Constrained optimization problems arise in various engineering system operations such as inventory management and electric power grids. However, the requirement to repeatedly solve such optimization problems with uncertain parameters poses a significant computational challenge. This work introduces a learning scheme using Bayesian Neural Networks (BNNs) to solve constrained optimization problems under limited labeled data and restricted model training times. We propose a semi-supervised BNN for this practical but complex regime, wherein training commences in a sandwiched fashion, alternating between a supervised learning step (using labeled data) for minimizing cost, and an unsupervised learning step (using unlabeled data) for enforcing constraint feasibility. Both supervised and unsupervised steps use a Bayesian approach, where Stochastic Variational Inference is employed for approximate Bayesian inference. We show that the proposed semi-supervised learning method outperforms conventional BNN and deep neural network (DNN) architectures on important non-convex constrained optimization problems from energy network operations, achieving up to a tenfold reduction in expected maximum equality gap and halving the optimality and inequality (feasibility) gaps, without requiring any correction or projection step. By leveraging the BNN's ability to provide posterior samples at minimal computational cost, we demonstrate that a Selection via Posterior (SvP) scheme can further reduce equality gaps by more than 10%. We also provide tight and practically meaningful probabilistic confidence bounds that can be constructed using a low number of labeled testing data and readily adapted to other applications.

This article presents a deep reinforcement learning-based approach to tackle a persistent surveillance mission requiring a single unmanned aerial vehicle initially stationed at a depot with fuel or time-of-flight constraints to repeatedly visit a set of targets with equal priority. Owing to the vehicle's fuel or time-of-flight constraints, the vehicle must be regularly refueled, or its battery must be recharged at the depot. The objective of the problem is to determine an optimal sequence of visits to the targets that minimizes the maximum time elapsed between successive visits to any target while ensuring that the vehicle never runs out of fuel or charge. We present a deep reinforcement learning algorithm to solve this problem and present the results of numerical experiments that corroborate the effectiveness of this approach in comparison with common-sense greedy heuristics.

The Multiple Traveling Salesman Problem (MTSP) with a single depot is a generalization of the well-known Traveling Salesman Problem (TSP) that involves an additional parameter, namely, the number of salesmen. In the MTSP, several salesmen at the depot need to visit a set of interconnected targets, such that each target is visited precisely once by at most one salesman while minimizing the total length of their tours. An equally important variant of the MTSP, the min-max MTSP, aims to distribute the workload (length of the individual tours) among salesmen by requiring the longest tour of all the salesmen to be as short as possible, i.e., minimizing the maximum tour length among all salesmen. The min-max MTSP appears in real-life applications to ensure a good balance of workloads for the salesmen. It is known in the literature that the min-max MTSP is notoriously difficult to solve to optimality due to the poor lower bounds its linear relaxations provide. In this paper, we formulate two novel parametric variants of the MTSP called the "fair-MTSP". One variant is formulated as a Mixed-Integer Second Order Cone Program (MISOCP), and the other as a Mixed Integer Linear Program (MILP). Both focus on enforcing the workloads for the salesmen to be equitable, i.e., the distribution of tour lengths for the salesmen to be fair while minimizing the total cost of their tours. We present algorithms to solve the two variants of the fair-MTSP to global optimality and computational results on benchmark and real-world test instances that make a case for fair-MTSP as a viable alternative to the min-max MTSP.