🤖 AI Summary
This work addresses the challenge of explicitly characterizing the iteration complexity in data-driven inverse optimization for estimating the objective function parameters of integer linear programs. By analyzing the geometric properties of a suboptimality-based loss function, the authors employ projected subgradient descent to achieve exact consistency with observed data within a finite number of steps. The key contribution lies in providing, for the first time, an explicit polynomial bound on the required number of iterations in terms of the sample size, feature dimension, feature range, and structural properties of the constraint matrix—thereby overcoming the prior reliance on unknown geometric constants. Leveraging fundamental quantities such as the Lipschitz continuity of the loss function and the diameter of the weight set, the study establishes a theoretically transparent and practically informative upper bound on the iterations needed to attain exact consistency.
📝 Abstract
A data-driven inverse optimization problem (DDIOP) is the problem of estimating the objective-function parameters (weights) that explain observed optimal-solution data, and it arises in many applications, including integer linear programming (ILP). It is known that, by applying gradient-based optimization methods to the suboptimality loss, the inverse optimization of ILPs can be solved exactly within finitely many oracle iterations, and that the required number of iterations is bounded as $T=O(1/γ(\ell_{\mathrm{sub}})^2)$ in terms of a problem-dependent geometric constant $γ(\ell_{\mathrm{sub}})$. However, no means of bounding $γ(\ell_{\mathrm{sub}})$ from below as a function of the problem size has been available, and hence the number of iterations could not be given as an explicit function of the problem size. We therefore give, when the forward problem is an integer linear program (ILP), the number of iterations sufficient for projected subgradient descent applied to the suboptimality loss to achieve exact consistency with the observed data, as a fully explicit function of the number of samples, the dimension of the features, the ranges of the features, and the structure of the constraint coefficient matrix, up to polynomial factors in the basic constants (the diameter of the weight set, the step-size parameter, and the Lipschitz constant of the suboptimality loss).