We can compress a rectifier network while exactly preserving its underlying functionality with respect to a given input domain if some of its neurons are stable.

Bundle pricing refers to designing several product combinations (i.e., bundles) and determining their prices in order to maximize the expected profit. It is a classic problem in revenue management and arises in many industries, such as e-commerce, tourism, and video games. However, the problem is typically intractable due to the exponential number of candidate bundles. In this paper, we explore the usage of graph convolutional networks (GCNs) in solving the bundle pricing problem. Specifically, we first develop a graph representation of the mixed bundling model (where every possible bundle is assigned with a specific price) and then train a GCN to learn the latent patterns of optimal bundles. Based on the trained GCN, we propose two inference strategies to derive high-quality feasible solutions. A local-search technique is further proposed to improve the solution quality. Numerical experiments validate the effectiveness and efficiency of our proposed GCN-based framework. Using a GCN trained on instances with 5 products, our methods consistently achieve near-optimal solutions (better than 97%) with only a fraction of computational time for problems of small to medium size. It also achieves superior solutions for larger size of problems compared with other heuristic methods such as bundle size pricing (BSP). The method can also provide high quality solutions for instances with more than 30 products even for the challenging cases where product utilities are non-additive.

Counterfactual explanation (CE) is a widely used post-hoc method that provides individuals with actionable changes to alter an unfavorable prediction from a machine learning model. Plausible CE methods improve realism by considering data distribution characteristics, but their optimization models introduce a large number of constraints, leading to high computational cost. In this work, we revisit the DACE framework and propose a refined Mixed-Integer Linear Programming (MILP) formulation that significantly reduces the number of constraints in the local outlier factor (LOF) objective component. We also apply the method to a linear SVM classifier with standard scaler. The experimental results show that our approach achieves faster solving times while maintaining explanation quality. These results demonstrate the promise of more efficient LOF modeling in counterfactual explanation and data science applications.

Integer programming lies at the heart of crucial combinatorial optimization tasks but remains challenging due to its NP-hard nature. An effective approach for practically solving integer programs is the manual design of acceleration cuts, i.e. inequalities that improve solver performance. However, this creative process demands deep expertise and is yet to be automated. Our proposed framework, EvoCut, automates the generation of acceleration cuts by combining large language models (LLMs) with an evolutionary search. EvoCut (i) initializes a diverse population of candidate cuts via an LLM-based initializer agent; (ii) for each cut empirically evaluates both preservation of the optimal solution and its ability to cut off fractional solutions across a verification set; and (iii) iteratively refines the population through evolutionary crossover and mutation agents. We quantify each cut's utility by its relative reduction in the solver's optimality gap. Our comparisons against standard integer programming practice show that EvoCut reduces optimality gap by 17-57% within a fixed time. It obtains the same solutions up to 4 times as fast, and obtains higher-quality solutions within the same time limit. Requiring no human expert input, EvoCut reliably generates, improves, and empirically verifies cuts that generalize to unseen instances. The code is available at https://github.com/milad1378yz/EvoCut.

If none is provided by the callback, the MILP solver accepts the solution as feasible. In our case, we use a lazy constraint callback for a slightly different purpose. For ease of explanation, they are in reverse order of appearance. This operation is performed in line 25. The case in which an entire layer is stably inactive is considered separately.

Recently, a novel two-phase framework named mol-infer for inference of chemical compounds with prescribed abstract structures and desired property values has been proposed. The framework mol-infer is primarily based on using mixed integer linear programming (MILP) to simulate the computational process of machine learning methods and describe the necessary and sufficient conditions to ensure such a chemical graph exists. The existing approaches usually first convert the chemical compounds into handcrafted feature vectors to construct prediction functions, but because of the limit on the kinds of descriptors originated from the need for tractability in the MILP formulation, the learning performances on datasets of some properties are not good enough. A lack of good learning performance can greatly lower the quality of the inferred chemical graphs, and thus improving learning performance is of great importance. On the other hand, graph neural networks (GNN) offer a promising machine learning method to directly utilize the chemical graphs as the input, and many existing GNN-based approaches to the molecular property prediction problem have shown that they can enjoy better learning performances compared to the traditional approaches that are based on feature vectors. In this study, we develop a molecular inference framework based on mol-infer, namely mol-infer-GNN, that utilizes GNN as the learning method while keeping the great flexibility originated from the two-layered model on the abstract structure of the chemical graph to be inferred. We conducted computational experiments on the QM9 dataset to show that our proposed GNN model can obtain satisfying learning performances for some properties despite its simple structure, and can infer small chemical graphs comprising up to 20 non-hydrogen atoms within reasonable computational time.

This work presents novel algorithms for learning Bayesian networks of bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed integer linear programming formulations for structure learning and treewidth computation. The approximate method consists in sampling k-trees (maximal graphs of treewidth k), and subsequently selecting, exactly or approximately, the best structure whose moral graph is a subgraph of that k-tree. The approaches are empirically compared to each other and to state-of-the-art methods on a collection of public data sets with up to 100 variables.

Mission planning often involves optimising the use of ISR (Intelligence, Surveillance and Reconnaissance) assets in order to achieve a set of mission objectives within allowed parameters subject to constraints. The missions of interest here, involve routing multiple UAVs visiting multiple targets, utilising sensors to capture data relating to each target. Finding such solutions is often an NP-Hard problem and cannot be solved efficiently on classical computers. Furthermore, during the mission new constraints and objectives may arise, requiring a new solution to be computed within a short time period. To achieve this we investigate near term quantum algorithms that have the potential to offer speed-ups against current classical methods. We demonstrate how a large family of these problems can be formulated as a Mixed Integer Linear Program (MILP) and then converted to a Quadratic Unconstrained Binary Optimisation (QUBO). The formulation provided is versatile and can be adapted for many different constraints with clear qubit scaling provided. We discuss the results of solving the QUBO formulation using commercial quantum annealers and compare the solutions to current edge classical solvers. We also analyse the results from solving the QUBO using Quantum Approximate Optimisation Algorithms (QAOA) and discuss their results. Finally, we also provide efficient methods to encode to the problem into the Variational Quantum Eigensolver (VQE) formalism, where we have tailored the ansatz to the problem making efficient use of the qubits available.

This work presents novel algorithms for learning Bayesian networks of bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed integer linear programming formulations for structure learning and treewidth computation. The approximate method consists in sampling k-trees (maximal graphs of treewidth k), and subsequently selecting, exactly or approximately, the best structure whose moral graph is a subgraph of that k-tree. The approaches are empirically compared to each other and to state-of-the-art methods on a collection of public data sets with up to 100 variables.

Constructing first-principles models is usually a challenging and time-consuming task due to the complexity of the real-life processes. On the other hand, data-driven modeling, and in particular neural network models often suffer from issues such as overfitting and lack of useful and highquality data. At the same time, embedding trained machine learning models directly into the optimization problems has become an effective and state-of-the-art approach for surrogate optimization, whose performance can be improved by physics-informed training. In this study, it is proposed to upgrade piece-wise linear neural network models with physics informed knowledge for optimization problems with neural network models embedded. In addition to using widely accepted and naturally piece-wise linear rectified linear unit (ReLU) activation functions, this study also suggests piece-wise linear approximations for the hyperbolic tangent activation function to widen the domain. Optimization of three case studies, a blending process, an industrial distillation column and a crude oil column are investigated. For all cases, physics-informed trained neural network based optimal results are closer to global optimality. Finally, associated CPU times for the optimization problems are much shorter than the standard optimization results.