Learning in Inverse Optimization: Incenter Cost, Augmented Suboptimality Loss, and Algorithms

TL;DR

This work advances inverse optimization by introducing the incenter cost vector as a robust, tractable alternative to the circumcenter, and by formulating Augmented Suboptimality Loss (ASL) to handle inconsistent data. It develops convex reformulations and ellipsoidal variants, and pairs them with a novel first-order algorithm, Stochastic Approximate Mirror Descent (SAMD), to efficiently solve IO problems with large or mixed-integer feasible sets. The authors provide a practical Python package, InvOpt, and demonstrate through extensive experiments that incenter-based approaches and ASL yield superior out-of-sample performance and computational efficiency compared with existing methods. The results bridge IO with structured prediction and offer scalable tools for applications in routing, economics, and beyond.

Abstract

In Inverse Optimization (IO), an expert agent solves an optimization problem parametric in an exogenous signal. From a learning perspective, the goal is to learn the expert's cost function given a dataset of signals and corresponding optimal actions. Motivated by the geometry of the IO set of consistent cost vectors, we introduce the "incenter" concept, a new notion akin to circumcenter recently proposed by Besbes et al. (2023). Discussing the geometric and robustness interpretation of the incenter cost vector, we develop corresponding tractable convex reformulations, which are in contrast with the circumcenter, which we show is equivalent to an intractable optimization program. We further propose a novel loss function called Augmented Suboptimality Loss (ASL), a relaxation of the incenter concept for problems with inconsistent data. Exploiting the structure of the ASL, we propose a novel first-order algorithm, which we name Stochastic Approximate Mirror Descent. This algorithm combines stochastic and approximate subgradient evaluations, together with mirror descent update steps, which is provably efficient for the IO problems with discrete feasible sets with high cardinality. We implement the IO approaches developed in this paper as a Python package called InvOpt. Our numerical experiments are reproducible, and the underlying source code is available as examples in the InvOpt package.

Figures (12)

• Figure 1: Geometrical visualization of the circumcenter and incenter vectors. The green regions are the intersection of $\mathbb{C}$ with the sphere and the blue cones are revolution cones with an aperture angle equal to the optimal value of \ref{['eq:circumcenter']} and \ref{['eq:incenter']}.
• Figure 2: Geometrical illustration of Corollary \ref{['coro:incenter_convex_reformulation']} in a simple 2D example. The grey region represents $\mathbb{C}$, the green region represents the feasible set of \ref{['eq:incenter_reformulation']}, and the red region represents the feasible set of \ref{['eq:incenter_convex_reformulation']}. As can be seen, the optimal solution of \ref{['eq:incenter_convex_reformulation']} can be interpreted as the smallest norm vector inside an inner cone with boundaries 1 unit away from the boundaries of $\mathbb{C}$. Also, it can be seen that by normalizing $\bar{\theta}^{\text{in}}$, we retrieve $\theta^{\text{in}}$, an optimal solution of \ref{['eq:incenter_reformulation']}.
• Figure 3: Out-of-sample results for consistent data scenario.
• Figure 4: Out-of-sample results for inconsistent data scenario.
• Figure 5: Out-of-sample results for the mixed-integer feasible set scenario.

Theorems & Definitions (23)

• Example 2.1: Quadratic hypothesis
• Definition 2.2: Consistent cost vectors
• Definition 3.1: Circumcenter besbes2023contextual
• Theorem 3.2: Intractability of circumcenter
• Definition 3.3: Incenter
• Remark 3.4: Robustness: circumcenter vs. incenter
• Theorem 3.5: Incenter reformulation
• Corollary 3.6: Incenter convex characterization
• Remark 3.7: Connection with extremal volume balls
• Definition 4.1: Augmented Suboptimality Loss