Semi-infinite Nonconvex Constrained Min-Max Optimization

1. The number of steps for the multi-step gradient ascent should depend on constants like $\eta_\psi$ or $c_\psi$. Additionally, the required number of steps seems related to the value of $\theta$. How can the exact number of steps, $M_k$, be practically selected?

R1. We thank the reviewer for their comment. The multi-step parameters $M_k$ and $N_k$ indeed depend on problem-specific constants associated with the maximization component, such as $\eta_\psi$ and $c_\psi$. Details regarding the selection of these parameters are provided in Appendix Theorem A.9 ( Restatement of Theorem 4.2). Additionally, the dependence of $M_k$ and $N_k$ on the parameter $\theta$ has an intuitive interpretation: $\theta$ reflects the complexity of the infinite constraint set, where values of $\theta$ closer to 0 indicate easier subproblems and faster convergence, while values near 1 correspond to more challenging scenarios and slower convergence rates -- See Remark 4.1.

In practice, although the exact problem constants may be unknown, the theoretical results in Theorem 4.2 suggest that $M_k$ and $N_k$ should grow at a rate of at least $\mathcal{O}(\log(k+2))$ or $\mathcal{O}(\log(T+1))$, where $T$ denotes the total number of iterations. One practical strategy is to choose a slightly more aggressive growth, e.g., $\lceil (k+2)^{0.1}\rceil$ or $\lceil \log^2(k+2)\rceil$, ensuring that beyond some iteration index $T_0$, their values exceed the theoretical threshold. As a result, the convergence guarantees of the proposed method remain valid after a certain warm-up phase.

2. The performance in the Multi-MNIST experiment described in Figure 1 exhibits significant oscillation. It would be helpful if the authors could provide some explanation for this behavior.

R2. This behavior indeed stems from the inherent complexity of the problem. As noted in lines 359–361, our robust Multi-Task Learning formulation gives rise to a challenging nonconvex optimization problem with infinitely many functional constraints. Although our method is supported by provable convergence guarantees, the observed oscillations are a natural manifestation of the algorithm's iterative efforts to balance three competing goals: minimizing the objective function, enforcing feasibility with respect to the robust constraints, and satisfying the KKT optimality conditions (including stationarity and complementarity slackness). These oscillations reflect the algorithm's trajectory as it alternates between improving feasibility and achieving stationarity. This behavior commonly arises in both convex and nonconvex optimization problems with functional constraints, e.g., see the experiments in [Ref1].

[Ref1] Yazdandoost Hamedani, Erfan, and Necdet Serhat Aybat. "A primal-dual algorithm with line search for general convex-concave saddle point problems." SIAM Journal on Optimization 31.2 (2021): 1299-1329.

3. (Minor) There appears to be a typo on line 240; it should likely be "whether."

R3. We thank the reviewer for pointing out this typo. All typos will be corrected in the revised version of the manuscript.

We hope our responses address your concerns and would be happy to clarify anything further if needed.

1. How it compares to discretization or cutting surface approaches in the convex setting? Even for non-convex cases, simple discretization should be possible. Shouldn't it be used as a baseline in the experiments for comparing both performance and efficiency?

R1. We appreciate the reviewer's comment. Classical SIP methods focus on minimization problems, and we are not aware of any method that can address min-max structure in the objective function in a nonconvex setting. Moreover, those that can handle nonconvex problems generally focus on a discretization technique, however, these methods suffer from the curse of dimensionality by construction, as the number of required grid points to accurately represent the infinite constraint naturally grows with the dimension of the constraint space and is not effectively implementable on our numerical examples involving neural networks in high dimension (dim($x$)$\approx 2M$). On the other hand, our proposed method offers a clear advantage by relying solely on first-order information. Other common SIP methods, such as cutting planes or exchange methods, rely on a convexity assumption and do not offer convergence guarantees in the non-convex setting. Common constrained min-max methods, such as gradient-based primal-dual methods require the constraint set to be tractable via projection or evaluation. However, in our semi-infinite constraint setting, there are no guarantees that either will hold.

