Multiobjective Bilevel Optimization

• Multiobjective bilevel optimization is a hierarchical framework that optimizes both upper‐ and lower-level objective vectors under interdependent constraints.
• Methodologies include scalarization, value-function reformulations, and evolutionary algorithms to handle Pareto-dominant, set-valued solution mappings.
• Applications in robust machine learning, neural architecture search, and engineering design demonstrate reduced function evaluations and enhanced trade-off analysis.

A multiobjective bilevel optimization problem is a hierarchical optimization framework in which both the upper-level (leader) and the lower-level (follower) agents optimize their respective objective vectors, subject to interdependent decision and feasibility constraints. These problems generalize standard bilevel optimization by allowing multi-criteria trade-offs at both levels, and appear in diverse applications such as robust machine learning, adversarial defense, game theory, neural architecture search, resource allocation, and many engineering design settings. The increased dimensionality and set-valued solution mappings induced by Pareto dominance at one or both levels create significant theoretical and algorithmic challenges, necessitating the development of new reformulations, optimality conditions, and specialized heuristics.

1. Formal Problem Statement and Taxonomy

The general bilevel multiobjective problem (BLMOP) is formulated as follows:

$$\begin{aligned} &\min_{x_1 \in X_1} F_U(x_1, x_2) = \bigl(F_U^1(x_1, x_2), \ldots, F_U^m(x_1, x_2)\bigr) \ &\quad \text{subject to} \;\;\; x_2 \in \mathrm{PS}(x_1) \ \end{aligned}$$

where the lower-level Pareto set mapping $\mathrm{PS}(x_1)$ is given by

$$\mathrm{PS}(x_1) = \left\{ x_2 \in X_2 \;\middle|\; \nexists x_2' \in X_2: f_L(x_1, x_2') \preceq f_L(x_1, x_2) \right\}$$

and $f_L: X_1 \times X_2 \to \mathbb{R}^n$ is the lower-level (vectorial) objective with $\preceq$ denoting Pareto dominance.

Classification axes for BLMOPs include:

• Vectorial objectives at upper/lower or both levels (fully vectorial or semi-vectorial)
• Optimistic (follower selects the UL-best among LL Pareto-optimal points) vs pessimistic (follower selects worst for UL) formulations
• Problem convexity or linearity; continuous vs mixed-integer structure
• Number of objectives, and the mapping cardinality $x_1 \mapsto \mathrm{PS}(x_1)$

Frameworks with robust (min–max or min–max–max) structure are prevalent in adversarial contexts, e.g., poisoning attacks in machine learning (Carnerero-Cano et al., 2020, Carnerero-Cano et al., 2023), robust representation learning (Gu et al., 2022), and fair multi-task learning (Chen et al., 2023, Wang et al., 5 Sep 2024).

2. Mathematical Reformulations: Value Function and Frontier Mapping

Standard scalarization is inadequate to encode the feasible set determined by Pareto-efficient responses at the lower level. The frontier mapping $\Psi(x)$, generalizing the classical value function, replaces the lower-level minimizer by the set of all efficient (under ordering cone $C$) outcome vectors:

$\Psi(x) := \mathrm{Eff}(f(x, Y(x)); C)$

where

$$\mathrm{Eff}(S; C) = \{z \in S : \nexists z' \in S, \; z' \leq_C z, \; z' \neq z \}$$

Efficient lower-level solution mapping: $S(x) = \{ y \in Y(x) : f(x, y) \in \Psi(x) \}$ Thus, the single-level value function reformulation for optimistic BLMOP is: $$\min_{x\in X,\, y\in Y(x)} F(x, y) \quad \text{s.t.} \;\; f(x, y) \in \Psi(x)$$ Extensions cover weak efficiency, graph-closedness of solution maps, and alternative horses such as risk-neutral (expectation), risk-averse (worst-case), and intermediate solution concepts (Lafhim et al., 2021, Hoff et al., 2023, Giovannelli et al., 2023).

Necessary optimality conditions are derived using Mordukhovich-type coderivative inclusions, relying on calmness or generalized value-function constraint qualifications (GVFCQ) (Lafhim et al., 2021). The extension to vector-valued and even fractional objectives is addressed with nonsmooth variational tools and dual formulations (Lara et al., 22 Nov 2025).

