This paper gives an overview of the development of Mixed-Integer Nonlinear Programming (MINLP) and Generalized Disjunctive Programming (GDP) over the past fifty years. We cover key methods, algorithms, and techniques for solving MINLPs and GDPs, focusing on both the modeling framework and solution techniques. We provide historical perspectives, highlight the key features and major challenges, and aim to give an in-depth introduction to the fields. We also discuss some future research directions. The paper is aimed at readers who are familiar with Mixed-Integer Linear Programming but are not experts on MINLP or GDP.

Multimodal mobility systems provide seamless travel by integrating different types of transportation modes. Most existing studies model service operations and users’ travel choices independently or iteratively and constrained with pre-defined multimodal travel options. The paper proposes a choice-based optimization approach that optimizes service operations with explicitly embedded travelers’ choices described by the multinomial logit (MNL) model. It allows the flexible combination of travel modes and routes in multimodal mobility systems. We propose a computationally efficient linearization method for transformed MNL constraints with bounded errors to solve the choice-based optimization model. The model is validated using a mobility on demand and public transport network by comparing it with a simulation sampling-based MNL linearization method. The results show that the mixed-integer formulation provides a high-quality solution in terms of both the estimated choice probability errors and computational speed. We also conduct an error analysis and a sensitivity analysis to explore the behavior of the proposed approach. The real-world case study in Stockholm further illustrates that the analytical formulation achieves a better system operation performance than the traditional iterative supply–demand updating optimization method. The choice-based optimization model and solution formulation are highly adaptable for operations decision support integrating stochastic travel choices in multimodal mobility systems.

Clustering is one of the most fundamental tools in data science and machine learning, and k-means clustering is one of the most common of such methods. There is a variety of approximate algorithms for the k-means problem, but only a few methods compute the globally optimal solution, as it is in general NP-hard. In this paper, we consider the k-means problem for instances with low-dimensional data and formulate it as a structured concave assignment problem. This allows us to exploit the low-dimensional structure and solve the problem to global optimality for very large data sets with several clusters, complementing and outperforming state-of-the-art for this class of problems. The method builds on iteratively solving a small concave problem and a large linear programming or assignment problem. This gives a sequence of feasible solutions along with bounds, which we show converges to a zero optimality gap. The paper combines methods from global optimization to accelerate the procedure, and we provide numerical results on synthetic data and real-world data.

Particle filter (PF) has proven effective for nonlinear identification scenarios; however, its performance in high-dimensional problems is often limited by the curse of dimensionality. To overcome this challenge, block particle filter (BPF) is proposed to reformulate a high-dimensional model into several blocks, so the identification of one high-dimensional system can be simplified into that for many lower-dimensional blocks. However, due to the coupling between blocks, the likelihood function for each state subgroup depends not only on its own state components (components of interest) but also on the components of its neighboring subgroups (nuisance components)—a dependency that BPF does not address. In order to extend BPF to coupled systems, we design likelihood functions, including plug-in, profile, and marginal likelihoods, that can incorporate nuisance components within each block. We demonstrate the state and parameter estimation performance of BPF with each likelihood through a numerical example of a forty-story Bouc-Wen frame structure under ground motion. We also design the BPF in a differentiable manner, integrate it into a deep learning architecture, and evaluate its performance on three datasets: the open-source Electricity Transformer Temperature (ETT) dataset, an open-source cutter wear dataset, and the AstraZeneca bearing dataset. The code is available at https://github.com/TianZLi/Likelihood-in-BPF.

Prognostic methods broadly fall into two categories—model-based and data-driven—both of which have shown effectiveness across a range of engineering applications. Model-based approaches require an explicit representation of the degradation process, defining failure as the point when the physical damage state exceeds a predetermined threshold. Data-driven methods, on the other hand, leverage sensor data to directly predict end-of-life (EOL) or related prognostic information. Although both approaches offer insights that could be complementary and potentially fused, most existing fusion methods either combine the outputs from multiple methods or adopt a data-driven method to assist the model-based method. To further enhance the prognostic performance, this study proposes a fusion-based prognostic approach in which the output of one method is actively used to update the model of the other through either the crossover operator or the likelihood function. The proposed approach is validated using both an aluminum fatigue dataset and the Prognostics and Health Management (PHM) 2010 cutter wear dataset, demonstrating improved prognostic accuracy compared to either method used independently.