For the baseline comparison, we consider an additive robust MTL model where the objective function is the weighted summation of task-specific losses. More specifically, we consider solving $\min_{x}\max_{y\in \Delta_n}\max_{w\in\Delta_m}\phi(x,y)+\rho\psi(x,w)$ where $\rho>0$ is the weight parameter. This problem can be solved via min-max optimization methods such as gradient descent multi-step ascent (GDMA) [Ref 1,Ref 2]. We compare the performance of our method GDMA for different values of $\rho$ in terms of the first task loss, i.e, $\phi(x_k,y_k)$, and feasibility $[\psi(x_k,w_k)-r]_+$, where $r$ is the single task threshold as discussed in our paper. Below you can find the result of the experiment on Multi-MNIST dataset.

iDB-PD (our method)

iteration = 0001: Obj Loss = 2.4279 Feasibility = 1.0807

iteration = 1000: Obj Loss = 2.8115 Feasibility = 0.2393

iteration = 2000: Obj Loss = 2.7242 Feasibility = 0.0387

iteration = 3000: Obj Loss = 2.5080 Feasibility = 0.0010

GDMA ($\rho=1$)

iteration = 0001: Obj Loss = 2.4073 Feasibility = 1.0576

iteration = 1000: Obj Loss = 2.2709 Feasibility = 0.9157

iteration = 2000: Obj Loss = 2.1412 Feasibility = 0.7698

iteration = 3000: Obj Loss = 2.0615 Feasibility = 0.6823

GDMA ($\rho=2$)

iteration = 0001: Obj Loss = 2.4556 Feasibility = 1.1168

iteration = 1000: Obj Loss = 2.3824 Feasibility = 0.6930

iteration = 2000: Obj Loss = 2.2876 Feasibility = 0.5165

iteration = 3000: Obj Loss = 2.2276 Feasibility = 0.3941

GDMA ($\rho=5$)

iteration = 0001: Obj Loss = 2.4700 Feasibility = 1.0810

iteration = 1000: Obj Loss = 2.6608 Feasibility = 0.3444

iteration = 2000: Obj Loss = 2.6859 Feasibility = 0.0352

iteration = 3000: Obj Loss = 2.7191 Feasibility = 0.0000

GDMA ($\rho=10$)

iteration = 0001: Obj Loss = 2.4803 Feasibility = 1.0565

iteration = 1000: Obj Loss = 3.1297 Feasibility = 0.0000

iteration = 2000: Obj Loss = 3.4119 Feasibility = 0.0000

iteration = 3000: Obj Loss = 3.5966 Feasibility = 0.0000

We observe that our proposed approach reaches the lowest value for the first task loss among the approaches that satisfy the constraint within an accuracy $\epsilon=10^{-3}$. Note that while GDMA with $\rho=1$ yields a lower objective loss, it fails to meet the feasibility requirement within the desired accuracy threshold. The full details and corresponding plots for the experiment will be added to the revised manuscript.

[Ref 1] Jin, Chi, Praneeth Netrapalli, and Michael Jordan. "What is local optimality in nonconvex-nonconcave minimax optimization?." International conference on machine learning. PMLR, 2020.

[Ref 2] Nouiehed, Maher, et al. "Solving a class of non-convex min-max games using iterative first order methods." Advances in Neural Information Processing Systems 32 (2019).

2. It is not evident whether this setup provides clear advantages over more standard approaches, such as minimizing the sum of task-specific losses.

R2. We appreciate the reviewer’s comment. In our revised experiments, we include a comparison with the robust formulation that minimizes the sum of task-specific losses. As detailed in our response to Question 1, our proposed method, iDB-PD, consistently achieves lower loss values across both tasks compared to GDMA with $\rho = 1$, highlighting the effectiveness of our formulation.

Furthermore, recent studies [Ref 3, Ref 4] have demonstrated the benefits of explicitly modeling task priorities in multi-task learning. Minimizing the unweighted sum of task-specific losses inherently assumes equal importance for all tasks, which limits the model’s ability to adapt to task-specific needs or real-world constraints where some tasks may be more critical. In contrast, our robust MTL formulation enables more flexible and effective optimization across heterogeneous tasks.

[Ref 3] Cheng, Zhengxing, et al. "No More Tuning: Prioritized Multi-Task Learning with Lagrangian Differential Multiplier Methods." Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 39. No. 11. 2025.

[Ref 4] Guo, Michelle, et al. "Dynamic task prioritization for multitask learning." Proceedings of the European conference on computer vision (ECCV). 2018.

3. In the experimental section, the author(s) did not verify whether the chosen values for these hyperparameters satisfy the theoretical assumptions.

R3. Details of our experimental setup and hyperparameter selection are provided in Appendix A.5. We selected the hyperparameters based on Theorem 4.2 and performed a grid search over a carefully chosen range of values, which is a common approach in the optimization literature, ensuring both theoretical consistency and empirical robustness.

