances use fast rotations, strong imbalances, shocks, and active $x_{II,1}$8-based time-linkage (Shao et al., 4 Jan 2026). This tiering makes the benchmark suitable both for baseline algorithm comparison and for stress-testing specialized methods such as prediction-based, knowledge-guided, or memory-based DMOO solvers.

The framework is also accompanied by a reference implementation, pydmoo, at https://github.com/dynoptimization/pydmoo (Shao et al., 4 Jan 2026). The benchmark specification recommends logging per-time parameters, Pareto-front samples, and obtained populations in formats such as CSV or HDF5 for reproducible analysis (Shao et al., 4 Jan 2026).

Its limitations are also explicitly stated. Current formulations are continuous and mostly unconstrained; mixed-integer or combinatorial DMOPs would require separate design choices. Stochastic objectives and noisy evaluations are not deeply modeled. Changing the number of objectives or variables over time is conceptually supported but not yet canonically implemented. For some complex irregular settings, analytic Pareto-front ground truth may be unavailable and may require high-resolution approximation (Shao et al., 4 Jan 2026). These caveats mean that DynamicEn should be understood as a generalized continuous DMOO benchmark framework rather than a universal dynamic optimization testbed.

A common misconception would be to treat DynamicEn as simply another dynamic ZDT/DTLZ family. The framework does reuse factorized ideas familiar from those families, but its defining contribution lies in the combination of hypersurface Pareto-set motion, imbalance, rotations, perturbation irregularity, and explicit time-linkage (Shao et al., 4 Jan 2026). Another plausible misconception is that its realism comes at the cost of reproducibility. The framework instead emphasizes deterministic construction where possible, especially through fixed schedules and $x_{II,1}$9-digit perturbations (Shao et al., 4 Jan 2026).

In summary, DynamicEn is best understood as an extensible benchmark-generation framework for dynamic multi-objective optimization whose central purpose is to make temporal benchmark environments structurally richer, less biased toward simple inverse mappings, and more revealing of algorithmic strengths and weaknesses than conventional dynamic suites (Shao et al., 4 Jan 2026). Its significance lies less in any one test instance than in the benchmark design principles it formalizes for next-generation DMOO evaluation.