Inverse Multi-Objective Optimization

• Inverse multi-objective optimization is the process of recovering unknown objective functions, parameters, or trade-off structures from observed Pareto-critical decisions.
• The method employs techniques like KKT-based SVD inversion, empirical risk minimization, and online learning to manage noise and ensure solution accuracy.
• Recent advances include inverse design and inverse reinforcement learning, which map desired trade-offs to feasible, Pareto-optimal solutions in complex systems.

Inverse multi-objective optimization concerns the inverse problem of multi-objective decision making: from observed decisions that are assumed to be Pareto-optimal, Pareto-efficient, or Pareto-critical, one seeks to recover the objective vector, model parameters, constraints, or trade-off structure that could have generated them. In one canonical formulation, given decision vectors $X=\{x^1,\dots,x^N\}\subset\mathbb R^n$, the task is to find an objective vector $$F(x)=(f_1(x),\dots,f_m(x))\in C^1(\mathbb R^n,\mathbb R^m)$$ such that each $x^j$ is Pareto-critical for $\min_x F(x)$ (Gebken et al., 2019). In parameter-learning formulations, noisy observations $y_i$ are fitted to the efficient set $X_E(\theta)$ of a convex multi-objective decision-making problem through losses such as $l(y,\theta)=\min_{x\in X_E(\theta)}\|y-x\|_2^2$ (Dong et al., 2018). Recent engineering papers also use the term in an inverse-design sense, where the goal is to map desired objective trade-offs or task-preference queries back to feasible designs or Pareto-optimal decisions (Farias et al., 2024, Wei et al., 12 Nov 2025).

1. Problem classes and scope

The literature contains several distinct, but related, inverse tasks. Some formulations infer an unknown objective vector from a given set of Pareto-critical points; others infer parameters of a known multi-objective model from noisy observations; others test whether observed behavior is even consistent with Pareto optimality; and still others learn an inverse map from objective space to decision space for design purposes.

Strand	Inverse quantity	Representative papers
Objective reconstruction	$F(x)$ from Pareto-critical data	(Gebken et al., 2019)
Parameter inference	$\theta$ in a convex MOP	(Dong et al., 2018, Dong et al., 2020, Dong et al., 2020)
Trade-off-preserving recovery	weights $\alpha$ and Pareto-optimal $$F(x)=(f_1(x),\dots,f_m(x))\in C^1(\mathbb R^n,\mathbb R^m)$$0 near $$F(x)=(f_1(x),\dots,f_m(x))\in C^1(\mathbb R^n,\mathbb R^m)$$1	(Chan et al., 2017)
Coordination detection	utilities and Pareto weights from observed actions	(Snow et al., 2022, Snow et al., 2023)
Inverse design / inverse modeling	$$F(x)=(f_1(x),\dots,f_m(x))\in C^1(\mathbb R^n,\mathbb R^m)$$2 from desired objective values or task-preference inputs	(Zhang et al., 2024, Farias et al., 2024, Wei et al., 12 Nov 2025, Kadlec et al., 2024)

A recurrent forward model is the weighted-sum scalarization. For a parameterized convex multi-objective program with objectives $$F(x)=(f_1(x),\dots,f_m(x))\in C^1(\mathbb R^n,\mathbb R^m)$$3 and feasible set $$F(x)=(f_1(x),\dots,f_m(x))\in C^1(\mathbb R^n,\mathbb R^m)$$4, one defines

$$F(x)=(f_1(x),\dots,f_m(x))\in C^1(\mathbb R^n,\mathbb R^m)$$5

Under convexity, this scalarization characterizes the efficient set through inclusions such as $$F(x)=(f_1(x),\dots,f_m(x))\in C^1(\mathbb R^n,\mathbb R^m)$$6 (Dong et al., 2018). In the online-learning formulation, for $$F(x)=(f_1(x),\dots,f_m(x))\in C^1(\mathbb R^n,\mathbb R^m)$$7 it generates all Pareto-efficient solutions (Dong et al., 2020).

2. Core mathematical formulations

A central route to inverse reconstruction is to invert necessary optimality conditions. For differentiable objectives, a point $$F(x)=(f_1(x),\dots,f_m(x))\in C^1(\mathbb R^n,\mathbb R^m)$$8 is extended Pareto-critical iff

$$F(x)=(f_1(x),\dots,f_m(x))\in C^1(\mathbb R^n,\mathbb R^m)$$9

The 2019 objective-reconstruction method treats the objective vector appearing in these multiobjective KKT conditions as the unknown. Each objective is expanded in a finite basis $x^j$0,

$x^j$1

which converts the inverse problem into a homogeneous linear system $x^j$2. Exact reconstruction is possible when $x^j$3; otherwise one solves

$x^j$4

through the SVD of $x^j$5, and the right-singular vector corresponding to the smallest singular value yields an approximate nullspace solution (Gebken et al., 2019). This formulation naturally accommodates noisy data through least squares, and the paper also notes Tikhonov regularization and $x^j$6-penalization as mechanisms for controlling overfitting or promoting sparsity.