4. In Section 3, $\zeta(x,w^*(x))$ only guarantees that at least one of the two terms in the indicator function is zero, but not necessarily both.

R4. We thank the reviewer for pointing this out. The correct statement should be: “then the point $x$ is feasible (i.e., the residual is zero) or a critical point of the constraint function.” However, under Assumption 2.3 and using the definition $\zeta(x, w) = ||\nabla_x \psi(x, w) [\psi(x, w)]_+||$, we can indeed conclude that $\zeta(x, w^*(x)) = 0$ implies that $x$ is feasible. We will clarify this point in the revised manuscript by expanding the explanation and explicitly connecting it to Assumption 2.3.

We hope our responses address your concerns and would be happy to clarify anything further if needed.

1. In the first paragraph of the introduction, several applications are mentioned without citations. Could references or concrete examples be added?

R1. We thank the reviewer for this helpful suggestion. We agree that the applications mentioned in the opening paragraph would benefit from concrete citations to better support the context and motivate the relevance of semi-infinite programming (SIP). Accordingly, we have revised the paragraph to explicitly cite representative works for each application area. Specifically, we now reference foundational works highlighting adversarial perturbations in AI models [Ref 1], [Ref 2], robust optimization in supply chain management [Ref 3], and safety and control under uncertainty in autonomous systems [Ref 4].

[Ref 1] Goodfellow, Ian J., Jonathon Shlens, and Christian Szegedy. "Explaining and harnessing adversarial examples." arXiv preprint arXiv:1412.6572 (2014).

[Ref 2] Zhang, Xingwei, Xiaolong Zheng, and Wenji Mao. "Adversarial perturbation defense on deep neural networks." ACM Computing Surveys (CSUR) 54.8 (2021): 1-36.

[Ref 3] Ben-Tal, Aharon, et al. "Retailer-supplier flexible commitments contracts: A robust optimization approach." Manufacturing and Service Operations Management 7.3 (2005): 248-271.

[Ref 4] Dutta, Raj Gautam, Xiaolong Guo, and Yier Jin. "Quantifying trust in autonomous system under uncertainties." 2016 29th IEEE International System-on-Chip Conference (SOCC). IEEE, 2016.

2. In lines 36–37, the problem structure is emphasized before motivating applications are introduced. Could a simplified application be included earlier?

R2. Thank you for the suggestion. We agree that introducing a motivating application earlier can improve readability. We will revise the manuscript to include a brief mention of relevant applications with appropriate references, before presenting the problem structure.

3. Line 42 introduces nonconvexity in $\phi$ and $\psi$, yet convergence relies on the Łojasiewicz inequality. Why is this assumption not highlighted sooner.

R3. Thank you for this observation. To clarify, Line 42 only motivates the development of new methods for addressing nonconvexity in both objective and constraint functions under min-max setting as a general direction. Moreover, Łojasiewicz condition is only required for the $\psi$ function with respect to $x$, which we explicitly discussed in the contribution section. We will revise the earlier sections to make this distinction more explicit and avoid potential confusion.

4. Assumption 2.2 sets $W\in \mathbb R^\ell$, but in Equation (1), $w\in W$. Please clarify the domain and generality of $W$.

R4. Thank you for pointing this out. We agree that the notation could be made clearer.

• When $\psi(x, \cdot)$ is $\eta_\psi$-strongly concave, $W$ can be any closed convex subset of $\mathbb{R}^\ell$.
• When $\psi(x, \cdot)$ is assumed to satisfy the Polyak-Łojasiewicz (PL) condition with constant $c_\psi$, we require that $W = \mathbb{R}^\ell$ to ensure the PL condition holds over the entire space.

We will revise the assumption to make this distinction more precise in the revised version.

5. Do the applications in Section 1.2 and experiments rigorously satisfy the PL or Łojasiewicz conditions?

R5. We thank the reviewer for this question. Both applications in Section 1.2 satisfy the Łojasiewicz inequality if the loss functions chosen are real analytical, definable in $o$-minimal structures, or differentiable sub-analytical functions. Common constraint loss choices in both problems include cross-entropy, squared loss, $\ell_1$, and Huber loss, all of which satisfy one of the above functional categories and therefore, the Łojasiewicz inequality as well. Our neural network in all of our experiments is connected by hyperbolic tangents with a softmax activation function, so the network satisfies the Łojasiewicz inequality, as we note in the discussion after Definition 2.1.

