Fractals are geometric shapes that can display complex and self-similar patterns found in nature (e.g., clouds and plants). Recent works in visual recognition have leveraged this property to create random fractal images for model pre-training. In this paper, we study the inverse problem -- given a target image (not necessarily a fractal), we aim to generate a fractal image that looks like it. We propose a novel approach that learns the parameters underlying a fractal image via gradient descent. We show that our approach can find fractal parameters of high visual quality and be compatible with different loss functions, opening up several potentials, e.g., learning fractals for downstream tasks, scientific understanding, etc.

Navigating mobile robots through environments shared with humans is challenging. From the perspective of the robot, humans are dynamic obstacles that must be avoided. These obstacles make the collision-free space nonconvex, which leads to two distinct passing behaviors per obstacle (passing left or right). For local planners, such as receding-horizon trajectory optimization, each behavior presents a local optimum in which the planner can get stuck. This may result in slow or unsafe motion even when a better plan exists. In this work, we identify trajectories for multiple locally optimal driving behaviors, by considering their topology. This identification is made consistent over successive iterations by propagating the topology information. The most suitable high-level trajectory guides a local optimization-based planner, resulting in fast and safe motion plans. We validate the proposed planner on a mobile robot in simulation and real-world experiments.

The Internet of Things (IoT) system generates massive high-speed temporally correlated streaming data and is often connected with online inference tasks under computational or energy constraints. Online analysis of these streaming time series data often faces a trade-off between statistical efficiency and computational cost. One important approach to balance this trade-off is sampling, where only a small portion of the sample is selected for the model fitting and update. Motivated by the demands of dynamic relationship analysis of IoT system, we study the data-dependent sample selection and online inference problem for a multi-dimensional streaming time series, aiming to provide low-cost real-time analysis of high-speed power grid electricity consumption data. Inspired by D-optimality criterion in design of experiments, we propose a class of online data reduction methods that achieve an optimal sampling criterion and improve the computational efficiency of the online analysis. We show that the optimal solution amounts to a strategy that is a mixture of Bernoulli sampling and leverage score sampling. The leverage score sampling involves auxiliary estimations that have a computational advantage over recursive least squares updates. Theoretical properties of the auxiliary estimations involved are also discussed. When applied to European power grid consumption data, the proposed leverage score based sampling methods outperform the benchmark sampling method in online estimation and prediction. The general applicability of the sampling-assisted online estimation method is assessed via simulation studies.

We address the problem of motion planning for four-way intersection crossings with right-of-ways. Road safety typically assigns liability to the follower in rear-end collisions and to the approaching vehicle required to yield in side crashes. As an alternative to previous models based on heuristic state machines, we propose a planning framework which changes the prediction model of other cars (e.g. their prototypical accelerations and decelerations) depending on the given longitudinal or lateral priority rules. Combined with a state-of-the-art trajectory optimization approach ROPT (Risk Optimization Method) this allows to find ego velocity profiles minimizing risks from curves and all involved vehicles while maximizing utility (needed time to arrive at a goal) and comfort (change and duration of acceleration) under the presence of regulatory conditions. Analytical and statistical evaluations show that our method is able to follow right-of-ways for a wide range of other vehicle behaviors and path geometries. Even when the other cars drive in a non-priority-compliant way, ROPT achieves good risk-comfort tradeoffs.

Identifying internal parameters for planning is crucial to maximizing the performance of a planner. However, automatically tuning internal parameters which are conditioned on the problem instance is especially challenging. A recent line of work focuses on learning planning parameter generators, but lack a consistent problem definition and software framework. This work proposes the unified planner optimization problem (POP) formulation, along with the Open Planner Optimization Framework (OPOF), a highly extensible software framework to specify and to solve these problems in a reusable manner.

While reinforcement learning produces very promising results for many applications, its main disadvantage is the lack of safety guarantees, which prevents its use in safety-critical systems. In this work, we address this issue by a safety shield for nonlinear continuous systems that solve reach-avoid tasks. Our safety shield prevents applying potentially unsafe actions from a reinforcement learning agent by projecting the proposed action to the closest safe action. This approach is called action projection and is implemented via mixed-integer optimization. The safety constraints for action projection are obtained by applying parameterized reachability analysis using polynomial zonotopes, which enables to accurately capture the nonlinear effects of the actions on the system. In contrast to other state-of-the-art approaches for action projection, our safety shield can efficiently handle input constraints and dynamic obstacles, eases incorporation of the spatial robot dimensions into the safety constraints, guarantees robust safety despite process noise and measurement errors, and is well suited for high-dimensional systems, as we demonstrate on several challenging benchmark systems.

Our research deals with the optimization version of the set partition problem, where the objective is to minimize the absolute difference between the sums of the two disjoint partitions. Although this problem is known to be NP-hard and requires exponential time to solve, we propose a less demanding version of this problem where the goal is to find a locally optimal solution. In our approach, we consider the local optimality in respect to any movement of at most two elements. To accomplish this, we developed an algorithm that can generate a locally optimal solution in at most $O(N^2)$ time and $O(N)$ space. Our algorithm can handle arbitrary input precisions and does not require positive or integer inputs. Hence, it can be applied in various problem scenarios with ease.

Operational planning and long-term planning for companies are more complex in recent years. Information changes fast, and the decision making is a hard task. Therefore, optimization algorithms (operations research) are used to find optimal solutions for these problems. Professionals in this field are one of the most valued in the market. The classes use examples that are created step by step, so we will create the algorithms together. Besides this Optimization with Python: Solve Operations Research Problems Course is more focused in mathematical approaches, you will also learn how to solve problems using artificial intelligence (AI), genetic algorithm, and particle swarm.

We study the combinatorial assignment domain, which includes combinatorial auctions and course allocation. The main challenge in this domain is that the bundle space grows exponentially in the number of items. To address this, several papers have recently proposed machine learning-based preference elicitation algorithms that aim to elicit only the most important information from agents. However, the main shortcoming of this prior work is that it does not model a mechanism's uncertainty over values for not yet elicited bundles. In this paper, we address this shortcoming by presenting a Bayesian optimization-based combinatorial assignment (BOCA) mechanism. Our key technical contribution is to integrate a method for capturing model uncertainty into an iterative combinatorial auction mechanism. Concretely, we design a new method for estimating an upper uncertainty bound that can be used to define an acquisition function to determine the next query to the agents. This enables the mechanism to properly explore (and not just exploit) the bundle space during its preference elicitation phase. We run computational experiments in several spectrum auction domains to evaluate BOCA's performance. Our results show that BOCA achieves higher allocative efficiency than state-of-the-art approaches.

In cooperative multi-agent robotic systems, coordination is necessary in order to complete a given task. Important examples include search and rescue, operations in hazardous environments, and environmental monitoring. Coordination, in turn, requires simultaneous satisfaction of safety critical constraints, in the form of state and input constraints, and a connectivity constraint, in order to ensure that at every time instant there exists a communication path between every pair of agents in the network. In this work, we present a model predictive controller that tackles the problem of performing multi-agent coordination while simultaneously satisfying safety critical and connectivity constraints. The former is formulated in the form of state and input constraints and the latter as a constraint on the second smallest eigenvalue of the associated communication graph Laplacian matrix, also known as Fiedler eigenvalue, which enforces the connectivity of the communication network. We propose a sequential quadratic programming formulation to solve the associated optimization problem that is amenable to distributed optimization, making the proposed solution suitable for control of multi-agent robotics systems relying on local computation. Finally, the effectiveness of the algorithm is highlighted with a numerical simulation.