Optimization equips engineers and scientists in a variety of fields with the ability to transcribe their problems into a generic formulation and receive optimal solutions with relative ease. Industries ranging from aerospace to robotics continue to benefit from advancements in optimization theory and the associated algorithmic developments. Nowadays, optimization is used in real time on autonomous systems acting in safety critical situations, such as self-driving vehicles. It has become increasingly more important to produce robust solutions by incorporating uncertainty into optimization programs. This paper provides a short survey about the state of the art in optimization under uncertainty. The paper begins with a brief overview of the main classes of optimization without uncertainty. The rest of the paper focuses on the different methods for handling both aleatoric and epistemic uncertainty. Many of the applications discussed in this paper are within the domain of control. The goal of this survey paper is to briefly touch upon the state of the art in a variety of different methods and refer the reader to other literature for more in-depth treatments of the topics discussed here.

The Traffic Alert Collision Avoidance System (TCAS) has been an integral part of the increased safety of air transport since it was federally mandated in the 1991 for all passenger carrying aircraft with more than 30 seats flying in U.S. airspace [1, 2]. TCAS led to a dramatic reduction in the occurrence of mid air collisions in modern aviation; however the heuristic based approach undertaken in TCAS has made it difficult to adapt the system to the evolving complexity of the National Airspace System (NAS), which includes new cooperative surveillance systems (e.g., ADS-B) and new vehicle entrants. In response, the Federal Aviation Administration (FAA) commissioned the development of a replacement for TCAS. This new system, referred to as the Next Generation Airborne Collision Avoidance System X (ACAS X), which is currently in development at MIT Lincoln Laboratory and John Hopkins Applied Physics Laboratory, is expected to integrate into multiple aircraft platforms and reduce nuisance alerts as well as reduce the risk of Near Mid Air Collisions (NMAC) [3]. ACAS X introduced several variants designed to reduce the risk of NMAC for a particular operation, such as commercial aviation (ACAS Xa) [4], large uncrewed aerial systems (ACAS Xu) [5], smaller uncrewed aerial vehicles (ACAS sXu) [6], and ACAS Xr which is under development for advanced air mobility and helicopter operations. Each variant adds capabilities and design considerations for the operational environment and platforms that will be commonly seen by the ACAS X equipped vehicle. For example, ACAS sXu introduced vehicle to vehicle surveillance to accommodate a future link that sUAS may use to interrogate and coordinate with each other. While, ACAS Xu added Remain Well Clear alerting due to its use in remotely piloted or autonomous UAS.

We consider the nonlinear inverse problem of learning a transition operator $\mathbf{A}$ from partial observations at different times, in particular from sparse observations of entries of its powers $\mathbf{A},\mathbf{A}^2,\cdots,\mathbf{A}^{T}$. This Spatio-Temporal Transition Operator Recovery problem is motivated by the recent interest in learning time-varying graph signals that are driven by graph operators depending on the underlying graph topology. We address the nonlinearity of the problem by embedding it into a higher-dimensional space of suitable block-Hankel matrices, where it becomes a low-rank matrix completion problem, even if $\mathbf{A}$ is of full rank. For both a uniform and an adaptive random space-time sampling model, we quantify the recoverability of the transition operator via suitable measures of incoherence of these block-Hankel embedding matrices. For graph transition operators these measures of incoherence depend on the interplay between the dynamics and the graph topology. We develop a suitable non-convex iterative reweighted least squares (IRLS) algorithm, establish its quadratic local convergence, and show that, in optimal scenarios, no more than $\mathcal{O}(rn \log(nT))$ space-time samples are sufficient to ensure accurate recovery of a rank-$r$ operator $\mathbf{A}$ of size $n \times n$. This establishes that spatial samples can be substituted by a comparable number of space-time samples. We provide an efficient implementation of the proposed IRLS algorithm with space complexity of order $O(r n T)$ and per-iteration time complexity linear in $n$. Numerical experiments for transition operators based on several graph models confirm that the theoretical findings accurately track empirical phase transitions, and illustrate the applicability and scalability of the proposed algorithm.