3. Solution Paradigms and Algorithms

Fundamental solution paradigms are outlined below:

Scalarization-Driven Approaches

Convert either upper or lower objective vectors (via weighted sums or $\epsilon$-constraint) to yield parametric families of scalar bilevel problems. For example, for LL objectives $f_L^j$: $$\min_{x_1,\lambda \in \Lambda,\,x_2} F_U(x_1, x_2) \;\text{s.t.}\; x_2 = \arg\min_{x_2'} \sum_{j=1}^n \lambda_j f_L^j(x_1, x_2')$$ with $\Lambda$ the simplex of convex combination weights.

Value Function/Frontier Reformulation

Replace lower-level minimization by a set-valued constraint on the efficient frontier, preserving multiobjective structure without scalarization.

Classical KKT-Based and MPEC Reduction

Apply KKT or strong duality conditions to substitute the lower-level program by complementarity constraints, yielding MPEC/MINLP reformulations tractable with mathematical programming only in convex or small-scale cases.

Risk-Neutral and Risk-Averse Formulations

For LL vector objectives, solve

• Risk-neutral: minimize upper-level cost averaged over $\lambda$, $\mathbb{E}_\lambda[f_U(x, y^*(x, \lambda))]$,
• Risk-averse: minimize the worst-case upper-level cost over the LL Pareto set, $\min_x \max_{y\in P(x)} f_U(x, y)$ with associated stochastic-gradient or subgradient methods (Giovannelli et al., 2023, Gu et al., 2022).

Evolutionary and Metaheuristic Methods

Use population-based algorithms (e.g., MOEA/D, NSGA-III, nested upper/lower evolutionary architectures), often combined with surrogates (e.g., Kriging, feedforward NNs for Pareto set prediction (Wang et al., 5 Sep 2024), or preference models for LL trade-off selection (Wang et al., 2023)) to address black-box and large-scale nonconvex BLMOPs.

Algorithmic complexity is exacerbated by the set-valued mapping $x_1 \mapsto \text{PS}(x_1)$; PSP-BLEMO (Wang et al., 5 Sep 2024) resolves this via helper variables and one-to-one mappings to train predictive neural networks for the LL Pareto set and embed them in the evolutionary loop.

4. Theoretical Properties: Existence, Optimality, and Duality

Existence and Closedness

Under mild lower-level graph-closedness and boundedness, efficient and weakly-efficient optimal solutions exist (Hoff et al., 2023, Lafhim et al., 2021). Linear vector-valued lower-level programs admit closed Pareto/frontier graphs and thus guarantee solution existence.

Necessary and Sufficient Optimality Conditions

Generalized Mordukhovich-stationarity conditions for BLMOPs are derived using coderivative calculus of the frontier (value) map and normal cone inclusions (Lafhim et al., 2021, Hoff et al., 2023). For multiobjective fractional BLMOPs, directional convexificators and nonsmooth Abadie-type CQ yield strong and weak duality theorems in the Mond-Weir sense, with conditions for pseudoconvex and quasiconvex objectives (Lara et al., 22 Nov 2025).

Subgradient and Hypergradient Methods

Single-loop subgradient methods (risk-averse, min–max structure) achieve $O(\epsilon^{-2})$ oracle complexity under mild smoothness plus strong convexity for the regularized LL or worst-case formulations (Chen et al., 2023). For robust, min–max, multiobjective BLO, the MORBiT and MORMA-SOBA algorithms provide convergence to first-order stationarity at $O(\sqrt{n}K^{-2/5})$ and $O(n^{5}\mu_{\lambda}^{-4}\epsilon^{-2})$ sample rates, with all hypergradients constructed via implicit differentiation or stochastic averages, and no inner-loop Hessian inversion (Gu et al., 2022, Chen et al., 2023).

5. Applications in Machine Learning and Engineering

Robust Hyperparameter and Poisoning-Resistant Learning

Multiobjective bilevel formulations designed for adversarial robustness, modeled as a multiobjective or min–max BLO between attacker and defender, reveal the regulatory role of $L_2$ regularization in mitigating poisoning attacks, with the learned hyperparameter increasing adaptively to counteract attack strength (Carnerero-Cano et al., 2020, Carnerero-Cano et al., 2023). Joint attacker–defender optimization is naturally cast as a saddle-point BLO with coupling through model parameters and hyperparameters, and the hypergradient-based coordinate updates accurately recover defense strategies not seen in constant-parameter baselines.