The real-time compliance of a family of movinghorizon simultaneous input-and-state estimation (MH-SISE) problems is assessed. Given the strict real-time requirements of practical applications, the use of numerical optimization techniques for estimation has been limited for systems with fast dynamics. Thus, we explore the use of solver code generation to improve solution times. By generating high-speed solver code tailored to this specific class of moving-horizon problems, substantial improvements in solution times compared to conventional solvers are observed. The proposed methods are evaluated through simulated benchmark tests on lumped-element models of rotating shafts to assess the real-time compliance with respect to measurement sampling rate, model size, and estimation horizon length.

We introduce PerfCam, an open source Proof-of-Concept (PoC) digital twinning framework that combines camera and sensory data with 3D Gaussian Splatting and computer vision models for digital twinning, object tracking, and Key Performance Indicators (KPIs) extraction in industrial production lines. By utilizing 3D reconstruction and Convolutional Neural Networks (CNNs), PerfCam offers a semi-automated approach to object tracking and spatial mapping, enabling highly accurate digital twins that capture real-time KPIs such as availability, performance, Overall Equipment Effectiveness (OEE), and rate of conveyor belts in the production line. We validate the effectiveness of PerfCam through a practical deployment within realistic test production lines in the pharmaceutical industry and contribute an openly published dataset to support further research and development in the field. The results demonstrate PerfCam’s ability to deliver actionable insights through its precise digital twin capabilities, underscoring its value as an effective tool for developing usable digital twins in smart manufacturing environments and extracting operational analytics.

We develop a class of mixed-integer formulations for disjunctive constraints intermediate to the big-M and convex hull formulations in terms of relaxation strength. The main idea is to capture the best of both the big-M and convex hull formulations: a computationally light formulation with a tight relaxation. The “P-split” formulations are based on a lifted transformation that splits convex additively separable constraints into P partitions and forms the convex hull of the linearized and partitioned disjunction. The “P-split” formulations are derived for disjunctive constraints with convex constraints within each disjunct, and we generalize the results for the case with nonconvex constraints within the disjuncts. We analyze the continuous relaxation of the P-split formulations and show that, under certain assumptions, the formulations form a hierarchy starting from a big-M equivalent and converging to the convex hull. We computationally compare the P-split formulations against big-M and convex hull formulations on 344 test instances. The test problems include K-means clustering, semi-supervised clustering, P_ball problems, and optimization over trained ReLU neural networks. The computational results show promising potential of the P-split formulations. For many of the test problems, P-split formulations are solved with a similar number of explored nodes as the convex hull formulation, while reducing the solution time by an order of magnitude and outperforming big-M both in time and number of explored nodes.

In the present article we propose a mixed-integer approximation of adjustable-robust optimization problems, that have both, continuous and discrete variables on the lowest level. As these trilevel problems are notoriously hard to solve, we restrict ourselves to weakly-connected instances. Our approach allows us to approximate, and in some cases exactly represent, the trilevel problem as a single-level mixed-integer problem. This allows us to leverage the computational efficiency of state-of-the-art mixed-integer programming solvers. We demonstrate the value of this approach by applying it to the optimization of power systems, particularly to the control of smart converters.

The presented work addresses two-stage stochastic programs (2SPs), a broadly applicable model to capture optimization problems subject to uncertain parameters with adjustable decision variables. In case the adjustable or second-stage variables contain discrete decisions, the corresponding 2SPs are known to be NP-complete. The standard approach of forming a single-stage deterministic equivalent problem can be computationally challenging even for small instances, as the number of variables and constraints scales with the number of scenarios. To avoid forming a potentially huge MILP problem, we build upon an approach of approximating the expected value of the second-stage problem by a neural network (NN) and encoding the resulting NN into the first-stage problem. The proposed algorithm alternates between optimizing the first-stage variables and retraining the NN. We demonstrate the value of our approach with the example of computing operating points in power systems by showing that the alternating approach provides improved first-stage decisions and a tighter approximation between the expected objective and its neural network approximation.