6. How does the algorithm's complexity compare to existing SIP or constrained min-max solvers, even heuristically?

R6. We appreciate the reviewer noting this. Classical SIP methods that can handle non-convex problems generally focus on a discretization technique. However, these methods suffer from the curse of dimensionality by construction, as the number of required grid points to accurately represent the infinite constraint naturally grows with the dimension of the constraint space. Given our numerical examples involve neural networks in high dimension (dim($x$)$\approx 2M$), our proposed method offers a clear advantage by relying solely on first-order information. Other common SIP methods, such as cutting planes or exchange methods, rely on a convexity assumption and do not offer convergence guarantees in the non-convex setting. Common constrained min-max methods, such as gradient-based primal-dual methods require the constraint set to be tractable via projection or evaluation. However, in our semi-infinite constraint setting, there are no guarantees that either will hold.

7. Is it possible to include baselines or ablation studies to better evaluate the practical performance of the method?

R7. We thank the reviewer for their suggestion. The main reason that we originally did not use any baseline method was that classical SIP methods mainly focus on minimization problems, and we are not aware of any method that can address a semi-infinite min-max structure in a nonconvex setting. For the baseline comparison, we consider an additive robust MTL model where the objective function is the weighted summation of task-specific losses. More specifically, we consider solving $\min_{x}\max_{y\in \Delta_n}\max_{w\in\Delta_m}\phi(x,y)+\rho\psi(x,w)$ where $\rho>0$ is the weight parameter. This problem can be solved via min-max optimization methods such as gradient descent multi-step ascent (GDMA) [Ref 5,Ref 6]. We compare the performance of our method GDMA for different values of $\rho$ in terms of the first task loss, i.e, $\phi(x_k,y_k)$, and feasibility $[\psi(x_k,w_k)-r]_+$, where $r$ is the single task threshold as discussed in our paper. Below you can find the result of the experiment on Multi-MNIST dataset.

iDB-PD (our method)

iteration = 0001: Obj Loss = 2.4279 Feasibility = 1.0807

iteration = 1000: Obj Loss = 2.8115 Feasibility = 0.2393

iteration = 2000: Obj Loss = 2.7242 Feasibility = 0.0387

iteration = 3000: Obj Loss = 2.5080 Feasibility = 0.0010

GDMA ($\rho=1$)

iteration = 0001: Obj Loss = 2.4073 Feasibility = 1.0576

iteration = 1000: Obj Loss = 2.2709 Feasibility = 0.9157

iteration = 2000: Obj Loss = 2.1412 Feasibility = 0.7698

iteration = 3000: Obj Loss = 2.0615 Feasibility = 0.6823

GDMA ($\rho=2$)

iteration = 0001: Obj Loss = 2.4556 Feasibility = 1.1168

iteration = 1000: Obj Loss = 2.3824 Feasibility = 0.6930

iteration = 2000: Obj Loss = 2.2876 Feasibility = 0.5165

iteration = 3000: Obj Loss = 2.2276 Feasibility = 0.3941

GDMA ($\rho=5$)

iteration = 0001: Obj Loss = 2.4700 Feasibility = 1.0810

iteration = 1000: Obj Loss = 2.6608 Feasibility = 0.3444

iteration = 2000: Obj Loss = 2.6859 Feasibility = 0.0352

iteration = 3000: Obj Loss = 2.7191 Feasibility = 0.0000

GDMA ($\rho=10$)

iteration = 0001: Obj Loss = 2.4803 Feasibility = 1.0565

iteration = 1000: Obj Loss = 3.1297 Feasibility = 0.0000

iteration = 2000: Obj Loss = 3.4119 Feasibility = 0.0000

iteration = 3000: Obj Loss = 3.5966 Feasibility = 0.0000

We observe that our proposed approach reaches the lowest value for the first task loss among the approaches that satisfy the constraint within an accuracy $\epsilon=10^{-3}$. Note that while GDMA with $\rho=1$ yields a lower objective loss, it fails to meet the feasibility requirement within the desired accuracy threshold. The full details and corresponding plots for the experiment will be added to the revised manuscript.

[Ref 5] Jin, Chi, Praneeth Netrapalli, and Michael Jordan. "What is local optimality in nonconvex-nonconcave minimax optimization?." International conference on machine learning. PMLR, 2020.

[Ref 6] Nouiehed, Maher, et al. "Solving a class of non-convex min-max games using iterative first order methods." Advances in Neural Information Processing Systems 32 (2019).

We hope our responses address your concerns and would be happy to clarify anything further if needed.

