computational complexity

Improving the convergence speed of adaptive filters is crucial for enhancing performance in applications involving highly correlated input signals. In this paper, we propose a novel method to improve the convergence performance of the affine projection LMS (AP-LMS) algorithm by incorporating a variable smoothing approach for the weight update matrix. The smoothing parameter is dynamically assigned based on the difference between the instantaneous and smoothed values of the weight update matrix. Simulation results for FIR system modeling demonstrate that the proposed algorithm achieves superior convergence performance in estimating system coefficients compared to competing adaptive algorithms for both stationary and non-stationary input signals.

The aim of this paper is to propose a convex risk measure in the framework of fuzzy random theory and verify its advantage over the conventional variance approach. For this purpose, this paper defines the quadratic deviation (QD) of fuzzy random variable as the mathematical expectation of QDs of fuzzy variables. As a result, the new risk criterion essentially describes the variation of a fuzzy random variable around its expected value. For triangular and trapezoidal fuzzy random variables as well as their linear combinations, we establish the analytical expressions of their QDs, and obtain the desirable convexity about the analytical expressions with respect to critical parameters. To explore the practical value of the proposed QD, we apply it to a portfolio selection problem to quantify the investment risk, and develop three mean-QD models to find the optimal allocation of the fund in different risky securities. Due to the convexity of our QD, the original three mean-QD models can be turned into their equivalent convex parametric quadratic programming problems, which can be solved by conventional optimization methods. The computational results clearly demonstrate that our new QD significantly reduces the computational complexity that cannot be avoided when variance is used as a risk criterion. Finally, the numerical comparison between the proposed mean-QD model and mean-variance model is conducted to show the consistency between the optimal results in both techniques. Meanwhile, the comparison between the proposed QD, variance, spread, and second moment is made to summarize the similarities and differences between them, distinguish these four risk criteria and determine their respective application scopes in decision systems.

The colorful paths and rainbow paths have been considered by several authors.A colorful directed path in a digraph $G$ is a directed path with $chi(G)$ vertices whose colors are different. A $v$-colorful directed path is such a directed path, starting from $v$. We prove that for a given $3$-regular triangle-free digraph $G$ determining whether there is a proper $chi(G)$ coloring of $G$ such that for every $v in V (G)$, there exists a $v$-colorful directed path is $ mathbf{NP} $ complete.