A second major formulation is empirical-risk minimization over the efficient set. Given noisy observations $x^j$7, the inverse learner minimizes the empirical risk

$x^j$8

Because $x^j$9 is difficult to manipulate explicitly, the 2018 framework approximates it by a finite union of weighted-sum solutions using sampled weights $\min_x F(x)$0, introducing efficient-point variables $\min_x F(x)$1 and binary assignment variables $\min_x F(x)$2 (Dong et al., 2018). The resulting model, termed IMOP-EMP-WS in the summary, enforces $\min_x F(x)$3 through KKT conditions or strong duality.

A third formulation begins with a possibly non-Pareto-optimal input $\min_x F(x)$4. The trade-off-preserving inverse model solves

$\min_x F(x)$5

along with feasibility constraints. At optimality, the Lagrange multipliers of these inequalities provide a valid weight vector $\min_x F(x)$6, and the corresponding $\min_x F(x)$7 is Pareto-optimal while preserving a prescribed trade-off pattern encoded by $\min_x F(x)$8 (Chan et al., 2017). Relative and absolute trade-off preservation arise as special cases.

3. Algorithms for estimation, streaming data, and robustness

For batch parameter inference, the 2018 framework develops two algorithmic routes. The first is an ADMM heuristic based on a nonconvex global-consensus splitting over groups of observations, with local copies $\min_x F(x)$9 and augmented-Lagrangian updates. The second is a clustering-based algorithm of K-means style: cluster the observations, solve a smaller inverse problem on cluster centroids, reassign observations to updated efficient points, and iterate until stabilization. The summary reports that each step decreases the sampled-loss objective and that only finitely many clusterings are possible, giving convergence to a local optimum (Dong et al., 2018).

For sequentially arrived data, the 2020 online-learning paper formulates the inverse task with a streaming loss

$y_i$0

where $y_i$1, and updates $y_i$2 by implicit proximal rules rather than explicit gradients. The full update solves

$y_i$3

while the accelerated update first assigns $y_i$4 to its nearest efficient sample under $y_i$5 and then solves a single subproblem for that index (Dong et al., 2020). Under compactness, strong convexity, and a convexity-of-loss assumption, the regret satisfies $y_i$6 with $y_i$7. In numerical experiments, the accelerated method is reported to be $y_i$8–$y_i$9 faster for large $X_E(\theta)$0.

Distributional robustness enters through the Wasserstein distributionally robust inverse multiobjective optimization model. Instead of minimizing empirical expected loss, the learner solves

$X_E(\theta)$1

where $X_E(\theta)$2 is a 1-Wasserstein ball around the empirical distribution $X_E(\theta)$3 (Dong et al., 2020). The paper derives a semi-infinite reformulation using dual variables $X_E(\theta)$4 and $X_E(\theta)$5, and then applies a cutting-plane exchange method: solve a master problem on finitely many sampled support points, identify the most-violating subproblems, add those points, and repeat until all violations are below a tolerance $X_E(\theta)$6. The algorithm returns a $X_E(\theta)$7-optimal solution in finite iterations, and the excess risk bound is $X_E(\theta)$8.

4. Revealed preferences, coordination detection, and inverse reinforcement learning

A nonparametric branch of inverse multi-objective optimization is grounded in revealed-preference theory. In the cognitive-radar setting, the network is said to coordinate if the agents’ emissions solve a Pareto-weighted utility maximization problem under a common resource constraint. The 2022 paper reformulates consistency with coordination through Afriat-type inequalities. Introducing feasible personalized quantities $X_E(\theta)$9, utility levels $l(y,\theta)=\min_{x\in X_E(\theta)}\|y-x\|_2^2$0, and positive multipliers $l(y,\theta)=\min_{x\in X_E(\theta)}\|y-x\|_2^2$1, the dataset is rationalizable iff these variables satisfy

$l(y,\theta)=\min_{x\in X_E(\theta)}\|y-x\|_2^2$2

for all agents and all observation pairs, together with feasibility relations linking the unobserved $l(y,\theta)=\min_{x\in X_E(\theta)}\|y-x\|_2^2$3 to the observed aggregate behavior (Snow et al., 2022). The feasibility problem can be encoded as a mixed-integer linear program, and whenever it is feasible, utilities are reconstructed by Afriat’s piecewise-linear formula

$l(y,\theta)=\min_{x\in X_E(\theta)}\|y-x\|_2^2$4

The 2023 statistical-detection paper adds noisy observations and turns rationalizability into a hypothesis test. For each agent, it solves a linear program for the minimal slack $l(y,\theta)=\min_{x\in X_E(\theta)}\|y-x\|_2^2$5 in the revealed-preference inequalities, defines $l(y,\theta)=\min_{x\in X_E(\theta)}\|y-x\|_2^2$6, estimates the null distribution of $l(y,\theta)=\min_{x\in X_E(\theta)}\|y-x\|_2^2$7 by simulation, and obtains a $l(y,\theta)=\min_{x\in X_E(\theta)}\|y-x\|_2^2$8-value $l(y,\theta)=\min_{x\in X_E(\theta)}\|y-x\|_2^2$9 (Snow et al., 2023). Under the null hypothesis of coordination, the paper states $F(x)$0, giving finite-sample Type-I control.

