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Recently a new least-squares primal-dual (LSPD) algorithm, that is impervious to degeneracy, has effectively been applied to solving linear programming problems by Barnes et al., 2002. In this paper, we show an application of LSPD to shortest path problems with nonnegative arc length is equivalent to the Dijkstra's algorithm. We also compare the LSPD algorithm with the conventional primal-dual algorithm in solving shortest path problems and show their difference due to degeneracy in solving the 1-1 shortest path problems.

The least-squares is a common and important method in linear regression. However, it often leads to overfitting phenomenon as dealing with high-dimensional problems, and various regularization schemes regarding prior information for specific problems are studied to make up such a deficiency. In the sense of Kendall’s $\tau$ from the community of nonparametric analysis, we establish a new model wherein the ordinary least-squares is equipped with perfect positive correlation constraint, sought to maintain the concordance of the rankings of the observations and the systematic components. By sorting the observations into an ascending order, we reduce the perfect positive correlation constraint into a linear inequality system. The resulting linearly constrained least-squares problem together with its dual problem is shown to be solvable. In particular, we introduce a mild assumption on the observations and the measurement matrix which rules out the zero vector from the optimal solution set. This indicates that our proposed model is statistically meaningful. To handle large-scale instances, we propose an efficient alternating direction method of multipliers (ADMM) to solve the proposed model from the dual perspective. The effectiveness of our model compared to ordinary least-squares is evaluated in terms of rank correlation coefficient between outputs and the systematic components, and the efficiency of our dual algorithm is demonstrated with the comparison to three efficient solvers via CVX in terms of computation time, solution accuracy and rank correlation coefficient.

Public health awareness programs have been a crucial strategy in mitigating the spread of emerging and re-emerging infectious disease outbreaks of public health significance such as COVID-19. This study adopts an Susceptible–Exposed–Infected–Recovered (SEIR) based model to assess the impact of public health awareness programs in mitigating the extent of the COVID-19 pandemic. The proposed model, which incorporates public health awareness programs, uses ABC fractional operator approach to study and analyze the transmission dynamics of SARS-CoV-2. It is possible to completely understand the dynamics of the model’s features because of the memory effect and hereditary qualities that exist in the fractional version. The fixed point theorem has been used to prove the presence of a unique solution, as well as the stability analysis of the model. The nonlinear least-squares method is used to estimate the parameters of the model based on the daily cumulative cases of the COVID-19 pandemic in Nigeria from March 29 to June 12, 2020. Through the use of simulations, the model’s best-suited parameters and the optimal ABC fractional-order parameter $\tau$ may be determined and optimized. The suggested model is proved to understand the virus’s dynamical behavior better than the integer-order version. In addition, numerous numerical simulations are run using an efficient numerical approach to provide further insight into the model’s features.

Design using second-moments is readily understood by engineers. The output means (first-moments) and covariances (second-moments) are expressed through the means and covariances of the inputs. Further, various performance indexes can be formulated in terms of the second-moments and used to measure the “goodness” of the system’s performance. This paper addresses the design of nonlinear dynamic systems with uncertainty in both the component parameters and the excitations. In order to reduce the computational effort needed for design iterations on the mechanistic model, meta-models are introduced as computationally efficient surrogates. Herein, a novel, differentiable, meta-model that finds the response of dynamic systems with simultaneous component and excitation uncertainty is presented. Operationally, a family of training excitations and sets of training parameters are chosen and stored in respective matrices. Both types of inputs must have some realistic bounds. The corresponding responses, produced by the mechanistic model, make use of all of the training parameter sets interleafed with the training excitations: the time-sampled results are stored in the response matrix. An application of singular value decomposition on the response matrix reveals a repeating pattern of sub-vectors in the left singular vectors. Each sub-vector (viewed as the output) is replaced by a least-squares meta-model that links in the parameter matrix. The result is a parameter-response matrix with the same number of rows as the excitation matrix. Finally, to complete the meta-model, another application of the least-squares paradigm links the excitation matrix to the columns of the parameter-response matrix. Performance indexes, and approximations of their means and covariances through Taylor series, provide cogent optimization measures. The required derivatives are easily obtained from the explicit form of the meta-model. The efficacy of the meta-model is shown through the design of a nonlinear, quarter automobile, system. The accuracy, increased computation speed and robustness of the methodology provide the impact of the work herein. The sources of errors are identified and ways to mitigate them are discussed.

Presented herein is a new computational strategy to eliminate the thermal effect in measured responses. The internal forces of a structure and its thermal load components are estimated alternately at each time step based on a recursive least-squares algorithm. The noise effect in the measured data and the estimated force in previous steps are removed with the Kalman filter. This approach directly models the thermal effects on the dynamic responses without the temperature information. This approach adopts force identification technique which has been researched extensively in the last two decades. The effectiveness of the proposed approach is validated via numerical studies with a bridge structure with or without noise in the measured data. The thermal load in the structural components can be estimated accurately with the proposed strategy with less than 8% error when there is 10% noise in the measured responses of the structure.

When dealing with complex and redundant data classification problems, many classifiers cannot provide high predictive accuracy and interpretability. We also find that the least-squares support vector classifiers (LSSVCs) hardly identify important instances and features from data, so they cannot give an interpretable prediction. Although the LSSVC has the properties of low bias and high robustness, its high variance often gives a poor predictive performance. In this paper, we propose an alternating minimization-based sparse least-squares classifier (AMSLC) approach in the framework of LSSVCs to address the aforementioned problems. Based on the reconstructed row- and column-wise kernel matrices, the sparsity-induced ${\ell }_0$-norm approximation function is introduced to the LSSVC model. By alternately solving two unconstrained quadratic optimization problems or two systems of linear equations, AMSLC can predict the class labels of given instances and extract the least number of important instances and features to obtain the interpretable classification. Compared with SVC, LSSVC, ${\ell }_1$-norm SVC (L1SVC), ${\ell }_0$-norm SVC (L0SVC), the least absolute shrinkage and selection operator classifier (LASSOC), and multiple kernel learning SVC (MKLSVC) on four real credit datasets, the experimental results show that the proposed AMSLC method generally obtains the best predictive accuracy and the interpretable classification with the minimum number of important instances and features.

A numerical method is presented to simulate dendritic crystal growth on a fixed mesh. The hyperbolic convective equation for the continuous phase field which identifies the solid and liquid phases is solved by the least-squares finite element method, and the heat conduction equation for the temperature field is solved by the Galerkin finite element method. Without special treatments this method can handle complicated interfacial shapes and physical features.