Search

Mixed Integer Linear Programming (MILP) is a pillar of mathematical optimization that offers a powerful modeling language for a wide range of applications. During the past decades, enormous algorithmic progress has been made in solving MILPs, and many commercial and academic software packages exist. Nevertheless, the availability of data, both from problem instances and from solvers, and the desire to solve new problems and larger (real-life) instances, trigger the need for continuing algorithmic development. MILP solvers use branch and bound as their main component. In recent years, there has been an explosive development in the use of machine learning algorithms for enhancing all main tasks involved in the branch-and-bound algorithm, such as primal heuristics, branching, cutting planes, node selection and solver configuration decisions. This paper presents a survey of such approaches, addressing the vision of integration of machine learning and mathematical optimization as complementary technologies, and how this integration can benefit MILP solving. In particular, we give detailed attention to machine learning algorithms that automatically optimize some metric of branch-and-bound efficiency. We also address how to represent MILPs in the context of applying learning algorithms, MILP benchmarks and software.

Robust Optimal Control (ROC) with adjustable uncertainties has proven to be effective in addressing critical challenges within modern energy networks, especially the reserve and provision problem. However, prior research on ROC with adjustable uncertainties has predominantly focused on the scenario of uncertainties modeled as continuous variables. In this paper, we explore ROC with binary adjustable uncertainties, where the uncertainties are modeled by binary decision variables, marking the first investigation of its kind. To tackle this new challenge, firstly we introduce a metric designed to quantitatively measure the extent of binary adjustable uncertainties. Then, to balance computational tractability and adaptability, we restrict control policies to be affine functions with respect to uncertainties, and propose a general design framework for ROC with binary adjustable uncertainties. To address the inherent computational demands of the original ROC problem, especially in large-scale applications, we employ strong duality (SD) and big-M-based reformulations to create a scalable and computationally efficient Mixed-Integer Linear Programming (MILP) formulation. Numerical simulations are conducted to showcase the performance of our proposed approach, demonstrating its applicability and effectiveness in handling binary adjustable uncertainties within the context of modern energy networks., Comment: 7 pages, submitted to the 22nd European Control Conference (ECC) 2024

Towards integrating renewable electricity generation sources into the grid, an important facilitator is the energy flexibility provided by buildings' thermal inertia. Most of the existing research follows a single-step price- or incentive-based scheme for unlocking the flexibility potential of buildings. In contrast, this paper proposes a novel two-step design approach for better harnessing buildings' energy flexibility. In a first step, a robust optimization model is formulated for assessing the energy flexibility of buildings in the presence of uncertain predictions of external conditions, such as ambient temperature, solar irradiation, etc. In a second step, energy flexibility is activated in response to a feasible demand response (DR) request from grid operators without violating indoor temperature constraints, even in the presence of uncertain external conditions. The proposed approach is tested on a high-fidelity Modelica simulator to evaluate its effectiveness. Simulation results show that, compared with price-based demand-side management, the proposed approach achieves greater energy reduction during peak hours., Comment: 8 pages, accepted to the 62nd IEEE Conference on Decision and Control (CDC 2023)

ReLU neural networks have been modelled as constraints in mixed integer linear programming (MILP), enabling surrogate-based optimisation in various domains and efficient solution of machine learning certification problems. However, previous works are mostly limited to MLPs. Graph neural networks (GNNs) can learn from non-euclidean data structures such as molecular structures efficiently and are thus highly relevant to computer-aided molecular design (CAMD). We propose a bilinear formulation for ReLU Graph Convolutional Neural Networks and a MILP formulation for ReLU GraphSAGE models. These formulations enable solving optimisation problems with trained GNNs embedded to global optimality. We apply our optimization approach to an illustrative CAMD case study where the formulations of the trained GNNs are used to design molecules with optimal boiling points.

Optimization models used to make discrete decisions often contain uncertain parameters that are context-dependent and estimated through prediction. To account for the quality of the decision made based on the prediction, decision-focused learning (end-to-end predict-then-optimize) aims at training the predictive model to minimize regret, i.e., the loss incurred by making a suboptimal decision. Despite the challenge of the gradient of this loss w.r.t. the predictive model parameters being zero almost everywhere for optimization problems with a linear objective, effective gradient-based learning approaches have been proposed to minimize the expected loss, using the empirical loss as a surrogate. However, empirical regret can be an ineffective surrogate because empirical optimal decisions can vary substantially from expected optimal decisions. To understand the impact of this deficiency, we evaluate the effect of aleatoric and epistemic uncertainty on the accuracy of empirical regret as a surrogate. Next, we propose three novel loss functions that approximate expected regret more robustly. Experimental results show that training two state-of-the-art decision-focused learning approaches using robust regret losses improves test-sample empirical regret in general while keeping computational time equivalent relative to the number of training epochs., Comment: 10 pages, 3 figures

