Differential-difference method with approximation of the inverse operator

The problem of finding an approximate solution of a nonlinear equation with operator decomposition is considered. For equations of this type, a nonlinear operator can be represented as the sum of two operators – differentiable and nondifferentiable. For numerical solving such an equation, a differential-difference method, which contains the sum of the derivative of the differentiable part and the divided difference of the nondifferentiable part of the nonlinear operator, is proposed. Also, the proposed iterative process does not require finding the inverse operator. Instead of inverting the operator, its one-step approximation is used. The analysis of the local convergence of the method under the Lipschitz condition for the first-order divided differences and the bounded second derivative is carried out and the order of convergence is established.

Gauss–Newton–Secant Method for Solving Nonlinear Least Squares Problems under Generalized Lipschitz Conditions

We develop a local convergence of an iterative method for solving nonlinear least squares problems with operator decomposition under the classical and generalized Lipschitz conditions. We consider the case of both zero and nonzero residuals and determine their convergence orders. We use two types of Lipschitz conditions (center and restricted region conditions) to study the convergence of the method. Moreover, we obtain a larger radius of convergence and tighter error estimates than in previous works. Hence, we extend the applicability of this method under the same computational effort.

Safe Multi-Agent Pathfinding with Time Uncertainty

In many real-world scenarios, the time it takes for a mobile agent, e.g., a robot, to move from one location to another may vary due to exogenous events and be difficult to predict accurately. Planning in such scenarios is challenging, especially in the context of Multi-Agent Pathfinding (MAPF), where the goal is to find paths to multiple agents and temporal coordination is necessary to avoid collisions. In this work, we consider a MAPF problem with this form of time uncertainty, where we are only given upper and lower bounds on the time it takes each agent to move. The objective is to find a safe solution, which is a solution that can be executed by all agents and is guaranteed to avoid collisions. We propose two complete and optimal algorithms for finding safe solutions based on well-known MAPF algorithms, namely, A* with Operator Decomposition (A* + OD) and Conflict-Based Search (CBS). Experimentally, we observe that on several standard MAPF grids the CBS-based algorithm performs better. We also explore the option of online replanning in this context, i.e., modifying the agents' plans during execution, to reduce the overall execution cost. We consider two online settings: (a) when an agent can sense the current time and its current location, and (b) when the agents can also communicate seamlessly during execution. For each setting, we propose a replanning algorithm and analyze its behavior theoretically and empirically. Our experimental evaluation confirms that indeed online replanning in both settings can significantly reduce solution cost.

The operator transformations in the quality theory of dpac’s mathematical designing

In this article the results on the quality theory of DPAC’s mathematical designing have been cited, which are connected with operator transformations. This are the elements of operator’s calculus by academician V.M.Glushkov, and also author’s results: on the operator decomposition of algorithmic graph-schemes, on the definition of operators, which are used in the researches on mathematical models of the DPAC’s compositional transformations.

Quantile Regression Learning with Coefficient Dependent lq-Regularizer

In this paper, We focus on conditional quantile regression learning algorithms based on the pinball loss and lq-regularizer with 1≤q≤2. Our main goal is to study the consistency of this kind of regularized quantile regression learning. With concentration inequality and operator decomposition techniques, we obtained satisfied error bounds and convergence rates.