Robust Optimization over Time Problems—Characterization and Literature Review
Abstract
:1. Introduction
2. Robust Optimization over Time
3. Proposed Framework
3.1. Dimension 1: Available Information
- Scenario 1 (S1): This is the case of ROOT problems where fitness functions from future environments can be evaluated exactly. Hence, it comprises a family of ROOT problems where the robustness of a solution can be quantified exactly. As mentioned earlier, the only source of uncertainty here is the black-box nature of the environment’s fitness function. So, in this type of ROOT problem, typical TMO algorithms like [4], ref. [14] could be applied without any major adaptation. In other words, problems belonging to this scenario are slightly different from traditional dynamic optimization problems, where tracking the optimum is the main goal. To better visualize this, consider the Averaged Fitness as the robustness definition and as the fitness function guiding the optimizer during the search. Since it is an aggregation of the present and future values of objective functions that can be evaluated exactly, the problem is reduced to a TMO problem with a single, dynamic objective function, i.e., the Averaged Fitness. In the case of robustness defined in terms of Survival Time, it is not trivial to transform problems from this scenario into TMO problems. The difficulty arises because, in theory, the robustness value could be infinite. This is the case of a problem with an endless number of changes (environments) in which it is possible to find a solution that both in the current and future environments maintains a fitness above the threshold V (Equation (3)). If the number of changes has a finite, computationally tractable limit, then it would be possible to construct an objective function that captures this definition in each environment t. Note that here, instead of using an average to aggregate the information from several environments, the values of the objective function are integers. For these reasons, this scenario is suitable for studying how the optimizer part of the algorithm copes with the uncertainty of the black box fitness function.
- Scenario 2 (S2): In this scenario, we consider ROOT problems with unknown future environments, but with past environments that can be evaluated exactly. Hence, the algorithm requires a forecasting model to estimate the future fitness values. Uncertainty is present here due to the future environments and the black box definition of fitness functions. From an experimental perspective, problems in this category are suitable to study forecasting models which help to evaluate future environments.
- Scenario 3 (S3): In this type of ROOT problem, neither the future environments nor the past environments can be evaluated precisely. This implies that algorithms must include mechanisms to not only forecast the future but also to store or approximate the past effectively. It is easy to infer that, of the three scenarios, this one encloses ROOT problems with the highest level of uncertainty. In addition to the performance of the optimizer and the forecasting model, problems in this category are suitable for studying function approximation models that properly represent past environments.
3.2. Dimension 2: Robustness Definition
3.3. Dimension 3: Number of Objectives
3.4. Dimension 4: Search Space Features
3.5. Dimension 5: Problem Source
4. Literature Review, Analysis, and Research Opportunities
(“robust optimization over time” OR “robust optimisation over time”)
4.1. Results
4.2. On the Effect of the Available Information
4.3. Research Opportunities
- 1.
- Defining and/or solving problems with discrete variables, with a moderate or high number of variables, and considering feasible regions unbounded or constrained;
- 2.
- Address more problems of type S2 or S3, rather than those of type S1. Although all three categories of problems have been addressed in the past, some recent contributions focus on the simpler S1 scenario (e.g., where future fitness functions can be accurately evaluated). As was stated earlier, this type of problem is in essence the same as the Tracking Moving Optima if we defined robustness in terms of Average Fitness. In our opinion, the true ROOT problems are those derived from the assumptions defining scenarios S2 and S3;
- 3.
- Exploring alternative robustness definitions. Although alternatives to the typical ones (e.g., Average Fitness and Survival Time) have appeared in recent years, these definitions are very specific to the problem being solved. Thus, there is room to propose alternatives that can efficiently guide the algorithm in the search for robust solutions.
- 4.
- Address more real-world (inspired) problems. Except for just a few works like [25,26,27,28,29,30], the rest were focused on artificial problems. Note that addressing a larger number of real-world problems is a way to increase the number of contributions that exploit the opportunities indicated above. This is because a large number of these problems involve discrete variables, various constraints, uncertainty about the available information, and very specific definitions of robustness.
