The C-DTLZ Problem Collection

Abstract

This package provides test suites for the C-DTLZ problems (Jain & Deb, 2014), a constrained version of the DTLZ problems (Deb et al., 2001). The DTLZ problems are a set of continuous multi-objective optimization problems consisting of seven types, each supporting a variable number of objectives and variables. The C-DTLZ problems extend the DTLZ problems by adding various types of constraints to some of them. The objective functions are wrapped from the DTLZ test suite in optproblems, while the constraint components are implemented separately according to the original paper (Jain & Deb, 2014).

Constraints in C-DTLZ Problems

There are three types of constraints in the C-DTLZ problems:

1. Type-1 Constraints: In these problems, the Pareto-optimal front of the original problem remains unchanged, but infeasible barriers are introduced along the path to the Pareto-optimal front, enabling the evaluation of an algorithm’s ability to overcome these infeasible regions. This type supports DTLZ1 and DTLZ3, referred to as C1-DTLZ1 and C1-DTLZ3, respectively.
2. Type-2 Constraints: In these problems, infeasible regions are introduced directly on the Pareto-optimal front, enabling the evaluation of an algorithm’s ability to handle disconnected Pareto-optimal fronts. This type supports only DTLZ2, referred to as C2-DTLZ2.
3. Type-3 Constraints: In these problems, multiple constraints are introduced so that the Pareto-optimal front of the corresponding unconstrained problem is no longer entirely optimal; instead, portions of the added constraint surfaces themselves constitute segments of the Pareto-optimal front. This type supports DTLZ1 and DTLZ4, referred to as C3-DTLZ1 and C3-DTLZ4, respectively.

APIs

class Problem(function_id: int, n_objectives: int, constraint_type: int, dimension: int | None = None, **kwargs: Any)

• function_id: Function ID of the DTLZ problem in [1, 4].
• n_objectives: Number of objectives.
• constraint_type: Type of constraints in [1, 3].
• dimension: Number of variables. If not provided, defaults to n_objectives + 4 for DTLZ1 and DTLZ4, or n_objectives + 9 for DTLZ2 and DTLZ3.
• kwargs: Arbitrary keyword arguments, please refer to the optproblems documentation for more details.

Note: Only specific combinations of constraint_type and function_id are supported:

• C1-DTLZ1: constraint_type=1, function_id=1
• C1-DTLZ3: constraint_type=1, function_id=3
• C2-DTLZ2: constraint_type=2, function_id=2
• C3-DTLZ1: constraint_type=3, function_id=1
• C3-DTLZ4: constraint_type=3, function_id=4

Methods and Properties

• search_space: Return the search space.
  • Returns: dict[str, optuna.distributions.BaseDistribution]
• directions: Return the optimization directions.
  • Returns: list[optuna.study.StudyDirection]
• __call__(trial: optuna.Trial): Evaluate the objective functions and return the objective values.
  • Args:
    • trial: Optuna trial object.
  • Returns: list[float]
• evaluate(params: dict[str, float]): Evaluate the objective functions and return the objective values.
  • Args:
    • params: Decision variable like {"x0": x0_value, "x1": x1_value, ..., "xn": xn_value}. The number of parameters must be equal to dimension.
  • Returns: list[float]
• evaluate_constraints(params: dict[str, float]): Evaluate the constraint functions and return the constraint values.
  • Args:
    • params: Decision variable like {"x0": x0_value, "x1": x1_value, ..., "xn": xn_value}. The number of parameters must be equal to dimension.
  • Returns: list[float]

The properties defined by optproblems are also available such as get_optimal_solutions.

Installation

Please install the optproblems package.

pip install -U optproblems

Example

import optuna
import optunahub


cdtlz = optunahub.load_module("benchmarks/dtlz_constrained")
c2dtlz2 = cdtlz.Problem(function_id=2, n_objectives=2, constraint_type=2, dimension=3)

study = optuna.create_study(
    sampler=optuna.samplers.GPSampler(seed=42, constraints_func=c2dtlz2.constraints_func, deterministic_objective=True),
    directions=c2dtlz2.directions,
)
study.optimize(c2dtlz2, n_trials=300)
optuna.visualization.plot_pareto_front(study).show()

The Pareto front obtained from the above code is illustrated as follows:

Reference

Jain, H. & Deb, K. (2014). An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point Based Nondominated Sorting Approach, Part II: Handling Constraints and Extending to an Adaptive Approach, vol. 18, no. 4, pp. 602-622.

Deb, K., Thiele, L., Laumanns, M., & Zitzler, E. (2001). Scalable Test Problems for Evolutionary Multi-Objective Optimization.