Inverse Optimization Procedure

• Inverse optimization is a rigorous framework that infers latent cost functions and constraints from observed decision data.
• It leverages forward and inverse models, KKT reformulations, and weighted-sum approximations to reconcile noisy, multiobjective decisions.
• Scalable algorithms like ADMM and clustering enable efficient parameter recovery and preference distribution estimation with robust statistical guarantees.

Inverse Optimization Procedure

Inverse optimization is a rigorous mathematical framework for inferring hidden parameters of an optimization model based on observed decisions or trajectories. The procedure inverts the conventional direction of optimization: given observed solutions generated by agents, systems, or physical processes—possibly under noise, population heterogeneity, or multiple objectives—it aims to reconstruct cost functions, constraints, or preference distributions that rationalize those observations. This paradigm provides foundational tools for preference elicitation, behavior modeling, system identification, and learning in operations research, statistics, and engineering. The following sections survey the main methodologies, theoretical underpinnings, computational structures, and example applications of inverse optimization procedures, with emphasis on modern statistical and algorithmic developments.

1. Mathematical Formulation: Forward and Inverse Models

Inverse optimization is built upon the structure of a forward optimization problem, denoted as a parameterized program:

• Single-objective case: $\min_{x\in X(\theta)} f(x,\theta)$, with %%%%1%%%%, $X(\theta)=\{x:g(x,\theta)\leq 0\}$, and unknown parameter vector $\theta$.
• Multiobjective extension: $$\min_{x\in X(\theta)} f(x,\theta) \equiv (f_1(x,\theta),\ldots,f_p(x,\theta))$$.

Assume observed decisions $\{y_i\}_{i=1}^N$ (possibly noisy). Each $y_i$ is assumed to be (approximately) generated by an optimal (or efficient, in the multiobjective case) solution $x^*(\theta)$ under unknown true $\theta_0$ and possibly unknown preferences (e.g., weights $w_i$ in multiobjective models).

The inverse optimization procedure defines a loss function that measures the fit between observed and model-generated decisions, for example:

• Single-objective loss: $$l(y,\theta) = \min_{x^* \in S(\theta)} ||y-x^*||^2_2$$, where $S(\theta)$ is the solution set.
• Multiobjective loss: $l(y,\theta) = \min_{x \in X_E(\theta)} ||y-x||^2_2$.

The population risk is $M(\theta) = \mathbb{E}[l(Y,\theta)]$, and empirical risk is $M^N(\theta) = (1/N)\sum_{i=1}^N l(y_i,\theta)$. The inverse optimization problem is then

$\min_{\theta \in \Theta} M^N(\theta)$

subject to the condition that feasible $x^*$ explain the data under the model parameter $\theta$.

For multiobjective models, the efficient set $X_E(\theta)$ is approximated by weighted-sum representations $S(w_k, \theta)$, with $w_k$ sampled from the simplex $W_p=\{w \in \mathbb{R}_+^p: \sum_{l} w_l = 1\}$.

2. Mathematical Reformulation and Model Structure

Inverse optimization procedures are reformulated to admit tractable computation and analysis.

• Single-level reformulation: By sampling $K$ representative weights $\{w_k\}$, the efficient set $X_E(\theta)$ is approximated by $\bigcup_{k=1}^K S(w_k,\theta)$. The assignment of observed $y_i$ to efficient solutions $x_k$ is tracked by binary variables $z_{ik}$.
• Optimization model (IMOP-EMP-WS):

$$\min_{\theta, \{x_k\}, \{z_{ik}\}} \frac{1}{N} \sum_{i=1}^N \| y_i - \sum_{k=1}^K z_{ik} x_k \|^2_2$$

subject to $x_k \in S(w_k, \theta)$, $\sum_{k=1}^K z_{ik} = 1$, $z_{ik} \in \{0,1\}$.

• KKT-based single-level reformulation: For convex $f$ and $g$, each $x_k \in S(w_k,\theta)$ is enforced by the KKT (Karush-Kuhn-Tucker) optimality conditions of the weighted-sum subproblem.

This yields a large-scale mixed-integer nonlinear program (MINLP) in $$(\theta, \{x_k\}, \{\text{dual variables}\}, \{z_{ik}\})$$. Direct solution of the full MINLP is intractable for large $N$ and $K$; hence, specialized scalable heuristics are developed.

3. Computational Algorithms: ADMM and Clustering-Based Methods

Two principal algorithmic approaches are used for solving large-scale inverse optimization problems in the multiobjective setting (Dong et al., 2018):

A. ADMM-Based Heuristic

• Partition the $N$ observations into $T$ disjoint blocks.
• Introduce local parameter copies $\theta^t$ for each block and consensus variable $\theta$ with dual variables $v^t$.
• Solve the augmented Lagrangian form:

$$\min_{\theta, \{\theta^t\}} \sum_{t=1}^T \sum_{i \in \text{block }t} l_K(y_i, \theta^t) + \rho \sum_{t=1}^T \|\theta^t-\theta\|_2^2$$

subject to $\theta^t = \theta$.