5. Conclusions and Future Works
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
Ref. | Source | Year | Scenario | Robust. Def. | N. Object. | Dimen. | Dec. Var. | Feas. Region | Prob. Source |
---|---|---|---|---|---|---|---|---|---|
[6] | Conference | 2010 | S3 | A.F. | Single | Low | Contin. | Bounded | Artificial |
[33] | Conference | 2012 | S1 | Others | Single | Low | Contin. | Constrain. | Artificial |
[34] | Conference | 2013 | S2 | S.T. | Single | Low | Contin. | Bounded | Artificial |
[25] | Conference | 2013 | S3 | Others | Single | Low | Contin. | Constrain. | Real-world inspired |
[13] | Journal | 2013 | S3 | A.F. | Single | Low | Contin. | Bounded | Artificial |
[21] | Conference | 2014 | S2 | A.F., S.T. | Many | Low | Contin. | Bounded | Artificial |
[17] | Journal | 2015 | S3 | S.T. | Many | Low | Contin. | Constrain. | Artificial |
[11] | Journal | 2015 | S3 | A.F., S.T. | Single | Low | Contin. | Bounded | Artificial |
[35] | Conference | 2015 | S3 | A.F., S.T. | Single | Low | Contin. | Bounded | Artificial |
[36] | Journal | 2017 | S3 | S.T. | Many | Low | Contin. | Constrain. | Artificial |
[37] | Journal | 2017 | S3 | S.T. | Many | Low | Contin. | Bounded | Artificial |
[38] | Conference | 2017 | S1 | A.F., S.T. | Single | Low | Contin. | Bounded | Artificial |
[26] | Journal | 2018 | S1 | Others | Many | Low | Mixed | Constrain. | Real-world inspired |
[31] | Conference | 2018 | S3 | A.F. | Single | Low | Contin. | Bounded | Artificial |
[39] | Conference | 2018 | S1 | Others | Single | Low | Contin. | Bounded | Artificial |
[40] | Conference | 2019 | S1 | A.F., S.T. | Single | Low | Contin. | Bounded | Artificial |
[41] | Conference | 2019 | S1 | A.F. | Single | High | Discrete | Constrain. | Real-world inspired |
[42] | Journal | 2019 | S3 | A.F. | Single | Low | Contin. | Bounded | Artificial |
[43] | Conference | 2019 | S1 | A.F. | Single | Low | Contin. | Bounded | Artificial |
[44] | Journal | 2019 | S3 | A.F., S.T. | Many | Low | Contin. | Constrain. | Artificial |
[45] | Conference | 2019 | S1 | A.F. | Single | Low | Contin. | Bounded | Artificial |
[27] | Conference | 2019 | S1 | Others | Single | Low | Contin. | Constrain. | Real-world |
[46] | Journal | 2019 | S3 | S.T. | Single | Low | Contin. | Bounded | Artificial |
[28] | Conference | 2020 | S1 | A.F. | Single | High | Discrete | Constrain. | Real-world inspired |
[47] | Journal | 2020 | S1 | A.F., S.T. | Single | Low | Contin. | Bounded | Artificial |
[32] | Conference | 2020 | S2 | A.F., S.T. | Single | Low | Contin. | Bounded | Artificial |
[48] | Journal | 2020 | S1 | S.T. | Single | Low | Contin. | Bounded | Artificial |
[49] | Journal | 2020 | S3 | A.F., S.T. | Many | Low | Contin. | Constrain. | Artificial |
[29] | Journal | 2020 | S1 | S.T. | Many | Low | Contin. | Bounded | Artificial |
[50] | Journal | 2020 | S3 | A.F. | Many | Low | Contin. | Constrain. | Real-world |
[51] | Journal | 2020 | S3 | Others | Single | Low | Contin. | Constrain. | Real-world inspired |
[30] | Journal | 2021 | S1 | Others | Single | Low | Contin. | Constrain. | Real-world |
[52] | Journal | 2022 | S3 | S.T. | Single | Low | Contin. | Bounded | Artificial |
[53] | Journal | 2023 | S3 | S.T. | Single | Low | Contin. | Bounded | Artificial |
[54] | Journal | 2023 | S3 | S.T. | Many | Low | Contin. | Constrain. | Artificial |
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Parameter | Values |
---|---|
Number of peaks (m) | 5 |
Dimension (D) | 2 |
Search space | |
Number of changes (environments) (N) | 30 |
Change frequency () | 1000 |
Change type | Small step (Equation (5)) |
Peaks’ height range (h) | |
Height severity () | 5.0 |
Peaks’ width range (w) | |
Width severity () | 0.5 |
Rotation angle range () | |
Angle severity () | 1.0 |
Time window (T) | 3 |
History size (period P) | 12 |
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Novoa-Hernández, P.; Puris, A.; Pelta, D.A. Robust Optimization over Time Problems—Characterization and Literature Review. Electronics 2023, 12, 4609. https://doi.org/10.3390/electronics12224609
Novoa-Hernández P, Puris A, Pelta DA. Robust Optimization over Time Problems—Characterization and Literature Review. Electronics. 2023; 12(22):4609. https://doi.org/10.3390/electronics12224609
Chicago/Turabian StyleNovoa-Hernández, Pavel, Amilkar Puris, and David A. Pelta. 2023. "Robust Optimization over Time Problems—Characterization and Literature Review" Electronics 12, no. 22: 4609. https://doi.org/10.3390/electronics12224609
APA StyleNovoa-Hernández, P., Puris, A., & Pelta, D. A. (2023). Robust Optimization over Time Problems—Characterization and Literature Review. Electronics, 12(22), 4609. https://doi.org/10.3390/electronics12224609