Inverse Optimization with Noisy Data

Inverse optimization refers to the inference of unknown parameters of an optimization problem based on knowledge of its optimal solutions. This paper considers inverse optimization in the setting where measurements of the optimal solutions of a convex optimization problem are corrupted by noise. We first provide a formulation for inverse optimization and prove it to be NP-hard. In contrast to existing methods, we show that the parameter estimates produced by our formulation are statistically consistent. Our approach involves combining a new duality-based reformulation for bilevel programs with a regularization scheme that smooths discontinuities in the formulation. Using epi-convergence theory, we show the regularization parameter can be adjusted to approximate the original inverse optimization problem to arbitrary accuracy, which we use to prove our consistency results. Next, we propose two solution algorithms based on our duality-based formulation. The first is an enumeration algorithm that is applicable to settings where the dimensionality of the parameter space is modest, and the second is a semiparametric approach that combines nonparametric statistics with a modified version of our formulation. These numerical algorithms are shown to maintain the statistical consistency of the underlying formulation. Lastly, using both synthetic and real data, we demonstrate that our approach performs competitively when compared with existing heuristics

A novel approach for bilevel programs based on Wolfe duality

This paper considers a bilevel program, which has many applications in practice. To develop effective numerical algorithms, it is generally necessary to transform the bilevel program into a single-level optimization problem. The most popular approach is to replace the lower-level program by its KKT conditions and then the bilevel program can be reformulated as a mathematical program with equilibrium constraints (MPEC for short). However, since the MPEC does not satisfy the Mangasarian-Fromovitz constraint qualification at any feasible point, the well-developed nonlinear programming theory cannot be applied to MPECs directly. In this paper, we apply the Wolfe duality to show that, under very mild conditions, the bilevel program is equivalent to a new single-level reformulation (WDP for short) in the globally and locally optimal sense. We give an example to show that, unlike the MPEC reformulation, WDP may satisfy the Mangasarian-Fromovitz constraint qualification at its feasible points. We give some properties of the WDP reformulation and the relations between the WDP and MPEC reformulations. We further propose a relaxation method for solving WDP and investigate its limiting behavior. Comprehensive numerical experiments indicate that, although solving WDP directly does not perform very well in our tests, the relaxation method based on the WDP reformulation is quite efficient

A Fenchel-Lagrange Duality Approach for a Bilevel Programming Problem with Extremal-Value Function

International audienceIn this paper, for a bilevel programming problem (S) with an extremal-value function, we first give its Fenchel-Lagrange dual problem. Under appropriate assumptions, we show that a strong duality holds between them. Then, we provide optimality conditions for (S) and its dual. Finally, we show that the resolution of the dual problem is equivalent to the resolution of a one-level convex minimization problem

Data-driven Inverse Optimization with Imperfect Information

In data-driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent's objective function that best explains a historical sequence of signals and corresponding optimal actions. We focus here on situations where the observer has imperfect information, that is, where the agent's true objective function is not contained in the search space of candidate objectives, where the agent suffers from bounded rationality or implementation errors, or where the observed signal-response pairs are corrupted by measurement noise. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the {\em predicted} decision ({\em i.e.}, the decision implied by a particular candidate objective) differs from the agent's {\em actual} response to a random signal. We show that our framework offers rigorous out-of-sample guarantees for different loss functions used to measure prediction errors and that the emerging inverse optimization problems can be exactly reformulated as (or safely approximated by) tractable convex programs when a new suboptimality loss function is used. We show through extensive numerical tests that the proposed distributionally robust approach to inverse optimization attains often better out-of-sample performance than the state-of-the-art approaches

Canonical duality theory and algorithm for solving bilevel knapsack problems with applications

A novel canonical duality theory (CDT) is presented for solving general bilevel mixed integer nonlinear optimization governed by linear and quadratic knapsack problems. It shows that the challenging knapsack problems can be solved analytically in term of their canonical dual solutions. The existence and uniqueness of these analytical solutions are proved. NP-hardness of the knapsack problems is discussed. A powerful CDT algorithm combined with an alternative iteration and a volume reduction method is proposed for solving the NP-hard bilevel knapsack problems. Application is illustrated by benchmark problems in optimal topology design. The performance and novelty of the proposed method are compared with the popular commercial codes. © 2013 IEEE

Globally Solving a Class of Bilevel Programs with Spatial Price Equilibrium Constraints

Bilevel programs with spatial price equilibrium constraints are strategic models that consider a price competition at the lower-level. These models find application in facility location-price models, optimal bidding in power networks, and integration of renewable energy sources in distribution networks. In this paper, for the case where the equilibrium at the lower level can be formulated as an optimization problem, we introduce an enhanced single-level formulation based on duality and show that its relaxation is stronger than the usual single-level formulation obtained using KKT conditions. Compared to the literature [1, 2], this new formulation is (i) computationally friendly to global solution strategies using branch-and-bound, and (ii) able to handle larger instance sizes. Further, we develop a heuristic procedure to find feasible solutions inside of the branch-and-bound tree that is effective on large-sized instances and produces solutions whose objective values are close to the relaxation bound. We demonstrate the benefits of this formulation and heuristic through an extensive numerical study on synthetic instances of Equilibrium Facility Location [3] and on standard IEEE bus networks for planning renewable generation capacity under uncertainty