The performance of individual evolutionary optimization algorithms is mostly measured in terms of statistics such as mean, median and standard deviation etc., computed over the best solutions obtained with few trails of the algorithm. To compare the performance of two algorithms, the values of these statistics are compared instead of comparing the solutions directly. This kind of comparison lacks direct comparison of solutions obtained with different algorithms. For instance, the comparison of best solutions (or worst solution) of two algorithms simply not possible. Moreover, ranking of algorithms is mostly done in terms of solution quality only, despite the fact that the convergence of algorithm is also an important factor. In this paper, a direct comparison approach is proposed to analyze the performance of evolutionary optimization algorithms. A direct comparison matrix called \emph{Prasatul Matrix} is prepared, which accounts direct comparison outcome of best solutions obtained with two algorithms for a specific number of trials. Five different performance measures are designed based on the prasatul matrix to evaluate the performance of algorithms in terms of Optimality and Comparability of solutions. These scores are utilized to develop a score-driven approach for comparing performance of multiple algorithms as well as for ranking both in the grounds of solution quality and convergence analysis. Proposed approach is analyzed with six evolutionary optimization algorithms on 25 benchmark functions. A non-parametric statistical analysis, namely Wilcoxon paired sum-rank test is also performed to verify the outcomes of proposed direct comparison approach.

Control of light through high numerical aperture (N.A.) objectives is a common requirement in microscopy, for example for engineering specific point spread functions for super-resolution imaging [1, 2], for generating target light distributions for optical stimulation [3, 4, 5], for optical tweezers [6, 7], or for aberration corrections in adaptive optics [8, 9, 10]. For controlling light in all these situations, computational modeling is the most versatile approach for finding a phase pattern that, when displayed on a spatial light modulator (SLM), results in the desired target light distribution in the focal volume. Light propagation through a microscope objective with high N.A. can accurately be described with the vectorial Debye-Wolf diffraction integral [11]. The Debye-Wolf integral takes into account the orientation of the electromagnectic field vector (polarization) which contributes to the shape of the focus in high N.A. objectives. Such effects can be exploited for high resolution imaging, for example with diffraction limited objects, such as single molecules or nanostructures [1, 12, 2]. However, inversion of the Debye-Wolf integral does not have a general closed-form solution [13], and one therefore typically resorts to numerical approaches for applications that aim to generate an intended target light distribution. A fast method for calculating the Debye-Wolf integral would therefore be useful across a range of applications, for example vectorial imaging [14], vectorial beam shaping for tight focusing [2] or superresolution computational imaging [15], as well as any light shaping or imaging applications also at lower resolution, that is where polarization effects have less of an impact.

In today's uncertain and competitive market, where enterprises are subjected to increasingly shortened product life-cycles and frequent volume changes, reconfigurable manufacturing systems (RMS) applications play a significant role in the manufacturing industry's success. Despite the advantages offered by RMS, achieving a high-efficiency degree constitutes a challenging task for stakeholders and decision-makers when they face the trade-off decisions inherent in these complex systems. This study addresses work tasks and resource allocations to workstations together with buffer capacity allocation in RMS. The aim is to simultaneously maximize throughput and minimize total buffer capacity under fluctuating production volumes and capacity changes while considering the stochastic behavior of the system. An enhanced simulation-based multi-objective optimization (SMO) approach with customized simulation and optimization components is proposed to address the abovementioned challenges. Apart from presenting the optimal solutions subject to volume and capacity changes, the proposed approach support decision-makers with discovered knowledge to further understand the RMS design. In particular, this study presents a problem-specific customized SMO combined with a novel flexible pattern mining method for optimizing RMS and conducting post-optimal analyzes. To this extent, this study demonstrates the benefits of applying SMO and knowledge discovery methods for fast decision-support and production planning of RMS.