1. The paper specifies that semi-infinite optimization is underdeveloped. I disagree with this statement. A lot of important citations and discussions are missing in this work. The citations in robust optimization are also not usual in my view, especially since Ben-Tal's book is missing.

R1. We would like to thank the reviewer for their thoughtful and constructive comment. Our intention was certainly not to overlook or diminish the extensive theoretical and algorithmic contributions in the SIP literature. Rather, we aimed to emphasize that, despite its rich development and broad applications in various areas such as robotic and control theory, comparatively less attention has been given to methods tailored for modern AI and Operations Research problems, which often involve different structural features (e.g., min-max objective functions) and require different computational approaches (e.g., fully first-order methods). To clarify this point, we will revise our statement as follows: "Despite its rich and extensive theoretical and algorithmic development, comparatively less attention has been devoted to designing efficient first-order methods for emerging applications in modern AI and Operations Research." Additionally, we expand our literature review to include more classical works in SIP, including those kindly suggested by the reviewer.

Our work introduces the first fully first-order algorithm for nonconvex semi-infinite constrained min-max optimization, that provides explicit global non-asymptotic convergence guarantees for finding an approximated stationary solution. Unlike prior SIP methods which are often second-order, require feasible initialization, and offer only asymptotic or local convergence results, our iDB-PD algorithm ensures convergence from any starting point, with provable iteration complexity bounds of $\mathcal{O}(\epsilon^{-3})$, $\mathcal{O}(\epsilon^{-6\theta})$, and $\mathcal{O}(\epsilon^{-3\theta/(1-\theta)})$ for stationarity, feasibility, and complementarity, respectively. In contrast, classical methods like exchange algorithms or adaptive discretization (e.g., [Ref 1], [Ref 2]) either lack convergence rate analysis or provide only local rates (e.g., quadratic convergence near optimality under strong assumptions), and robust optimization frameworks (e.g., Ben-Tal et al.) solve restricted convex subclasses without iterative rate analysis. Thus, our method uniquely bridges the gap between theoretical rigor and broad applicability, advancing the state of the art in semi-infinite programming.

[Ref 1] Seidel, Tobias, and Karl-Heinz Küfer. "An adaptive discretization method solving semi-infinite optimization problems with quadratic rate of convergence." Optimization 71.8 (2022): 2211-2239.

[Ref 2] Floudas, Christodoulos A., and Oliver Stein. "The adaptive convexification algorithm: a feasible point method for semi-infinite programming." SIAM Journal on Optimization 18.4 (2008): 1187-1208.

2. There are a lot of works in distributionally robust optimization. In those cases, the inner "convex maximization" problem can still be dualized with some modeling tricks. Such approaches are not really discussed in this work.

R2. We appreciate the reviewer’s comment. While we acknowledge the extensive literature on distributionally robust optimization (DRO), including approaches such as “regularization via mass transportation” where the inner maximization can be dualized under convexity assumptions, our work addresses a fundamentally different setting: nonconvex-concave semi-infinite min-max problems where such dualization does not lead to a tractable convex reformulation. Unlike prior DRO methods that rely on strong duality and convexity to enable tractable solutions, our goal is to develop a fully first-order method with global non-asymptotic convergence guarantees. We will explicitly incorporate this distinction and relevant DRO references in the revised paper.

3. Numerical experiments have no benchmark to compare against. In my view, to motivate the proposed sophisticated algorithms, we should get strong numerical evidence.

R3. Thank you for your comment and great suggestion. The main reason that we originally did not use any baseline method was that classical SIP methods mainly focus on minimization problems, and we are not aware of any method that can address a semi-infinite min-max structure in a nonconvex setting. Moreover, those that can handle non-convex problems generally focus on a discretization technique, however, these methods suffer from the curse of dimensionality by construction, as the number of required grid points to accurately represent the infinite constraint naturally grows with the dimension of the constraint space and is not effectively implementable on our numerical examples involving neural networks in high dimension (dim($x$)$\approx 2M$). For the baseline comparison, we consider an additive robust MTL model where the objective function is the weighted summation of task-specific losses. More specifically, we consider solving $\min_{x}\max_{y\in \Delta_n}\max_{w\in\Delta_m}\phi(x,y)+\rho\psi(x,w)$ where $\rho>0$ is the weight parameter. This problem can be solved via min-max optimization methods such as gradient descent multi-step ascent (GDMA) [Ref 3,Ref 4]. We compare the performance of our method GDMA for different values of $\rho$ in terms of the first task loss, i.e, $\phi(x_k,y_k)$, and feasibility $[\psi(x_k,w_k)-r]_+$, where $r$ is the single task threshold as discussed in our paper. Below you can find the result of the experiment on Multi-MNIST dataset.

