Some Random Fixed-Point Theorems for Weakly Contractive Random Operators in a Separable Banach Space

The aim of this paper is to prove a common random fixed-point and some random fixed-point theorems for random weakly contractive operators in separable Banach spaces. A random Mann iterative process is introduced to approximate the fixed point. Finally, the main result is supported by an example and used to prove the existence and the uniqueness of a solution of a nonlinear stochastic integral equation system.

A Modified Cuckoo Search Algorithm and Its Applications in Function Optimization

A modified Cuckoo search algorithm (MCS) is proposed in this paper to improve the accuracy of the algorithm’s convergence by implementing random operators and adapt the adjustment mechanism of the Levy Flight search step length. Comparative experiments reveal that MCS can effectively adjust the search mechanism in the high-dimensional function optimization and converge to the optimal global value.

Gaussian Elimination versus Greedy Methods for the Synthesis of Linear Reversible Circuits

Linear reversible circuits represent a subclass of reversible circuits with many applications in quantum computing. These circuits can be efficiently simulated by classical computers and their size is polynomially bounded by the number of qubits, making them a good candidate to deploy efficient methods to reduce computational costs. We propose a new algorithm for synthesizing any linear reversible operator by using an optimized version of the Gaussian elimination algorithm coupled with a tuned LU factorization. We also improve the scalability of purely greedy methods. Overall, on random operators, our algorithms improve the state-of-the-art methods for specific ranges of problem sizes: The custom Gaussian elimination algorithm provides the best results for large problem sizes (n > 150), while the purely greedy methods provide quasi optimal results when n < 30. On a benchmark of reversible functions, we manage to significantly reduce the CNOT count and the depth of the circuit while keeping other metrics of importance (T-count, T-depth) as low as possible.

Random Fixed Point Theorems and Applications to Random First-Order Vector-Valued Differential Equations

In this paper, we establish several random fixed point theorems for random operators satisfying some iterative condition w.r.t. a measure of noncompactness. We also discuss the case of monotone random operators in ordered Banach spaces. Our results extend several earlier works, including Itoh’s random fixed point theorem. As an application, we discuss the existence of random solutions to a class of random first-order vector-valued ordinary differential equations with lack of compactness.

Boundary Value Problem for Caputo–Fabrizio Random Fractional Differential Equations

This article deals with some existence of random solutions and Ulam stability results for a class of Caputo-Fabrizio random fractional differential equations with boundary conditions in Banach spaces. Our results are based on the fixed point theory and random operators. Two illustrative examples are presented in the last section.

A db-Scan Hybrid Algorithm: An Application to the Multidimensional Knapsack Problem

This article proposes a hybrid algorithm that makes use of the db-scan unsupervised learning technique to obtain binary versions of continuous swarm intelligence algorithms. These binary versions are then applied to large instances of the well-known multidimensional knapsack problem. The contribution of the db-scan operator to the binarization process is systematically studied. For this, two random operators are built that serve as a baseline for comparison. Once the contribution is established, the db-scan operator is compared with two other binarization methods that have satisfactorily solved the multidimensional knapsack problem. The first method uses the unsupervised learning technique k-means as a binarization method. The second makes use of transfer functions as a mechanism to generate binary versions. The results show that the hybrid algorithm using db-scan produces more consistent results compared to transfer function (TF) and random operators.

A Db-Scan Binarization Algorithm Applied to Matrix Covering Problems

The integration of machine learning techniques and metaheuristic algorithms is an area of interest due to the great potential for applications. In particular, using these hybrid techniques to solve combinatorial optimization problems (COPs) to improve the quality of the solutions and convergence times is of great interest in operations research. In this article, the db-scan unsupervised learning technique is explored with the goal of using it in the binarization process of continuous swarm intelligence metaheuristic algorithms. The contribution of the db-scan operator to the binarization process is analyzed systematically through the design of random operators. Additionally, the behavior of this algorithm is studied and compared with other binarization methods based on clusters and transfer functions (TFs). To verify the results, the well-known set covering problem is addressed, and a real-world problem is solved. The results show that the integration of the db-scan technique produces consistently better results in terms of computation time and quality of the solutions when compared with TFs and random operators. Furthermore, when it is compared with other clustering techniques, we see that it achieves significantly improved convergence times.