Agent-based modelling constitutes a versatile approach to representing and simulating complex systems. Studying large-scale systems is challenging because of the computational time required for the simulation runs: scaling is at least linear in system size (number of agents). Given the inherently modular nature of MABSs, parallel computing is a natural approach to overcoming this challenge. However, because of the shared information and communication between agents, parellelization is not simple. We present a protocol for shared-memory, parallel execution of MABSs. This approach is useful for models that can be formulated in terms of sequential computations, and that involve updates that are localized, in the sense of involving small numbers of agents. The protocol has a bottom-up and asynchronous nature, allowing it to deal with heterogeneous computation in an adaptive, yet graceful manner. We illustrate the potential performance gains on exemplar cultural dynamics and disease spreading MABSs., Comment: 12 pages, 3 figures; work presented at the 24th International Workshop on Multi-Agent-Based Simulation

Training neural networks (NNs) using combinatorial optimization solvers has gained attention in recent years. In low-data settings, state-of-the-art mixed integer linear programming solvers can train exactly a NN, avoiding intensive GPU-based training and hyper-parameter tuning and simultaneously training and sparsifying the network. We study the case of few-bit discrete-valued neural networks, both Binarized Neural Networks (BNNs), whose values are restricted to +-1, and Integer Neural Networks (INNs), whose values lie in a range {-P, ..., P}. Few-bit NNs receive increasing recognition due to their lightweight architecture and ability to run on low-power devices. This paper proposes new methods to improve the training of BNNs and INNs. Our contribution is a multi-objective ensemble approach based on training a single NN for each possible pair of classes and applying a majority voting scheme to predict the final output. Our approach results in training robust sparsified networks whose output is not affected by small perturbations on the input and whose number of active weights is as small as possible. We compare this BeMi approach to the current state-of-the-art in solver-based NN training and gradient-based training, focusing on BNN learning in few-shot contexts. We compare the benefits and drawbacks of INNs versus BNNs, bringing new light to the distribution of weights over the {-P, ..., P} interval. Finally, we compare multi-objective versus single-objective training of INNs, showing that robustness and network simplicity can be acquired simultaneously, thus obtaining better test performances. While the previous state-of-the-art approaches achieve an average accuracy of 51.1% on the MNIST dataset, the BeMi ensemble approach achieves an average accuracy of 68.4% when trained with 10 images per class and 81.8% when trained with 40 images per class, having up to 75.3% NN links removed., Comment: 22 pages, 6 figure, 5 tables

State-of-the-art Mixed Integer Linear Program (MILP) solvers combine systematic tree search with a plethora of hard-coded heuristics, such as the branching rule. The idea of learning branching rules from data has received increasing attention recently, and promising results have been obtained by learning fast approximations of the strong branching expert. In this work, we instead propose to learn branching rules from scratch via Reinforcement Learning (RL). We revisit the work of Etheve et al. (2020) and propose tree Markov Decision Processes, or tree MDPs, a generalization of temporal MDPs that provides a more suitable framework for learning to branch. We derive a tree policy gradient theorem, which exhibits a better credit assignment compared to its temporal counterpart. We demonstrate through computational experiments that tree MDPs improve the learning convergence, and offer a promising framework for tackling the learning-to-branch problem in MILPs., Comment: 10 pages, 2 figures, plus supplementary material

It is increasingly common to solve combinatorial optimisation problems that are partially-specified. We survey the case where the objective function or the relations between variables are not known or are only partially specified. The challenge is to learn them from available data, while taking into account a set of hard constraints that a solution must satisfy, and that solving the optimisation problem (esp. during learning) is computationally very demanding. This paper overviews four seemingly unrelated approaches, that can each be viewed as learning the objective function of a hard combinatorial optimisation problem: 1) surrogate-based optimisation, 2) empirical model learning, 3) decision-focused learning (`predict + optimise'), and 4) structured-output prediction. We formalise each learning paradigm, at first in the ways commonly found in the literature, and then bring the formalisations together in a compatible way using regret. We discuss the differences and interactions between these frameworks, highlight the opportunities for cross-fertilization and survey open directions.