• Update $\theta^t$, $\theta$, $v^t$ in alternating fashion.
• Each $\theta^t$ update solves an IMOP subproblem on a small batch.
• Empirical convergence in $O(100)$ iterations; substantial parallel speed-up.

B. Clustering-Based Heuristic (Kmeans-IMOP)

• Observe equivalence to $K$-means clustering: If cluster assignments $C_k$ are known, the objective simplifies as

$$(1/N) \sum_{k=1}^K |C_k| \left( \| \overline{y}_k - x_k \|^2_2 + \text{Var}(C_k) \right)$$

where $\overline{y}_k$ is cluster centroid.

• Alternate assignment of each $y_i$ to nearest $x_k$, and updating $x_k$ by solving a reduced IMOP for cluster centroids.
• Guaranteed monotonic descent and finite convergence to a local optimum.

Both methods scale to $N$ up to $10^5$ and $K$ up to $100$; direct MINLP is only feasible for $N, K \lesssim 20$.

4. Statistical Guarantees: Consistency, Identifiability, and Preference Recovery

Under convexity, boundedness, and regularity assumptions (Dong et al., 2018), the procedure enjoys the following statistical properties:

• Uniform law of large numbers: $$\sup_{\theta \in \Theta} | M_K^N(\theta) - M_K(\theta) | \xrightarrow{p} 0$$ as $N\to\infty$.
• Uniform convergence in $K$: $\sup_\theta | M_K(\theta) - M(\theta) | \to 0$ as $K\to\infty$, provided objective functions are strongly convex.
• Prediction consistency: Any empirical minimizer $\hat \theta_K^N$ satisfies $M(\hat \theta_K^N) \to M(\theta^*)$ in probability where $\theta^*$ minimizes $M$.
• Identifiability (Hausdorff-semi-distance): The model is identifiable at $\theta_0$ if for all $\theta\neq\theta_0$, $d_{sH}(X_E(\theta_0), X_E(\theta)) > 0$.
• Preference recovery: Under bijectivity ($w \mapsto S(w,\theta_0)$ one-to-one), recovered weights $w_k$ assigned to each $y_i$ converge to true $w_i$.
• Generalization bound: By Rademacher complexity, for minimizer $\hat\theta_K^N$, with probability $\ge 1 - \delta$,

$$M_K(\hat\theta_K^N) \le M_K^N(\hat\theta_K^N) + O(1/\sqrt{N})$$

These guarantees ensure estimator consistency, recovery of true parameters, and reliable estimation of population-wide preference heterogeneity.

5. Recovery of Population Preference Distributions

Beyond point estimation, the procedure supports population-level inference of preference distributions:

• After IMOP-EMP-WS solution, cluster assignments $z_{ik}$ yield $C_k = \{ i : z_{ik}=1 \}$.
• Each cluster corresponds to a sampled preference weight $w_k$.
• The empirical distribution of $\{w_k\}$, weighted by $|C_k|/N$, estimates the population distribution of $w$.
• Under identifiability and bijectivity, this empirical distribution converges to the true population distribution as $(N, K) \to \infty$.

This facilitates quantitative characterization of groupwise and aggregate variability in multiobjective tradeoff preferences, which is key in applications where individual-level precision is infeasible.

6. Numerical Case Studies and Computational Scaling

Several case studies demonstrate the empirical accuracy, scalability, and preference recovery of the procedure (Dong et al., 2018):

• Tri-objective linear program: Efficient faces are perfectly recovered; parameter errors decay to zero with $N, K$ increasing.
• Quadratic program (RHS- and objective-learning): Both parameter and predictive errors decay with $N, K$; ADMM yields $>10\times$ speedup over MINLP.
• Markowitz portfolio reconstruction ($n=8$ assets): Noisy optimal portfolios under sampled normal weights; recovered expected-returns $\hat r$ generate efficient frontiers indistinguishable from ground truth; inferred weight distributions match generating distributions.
• Bi-criteria traffic assignment (network of 6 nodes, 2 OD pairs): Observed link flows under varied preferences accurately yield estimated OD demands convergent to true values.

In all cases, clustering and ADMM heuristics solve $N\sim 10^5$, $K\sim 100$ instances in minutes. Direct MINLP is prohibitively slow beyond $N,K \sim 20$. Empirical tests validate the theoretical consistency and identifiability results.

7. Significance and Limitations

The inverse optimization procedure described provides a powerful, scalable framework for parameter estimation, preference distribution recovery, and denoising in multiobjective decision environments. Its design accommodates noisy observations, population heterogeneity, and computational constraints via carefully constructed loss functions, convex reformulations, and efficient heuristics. The statistical guarantees ensure robust estimation under realistic data-generating mechanisms.

A plausible implication is that the approach extends naturally to more general multi-criteria and population mixture models, as well as to domain-specific inverse decision reconstruction problems. Limitations include reliance on convexity, necessity for identifiability, and restriction to settings where the efficient set can be approximated by weighted-sum formulations. Scalability, however, is preserved through ADMM and K-means-inspired decomposition techniques, validating its applicability in large-scale empirical studies and practical behavioral modeling.