iDB-PD (our method)

iteration = 0001: Obj Loss = 2.4279 Feasibility = 1.0807

iteration = 1000: Obj Loss = 2.8115 Feasibility = 0.2393

iteration = 2000: Obj Loss = 2.7242 Feasibility = 0.0387

iteration = 3000: Obj Loss = 2.5080 Feasibility = 0.0010

GDMA ($\rho=1$)

iteration = 0001: Obj Loss = 2.4073 Feasibility = 1.0576

iteration = 1000: Obj Loss = 2.2709 Feasibility = 0.9157

iteration = 2000: Obj Loss = 2.1412 Feasibility = 0.7698

iteration = 3000: Obj Loss = 2.0615 Feasibility = 0.6823

GDMA ($\rho=2$)

iteration = 0001: Obj Loss = 2.4556 Feasibility = 1.1168

iteration = 1000: Obj Loss = 2.3824 Feasibility = 0.6930

iteration = 2000: Obj Loss = 2.2876 Feasibility = 0.5165

iteration = 3000: Obj Loss = 2.2276 Feasibility = 0.3941

GDMA ($\rho=5$)

iteration = 0001: Obj Loss = 2.4700 Feasibility = 1.0810

iteration = 1000: Obj Loss = 2.6608 Feasibility = 0.3444

iteration = 2000: Obj Loss = 2.6859 Feasibility = 0.0352

iteration = 3000: Obj Loss = 2.7191 Feasibility = 0.0000

GDMA ($\rho=10$)

iteration = 0001: Obj Loss = 2.4803 Feasibility = 1.0565

iteration = 1000: Obj Loss = 3.1297 Feasibility = 0.0000

iteration = 2000: Obj Loss = 3.4119 Feasibility = 0.0000

iteration = 3000: Obj Loss = 3.5966 Feasibility = 0.0000

We observe that our proposed approach reaches the lowest value for the first task loss among the approaches that satisfy the constraint within an accuracy $\epsilon=10^{-3}$. Note that while GDMA with $\rho=1$ yields a lower objective loss, it fails to meet the feasibility requirement within the desired accuracy threshold. The full details and corresponding plots for the experiment will be added to the revised manuscript.

[Ref3] Jin, Chi, Praneeth Netrapalli, and Michael Jordan. "What is local optimality in nonconvex-nonconcave minimax optimization?." International conference on machine learning. PMLR, 2020.

[Ref4] Nouiehed, Maher, et al. "Solving a class of non-convex min-max games using iterative first order methods." Advances in Neural Information Processing Systems 32 (2019).

4. Would Assumption 2.3 not hold for NNs in general?

R4. We thank the reviewer for this insightful question. While Assumption 2.3 holds in many practical NN architectures involving smooth components as noted in Remark 2.1, it does not extend to general NN architectures with nonsmooth components such as ReLU activation function, and one needs to use the smoothed variant, e.g., softplus, to ensure Assumption 2.3 holds.

5. Do the authors believe simplex constraints are representative enough for $\psi$ in the numerical experiments?

R5. We thank the reviewer for raising this question. As a simplex set describes a discrete probability distribution domain, it is a natural structure arising in distributionally robust models for capturing uncertainty when probabilities are unknown. Note that the regularization term in the robust MTL model considered in our numerical experiment finds the worst-case distributions in the neighborhood of the uniform distribution. This formulation is commonly used in DRO literature [Ref 5] and numerical examples for min-max optimization methods [Ref 6,Ref 7].
Furthermore, the simplex set satisfies convexity and compactness, which help make the resulting optimization problem computationally manageable.

[Ref 5] Namkoong, Hongseok, and John C. Duchi. "Stochastic gradient methods for distributionally robust optimization with f-divergences." Advances in neural information processing systems 29 (2016).

[Ref 6] Zhang, Xuan, Necdet Serhat Aybat, and Mert Gurbuzbalaban. "Sapd+: An accelerated stochastic method for nonconvex-concave minimax problems." Advances in Neural Information Processing Systems 35 (2022): 21668-21681.

[Ref 7] Mehta, Ronak, Jelena Diakonikolas, and Zaid Harchaoui. "DRAGO: Primal-Dual Coupled Variance Reduction for Faster Distributionally Robust Optimization." Advances in Neural Information Processing Systems 37 (2024): 134770-134825.

We hope our responses address your concerns and would be happy to clarify anything further if needed.