Inverse reinforcement learning introduces another route to inverse multi-objective recovery. In Wasserstein inverse reinforcement learning, expert actions are assumed to arise from an unknown scalarization weight $F(x)$1, and the learner minimizes

$F(x)$2

by projected subgradient updates

$F(x)$3

(Kitaoka et al., 2023). A companion proof paper establishes finite-iteration reward imitation and, under lexicographic tie-breaking, an action-imitation theorem linking zero subgradient, exact action recovery, and vanishing Wasserstein distance (Kitaoka et al., 2023).

5. Inverse design and inverse modeling

Engineering papers use inverse multi-objective optimization in a different sense: rather than inferring latent utilities or parameters from observed decisions, they seek a design $F(x)$4 that realizes desired multi-objective trade-offs. In microperforated-panel design, the inverse analysis maps acoustic performance and fabrication cost back to mixed discrete-continuous panel parameters. The framework couples a finite-element acoustic model with an adapted multi-objective particle swarm optimization algorithm, runs separate swarms for the discrete number of layers $F(x)$5, compares the resulting Pareto fronts by hypervolume, and returns non-dominated solutions that balance low-frequency sound absorption against fabrication cost (Zhang et al., 2024). The paper reports a final set of $F(x)$6 non-dominated solutions.

A more explicitly inverse-modeling formulation appears in IM-C-MOEA/D. There, one approximates the inverse map from objective vectors $F(x)$7 to feasible decisions $F(x)$8 by decomposed Gaussian-process models $F(x)$9, embeds them in a decomposition-based constrained evolutionary algorithm, and assigns offspring to subproblems through the weighted Tchebycheff scalarizing function $\theta$0 (Farias et al., 2024). Constraint handling follows the feasibility-based rules of Jain and Deb. The reported experiments cover real-world constrained multi-objective problems RWMOP1–35 and evaluate performance by hypervolume.

Parametric expensive multi-objective optimization extends the inverse-design idea to a family of tasks indexed by $\theta$1. The goal is to learn a direct inverse map

$\theta$2

so that unseen task-preference queries can be answered without new expensive evaluations (Wei et al., 12 Nov 2025). The framework alternates between acquisition-driven search using a task-aware Gaussian process and generative solution sampling using a conditional VAE or conditional DDPM. The paper states that after training, the learned inverse model is used for zero-shot solution prediction on unseen tasks.

Inverse antenna design provides yet another design-oriented usage. The Multi-objective Memetic Algorithm with Adaptive Weights combines NSGA-II, a rank-1 local search over binary shape variables, and adaptive weighting of objective functions to recover dense Pareto approximations for discrete topology optimization (Kadlec et al., 2024).

6. Identifiability, computational structure, and unresolved questions

Several papers emphasize that inverse multi-objective problems are often underdetermined. In the KKT-SVD reconstruction method, exact nontrivial solutions exist iff $\theta$3; in the underdetermined regime $\theta$4, a nonzero nullspace always exists, but some solutions may degenerate by dropping dependence on variables. Uniqueness is only up to scaling when the smallest singular value is simple, and ties among singular values require additional criteria such as sparsity, nonnegativity, or smoothness (Gebken et al., 2019).

For parameter inference, identifiability is formalized through separation of efficient sets. The 2018 paper defines identifiability at $\theta$5 by requiring $\theta$6 for every $\theta$7, where $\theta$8 is the Hausdorff semi-distance (Dong et al., 2018). Under noiseless data, strong convexity, and identifiability, the estimator is consistent. The same paper also derives a uniform law of large numbers in $\theta$9, persistence of empirical-risk minimizers, and recovery of preference distributions under bijectivity of the map $\alpha$0.

Computationally, the field spans linear algebra, LP/MILP, mixed-integer conic optimization, and semi-infinite optimization. The KKT-basis method reduces inversion to assembling $\alpha$1 at cost $\alpha$2 and computing an SVD of an $\alpha$3 matrix (Gebken et al., 2019). The revealed-preference coordination test is an MILP and is described as NP-hard in general in $\alpha$4, $\alpha$5, and $\alpha$6, though the radar paper states that for moderate $\alpha$7 and small $\alpha$8 it is readily solved in milliseconds (Snow et al., 2022). Online IMOP subproblems can be written via KKT conditions as mixed-integer conic programs (Dong et al., 2020), and Wasserstein-robust IMOP requires a cutting-plane method for a semi-infinite reformulation (Dong et al., 2020).

The literature also delineates several open issues. The finite-imitation WIRL analysis does not provide explicit iteration-complexity bounds in terms of the number of objectives or variables, and it leaves the non-realizable expert case, nonconvex $\alpha$9, nonlinear reward parameterizations, partial observability, and stochastic policies for future study (Kitaoka et al., 2023). A broader implication of the surveyed work is that inverse multi-objective optimization is not a single model class but a family of inverse problems organized around Pareto structure: reconstructing objectives, inferring parameters or utilities, statistically testing Pareto rationalizability, and learning inverse maps from desired trade-offs to feasible decisions.