The increase of electric vehicle (EV) adoption in recent years has correspondingly increased the importance of providing adequate charging infrastructure for EV users. For a charging service provider, the fundamental question is to determine the optimal location and sizing of charging stations with respect to a given objective and subject to budget and other practical constraints. Practical objectives include maximizing EV adoption as part of a public policy on electric transportation, and maximizing the profit gained from providing this service, in which case the price of charging may also be optimized. In this talk, we will present an overview of our work in this area and discuss open directions for future research.

When Machine Learning models are used in high stakes decision making settings, feedback to users on decisions may be requested. Assume given a trained binary Supervised Classification model, where the positive class is the "good" class. A counterfactual solution of a record is a point which is simultaneously close enough -according to a given metric- to the record and whose probability of belonging to the positive class -according to the classification model- is high enough. Finding counterfactual solutions naturally leads to biobjective optimization problems, whose structure -and thus the optimization tools available to address them- will depend on the different ingredients associated with the problem. In this talk some of these mathematical optimization problems will be analyzed.

It is known that all directed cycles necessary to reach an optimal network flow solution are observed on the so-called residual network. Each of these accommodates positive flow values and forms a direction. A degenerate pivot is induced when the selected cycle in fact does not exist. The concept of cycles can be transferred to linear programs and alternative necessary and sufficient optimality conditions are expressed on the linear programming residual problem. We propose a family of algorithms with non-degenerate pivots and also show that the local search heuristics for vehicle routing problems, such as 2-opt, 3-opt, swap, relocate… are indeed directed cycles on the (appropriate!) residual network.

Thanks to significant algorithmic advances in the field of computational bilevel optimization, today we can solve much larger and more complicated bilevel problems compared to what was possible two decades ago. In this talk, we will focus on one of the emerging and challenging classes of bilevel problems: bilevel optimization under uncertainty. We will discuss classical ways of addressing uncertainties in bilevel optimization using stochastic or robust optimization techniques. However, the sources of uncertainty in bilevel optimization can be much richer than for usual, single-level problems, since not only the problem’s data can be uncertain but also the (observation of the) decisions of the two players can be subject to uncertainty. Thus, we will also discuss bilevel optimization under limited observability, the area of problems considering only near-optimal decisions, and intermediate solution concepts between the optimistic and pessimistic cases. The talk is based on articles by [1, 2, 3].

[1] Yasmine Beck, Ivana Ljubic, Martin Schmidt, A Survey on Bilevel Optimization Under Uncertainty, European Journal of Operational Research, (2023). https://doi.org/10.1016/j.ejor.2023.01.008
[2] Yasmine Beck, Ivana Ljubic, Martin Schmidt, A Brief Introduction to Robust Bilevel Optimization, SIAG/OPT Views and News, 30(2), (2023). https://siagoptimization.github.io/assets/views/ViewsAndNews-30-2.pdf
[3] Yasmine Beck, Ivana Ljubic, Martin Schmidt, Exact Methods for Discrete τ-Robust Interdiction Problems with an Application to the Bilevel Knapsack Problem, Mathematical Programming Computation, (2023). https://doi.org/10.1007/s12532-023-00244-6

We introduce a novel approach to prescriptive analytics that leverages robust satisficing techniques to determine optimal decisions in situations of risk ambiguity and prediction uncertainty. Our decision model relies on a reward function that incorporates uncertain parameters, which can be partially predicted using available side information. However, the accuracy of the linear prediction model depends on the quality of regression coefficient estimates derived from the available data. To achieve a desired level of fragility, we begin by establishing a target relative to the predict-then-optimize objective and solve a residual-based robust satisficing model. Next, we solve a new estimation-fortified robust satisficing model that minimizes the influence of estimation uncertainty while ensuring that the estimated fragility of the solution in achieving a less ambitious guarding target falls below the level for the desired target. Our approach is supported by statistical justifications, and we propose tractable models for various scenarios, such as saddle functions, two-stage linear optimization problems, and decision-dependent predictions. We demonstrate the effectiveness of our approach through case studies involving a wine portfolio investment problem and a multi-product pricing problem using real-world data. Our numerical studies show that our approach outperforms the predict-then-optimize approach in achieving higher expected rewards when evaluated on the actual distribution. Notably, we observe significant improvements over the benchmarks, particularly in cases of limited data availability.

This is a joint work with Qinshen Tang, Minglong Zhou and Taozeng Zhu.

Lagrangian based methods have been well known for over 50 years. These methods are robust and can often handle optimization problems with complex geometries and structures involving nonsmooth, convex, and nonconvex data. These are prevalent in modern applications, and as such, Lagrangian based methods have attracted a strong revived research interest over the last decade. However, Lagrangian schemes often pose serious theoretical and computational challenges. This talk addresses some of these theoretical challenges by exploiting the problem's structure. In the convex case, the focus is on nonasymptotic global rate of convergence guarantees, which are derived here through a novel and simple approach. In the much harder nonconvex setting, we focus on the design and analysis of adaptive Lagrangian schemes for a comprehensive class of models with nonlinear functional composite structures. We highlight the main obstacles involved, some recent novel ways to tackle or avoid them, and the pillars of a convergence analysis which allows to achieve theoretical guarantees, including iteration complexity and pointwise global convergence.