Evolutionary Architecture and Multi-Task Learning

Neural architecture search for multi-task graph neural networks and multi-task deep learning are formulated as BL-MOPs, explicitly modeling the trade-offs at both the topology and learning parameter levels (Wang et al., 2023). Surrogate-assisted upper-level search over combined topology and preference vectors, along with LL parameter training under scalarized or preference-weighted losses, achieves state-of-the-art Pareto front coverage across diverse tasks.

Black-Box and Large-Scale Engineering Design

Surrogate-aided bilevel evolutionary frameworks, e.g., PSP-BLEMO, enable efficient search in black-box BLMOPs by learning mappings from upper-level designs to lower-level Pareto solutions, drastically reducing function evaluation cost for each new upper-level candidate (Wang et al., 5 Sep 2024).

6. Benchmarking, Metrics, and Experimental Insights

Standard benchmarking for BLMOPs uses the ECO test suite (Deb & Sinha), mixed-integer extensions, and synthetic multiobjective lower-level models. Dominant performance metrics include IGD (inverted generational distance), Hypervolume, Epsilon indicator, and solution spread.

Empirical results from recent frameworks (BLMOL, PSP-BLEMO, MORBiT, MORMA-SOBA) demonstrate:

• Superior IGD and hypervolume performance for predictor-based seeding relative to nested surrogate or GAN methods, notably under deceptive landscapes (Wang et al., 5 Sep 2024).
• Substantial reduction in function evaluations and runtime, with accurate estimation of Pareto-optimal tradeoffs at both levels.
• Monotonic adaptation of regularization or hyperparameters in adversarial contexts as the attack budget increases, providing empirical evidence of defense provided by proper multiobjective BLO formulations (Carnerero-Cano et al., 2020, Carnerero-Cano et al., 2023).
• Transferability and enhanced generalization in multi-task representation learning, favoring robust (min–max) objectives (Gu et al., 2022).

Method/Class	Key Feature	Sample Complexity
Scalarization + MPEC	Converts lower-level to scalar	Exponential (NP-hard)
Value-function	Frontier/set-valued constraint	Polynomial in small n
MORBiT/MORMA-SOBA	Single-loop min–max BLO	$O(\sqrt{n} K^{-2/5})$ / $O(n^5\mu_\lambda^{-4}\epsilon^{-2})$
PSP-BLEMO, BLMOL	Surrogate evolutionary, NNs	Empirically sublinear

7. Current Limitations and Research Directions

Critical challenges in BLMOP remain:

• Exact solution mappings are nonconvex, NP-hard, and set-valued even for convex or linear data (Pujara et al., 5 Nov 2025).
• Handling $n>2$ LL objectives for PSP-like predictors requires advanced helper-variable frameworks or multidimensional indexing; feedforward NNs may become inadequate (Wang et al., 5 Sep 2024).
• Surrogate accuracy in large design spaces and uncertainty quantification are open challenges.
• Theoretical convergence guarantees for metaheuristic frameworks are limited; most results are asymptotic or local and rely on strong assumptions (smoothness, domination).
• Robustness against adversarial, stochastic, or noncooperative LL behavior necessitates risk-neutral and risk-averse formulations; algorithmic theory for stochastic and federated settings is developing.

Open research avenues include:

• Hybridization of surrogate models (NN and GP) with uncertainty quantification
• Algorithmic advances in single-loop Hessian-inversion-free methods matching single-level lower bounds
• Pareto set prediction for high-dimensional and multiobjective lower-level spaces
• Advanced coderivative calculus for stationarity conditions with nonconvex, nonsmooth, and fractional objectives (Lara et al., 22 Nov 2025)
• Applications to emergent fields such as distributed/federated learning, fair meta-learning, and automated decomposition for very large-scale BLMOPs

The literature points to the continued need for both theoretical and algorithmic innovation to make large-scale multiobjective bilevel optimization tractable in real-world settings (Pujara et al., 5 Nov 2025, Wang et al., 5 Sep 2024).