This paper proposes embedded Gaussian Process Barrier States (GP-BaS), a methodology to safely control unmodeled dynamics of nonlinear system using Bayesian learning. Gaussian Processes (GPs) are used to model the dynamics of the safety-critical system, which is subsequently used in the GP-BaS model. We derive the barrier state dynamics utilizing the GP posterior, which is used to construct a safety embedded Gaussian process dynamical model (GPDM). We show that the safety-critical system can be controlled to remain inside the safe region as long as we can design a controller that renders the BaS-GPDM's trajectories bounded (or asymptotically stable). The proposed approach overcomes various limitations in early attempts at combining GPs with barrier functions due to the abstention of restrictive assumptions such as linearity of the system with respect to control, relative degree of the constraints and number or nature of constraints. This work is implemented on various examples for trajectory optimization and control including optimal stabilization of unstable linear system and safe trajectory optimization of a Dubins vehicle navigating through an obstacle course and on a quadrotor in an obstacle avoidance task using GP differentiable dynamic programming (GP-DDP). The proposed framework is capable of maintaining safe optimization and control of unmodeled dynamics and is purely data driven.

Autonomous robotic systems are of particular interest for many fields, especially those that can be dangerous for human intervention like search and rescue, and maintenance on rigs. However, motion planning in unstructured environment is still a hard problem for legged robots and their success depends largely on their ability to plan their paths robustly. Moreover, the method in which a controller deals with obstacles has great consequences on the planned trajectory, and these optimizations are quintessential in generating agile motions for real-world robots. Trajectory optimization is a common practice for generating motion for legged systems [1, 2, 3], since it can produce optimal trajectories which satisfy the physical and environmental constraints of the robot. However, the solution from trajectory optimization is only valid for a particular pair of initial and target positions, and one needs to re-plan if the pair changes.

Marketing is an important mechanism to increase user engagement and improve platform revenue, and heterogeneous causal learning can help develop more effective strategies. Most decision-making problems in marketing can be formulated as resource allocation problems and have been studied for decades. Existing works usually divide the solution procedure into two fully decoupled stages, i.e., machine learning (ML) and operation research (OR) -- the first stage predicts the model parameters and they are fed to the optimization in the second stage. However, the error of the predicted parameters in ML cannot be respected and a series of complex mathematical operations in OR lead to the increased accumulative errors. Essentially, the improved precision on the prediction parameters may not have a positive correlation on the final solution due to the side-effect from the decoupled design. In this paper, we propose a novel approach for solving resource allocation problems to mitigate the side-effects. Our key intuition is that we introduce the decision factor to establish a bridge between ML and OR such that the solution can be directly obtained in OR by only performing the sorting or comparison operations on the decision factor. Furthermore, we design a customized loss function that can conduct direct heterogeneous causal learning on the decision factor, an unbiased estimation of which can be guaranteed when the loss converges. As a case study, we apply our approach to two crucial problems in marketing: the binary treatment assignment problem and the budget allocation problem with multiple treatments. Both large-scale simulations and online A/B Tests demonstrate that our approach achieves significant improvement compared with state-of-the-art.

Applications such as employees sharing office spaces over a workweek can be modeled as problems where agents are matched to resources over multiple rounds. Agents' requirements limit the set of compatible resources and the rounds in which they want to be matched. Viewing such an application as a multi-round matching problem on a bipartite compatibility graph between agents and resources, we show that a solution (i.e., a set of matchings, with one matching per round) can be found efficiently if one exists. To cope with situations where a solution does not exist, we consider two extensions. In the first extension, a benefit function is defined for each agent and the objective is to find a multi-round matching to maximize the total benefit. For a general class of benefit functions satisfying certain properties (including diminishing returns), we show that this multi-round matching problem is efficiently solvable. This class includes utilitarian and Rawlsian welfare functions. For another benefit function, we show that the maximization problem is NP-hard. In the second extension, the objective is to generate advice to each agent (i.e., a subset of requirements to be relaxed) subject to a budget constraint so that the agent can be matched. We show that this budget-constrained advice generation problem is NP-hard. For this problem, we develop an integer linear programming formulation as well as a heuristic based on local search. We experimentally evaluate our algorithms on synthetic networks and apply them to two real-world situations: shared office spaces and matching courses to classrooms.