The following notations are used throughout this paper. We demonstrate the application of Algo. 1 using Examples 1 (Figure 1b), 2 (Figure 1a), and 3 (Figure 1b).

In offline multi-objective optimization (MOO), we leverage an offline dataset of designs and their associated labels to simultaneously minimize multiple objectives. This setting more closely mirrors complex real-world problems compared to single-objective optimization. Recent works mainly employ evolutionary algorithms and Bayesian optimization, with limited attention given to the generative modeling capabilities inherent in such data. In this study, we explore generative modeling in offline MOO through flow matching, noted for its effectiveness and efficiency. We introduce ParetoFlow, specifically designed to guide flow sampling to approximate the Pareto front. Traditional predictor (classifier) guidance is inadequate for this purpose because it models only a single objective. In response, we propose a multi-objective predictor guidance module that assigns each sample a weight vector, representing a weighted distribution across multiple objective predictions. A local filtering scheme is introduced to address non-convex Pareto fronts. These weights uniformly cover the entire objective space, effectively directing sample generation towards the Pareto front. Since distributions with similar weights tend to generate similar samples, we introduce a neighboring evolution module to foster knowledge sharing among neighboring distributions. This module generates offspring from these distributions, and selects the most promising one for the next iteration. Our method achieves state-of-the-art performance across various tasks.

Studying the set of exact solutions of a system of polynomial equations largely depends on a single iterative algorithm, known as Buchberger's algorithm. Optimized versions of this algorithm are crucial for many computer algebra systems (e.g., Mathematica, Maple, Sage). We introduce a new approach to Buchberger's algorithm that uses reinforcement learning agents to perform S-pair selection, a key step in the algorithm. We then study how the difficulty of the problem depends on the choices of domain and distribution of polynomials, about which little is known. Finally, we train a policy model using proximal policy optimization (PPO) to learn S-pair selection strategies for random systems of binomial equations. In certain domains, the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of algorithms in symbolic computation.

Wilson et al. propose a measure of flexibility in project scheduling problems and propose several ways of distributing flexibility over tasks without overrunning the deadline. These schedules prove quite robust: delays of some tasks do not necessarily lead to delays of subsequent tasks. The number of tasks that finish late depends, among others, on the way of distributing flexibility. In this paper I study the different flexibility distributions proposed by Wilson et al. and the differences in number of violations (tasks that finish too late). I show one factor in the instances that causes differences in the number of violations, as well as two properties of the flexibility distribution that cause them to behave differently. Based on these findings, I propose three new flexibility distributions. Depending on the nature of the delays, these new flexibility distributions perform as good as or better than the distributions by Wilson et al.