Recent work has shown potential in using Mixed Integer Programming (MIP) solvers to optimize certain aspects of neural networks (NNs). However the intriguing approach of training NNs with MIP solvers is under-explored. State-of-the-art-methods to train NNs are typically gradient-based and require significant data, computation on GPUs, and extensive hyper-parameter tuning. In contrast, training with MIP solvers does not require GPUs or heavy hyper-parameter tuning, but currently cannot handle anything but small amounts of data. This article builds on recent advances that train binarized NNs using MIP solvers. We go beyond current work by formulating new MIP models which improve training efficiency and which can train the important class of integer-valued neural networks (INNs). We provide two novel methods to further the potential significance of using MIP to train NNs. The first method optimizes the number of neurons in the NN while training. This reduces the need for deciding on network architecture before training. The second method addresses the amount of training data which MIP can feasibly handle: we provide a batch training method that dramatically increases the amount of data that MIP solvers can use to train. We thus provide a promising step towards using much more data than before when training NNs using MIP models. Experimental results on two real-world data-limited datasets demonstrate that our approach strongly outperforms the previous state of the art in training NN with MIP, in terms of accuracy, training time and amount of data. Our methodology is proficient at training NNs when minimal training data is available, and at training with minimal memory requirements -- which is potentially valuable for deploying to low-memory devices.

In line with the growing trend of using machine learning to help solve combinatorial optimisation problems, one promising idea is to improve node selection within a mixed integer programming (MIP) branch-and-bound tree by using a learned policy. Previous work using imitation learning indicates the feasibility of acquiring a node selection policy, by learning an adaptive node searching order. In contrast, our imitation learning policy is focused solely on learning which of a node's children to select. We present an offline method to learn such a policy in two settings: one that comprises a heuristic by committing to pruning of nodes; one that is exact and backtracks from a leaf to guarantee finding the optimal integer solution. The former setting corresponds to a child selector during plunging, while the latter is akin to a diving heuristic. We apply the policy within the popular open-source solver SCIP, in both heuristic and exact settings. Empirical results on five MIP datasets indicate that our node selection policy leads to solutions significantly more quickly than the state-of-the-art precedent in the literature. While we do not beat the highly-optimised SCIP state-of-practice baseline node selector in terms of solving time on exact solutions, our heuristic policies have a consistently better optimality gap than all baselines, if the accuracy of the predictive model is sufficient. Further, the results also indicate that, when a time limit is applied, our heuristic method finds better solutions than all baselines in the majority of problems tested. We explain the results by showing that the learned policies have imitated the SCIP baseline, but without the latter's early plunge abort. Our recommendation is that, despite the clear improvements over the literature, this kind of MIP child selector is better seen in a broader approach using learning in MIP branch-and-bound tree decisions., Comment: Authors' version, not publisher's final version which is available at DOI

A computational system is called autonomous if it is able to make its own decisions, or take its own actions, without human supervision or control. The capability and spread of such systems have reached the point where they are beginning to touch much of everyday life. However, regulators grapple with how to deal with autonomous systems, for example how could we certify an Unmanned Aerial System for autonomous use in civilian airspace? We here analyse what is needed in order to provide verified reliable behaviour of an autonomous system, analyse what can be done as the state-of-the-art in automated verification, and propose a roadmap towards developing regulatory guidelines, including articulating challenges to researchers, to engineers, and to regulators. Case studies in seven distinct domains illustrate the article., Comment: 66 pages

The Order Acceptance and Scheduling (OAS) problem describes a class of real-world problems such as in smart manufacturing and satellite scheduling. This problem consists of simultaneously selecting a subset of orders to be processed as well as determining the associated schedule. A common generalization includes sequence-dependent setup times and time windows. A novel memetic algorithm for this problem, called Sparrow, comprises a hybridization of biased random key genetic algorithm (BRKGA) and adaptive large neighbourhood search (ALNS). Sparrow integrates the exploration ability of BRKGA and the exploitation ability of ALNS. On a set of standard benchmark instances, this algorithm obtains better-quality solutions with runtimes comparable to state-of-the-art algorithms. To further understand the strengths and weaknesses of these algorithms, their performance is also compared on a set of new benchmark instances with more realistic properties. We conclude that Sparrow is distinguished by its ability to solve difficult instances from the OAS literature, and that the hybrid steady-state genetic algorithm (HSSGA) performs well on large instances in terms of optimality gap, although taking more time than Sparrow.

Constraint Programming (CP) has proved an effective paradigm to model and solve difficult combinatorial satisfaction and optimisation problems from disparate domains. Many such problems arising from the commercial world are permeated by data uncertainty. Existing CP approaches that accommodate uncertainty are less suited to uncertainty arising due to incomplete and erroneous data, because they do not build reliable models and solutions guaranteed to address the user's genuine problem as she perceives it. Other fields such as reliable computation offer combinations of models and associated methods to handle these types of uncertain data, but lack an expressive framework characterising the resolution methodology independently of the model. We present a unifying framework that extends the CP formalism in both model and solutions, to tackle ill-defined combinatorial problems with incomplete or erroneous data. The certainty closure framework brings together modelling and solving methodologies from different fields into the CP paradigm to provide reliable and efficient approches for uncertain constraint problems. We demonstrate the applicability of the framework on a case study in network diagnosis. We define resolution forms that give generic templates, and their associated operational semantics, to derive practical solution methods for reliable solutions., Comment: Revised version