# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

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## Numerical Algebra, Control and Optimization

June 2021 , Volume 11 , Issue 2

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2021, 11(2): 143-207 doi: 10.3934/naco.2020021 +[Abstract](2504) +[HTML](714) +[PDF](1728.83KB)
Abstract:

The controlled motion of a rolling ball actuated by internal point masses that move along arbitrarily-shaped rails fixed within the ball is considered. The controlled equations of motion are solved numerically using a predictor-corrector continuation method, starting from an initial solution obtained via a direct method, to realize trajectory tracking and obstacle avoidance maneuvers.

2021, 11(2): 209-220 doi: 10.3934/naco.2020022 +[Abstract](1943) +[HTML](575) +[PDF](432.46KB)
Abstract:

In this paper we consider non-cooperative game problem based on the Malfatti's problem. This problem is a special case of generalized Nash equilibrium problems with nonconvex shared constraints. Some numerical results are provided.

2021, 11(2): 221-253 doi: 10.3934/naco.2020023 +[Abstract](3853) +[HTML](1766) +[PDF](1253.34KB)
Abstract:

One of the challenges facing supply chain designers is designing a sustainable and resilient supply chain network. The present study considers a closed-loop supply chain by taking into account sustainability, resilience, robustness, and risk aversion for the first time. The study suggests a two-stage mixed-integer linear programming model for the problem. Further, the robust counterpart model is used to handle uncertainties. Furthermore, conditional value at risk criterion in the model is considered in order to create real-life conditions. The sustainability goals addressed in the present study include minimizing the costs, \begin{document}$\text{CO}_2$\end{document} emission, and energy, along with maximizing employment. In addition, effective environmental and social life-cycle evaluations are provided to assess the associated effects of the model on society, environment, and energy consumption. The model aims to answer the questions regarding the establishment of facilities and amount of transported goods between facilities. The model is implemented in a car assembler company in Iran. Based on the results, several managerial insights are offered to the decision-makers. Due to the complexity of the problem, a constraint relaxation is applied to produce quality upper and lower bounds in medium and large-scale models. Moreover, the LP-Metric method is used to merge the objectives to attain an optimal solution. The results revealed that the robust counterpart provides a better estimation of the total cost, pollution, energy consumption, and employment level compared to the basic model.

2021, 11(2): 255-267 doi: 10.3934/naco.2020024 +[Abstract](2550) +[HTML](659) +[PDF](1262.45KB)
Abstract:

Regularized Multidimensional Scaling with Radial basis function (RMDS) is a nonlinear variant of classical Multi-Dimensional Scaling (cMDS). A key issue that has been addressed in RMDS is the effective selection of centers of the radial basis functions that plays a very important role in reducing the dimension preserving the structure of the data in higher dimensional space. RMDS uses data in unsupervised settings that means RMDS does not use any prior information of the dataset. This article is concerned on the supervised setting. Here we have incorporated the class information of some members of data to the RMDS model. The class separability term improved the method RMDS significantly and also outperforms other discriminant analysis methods such as Linear discriminant analysis (LDA) which is documented through numerical experiments.

2021, 11(2): 269-281 doi: 10.3934/naco.2020025 +[Abstract](1860) +[HTML](600) +[PDF](651.63KB)
Abstract:

We apply a space adaptive interior penalty discontinuous Galerkin method for solving advective Allen-Cahn equation with expanding and contracting velocity fields. The advective Allen-Cahn equation is first discretized in time and the resulting semi-linear elliptic PDE is solved by an adaptive algorithm using a residual-based a posteriori error estimator. The a posteriori error estimator contains additional terms due to the non-divergence-free velocity field. Numerical examples demonstrate the effectiveness and accuracy of the adaptive approach by resolving the sharp layers accurately.

2021, 11(2): 283-305 doi: 10.3934/naco.2020026 +[Abstract](1940) +[HTML](665) +[PDF](1742.52KB)
Abstract:

There are a limited number of user-friendly, publicly available optimal control software packages that are designed to accommodate problems with delays. GPOPS-Ⅱ is a well developed MATLAB based optimal control code that was not originally designed to accommodate problems with delays. The use of GPOPS-Ⅱ on optimal control problems with delays is examined for the first time. The use of various formulations of delayed optimal control problems is also discussed. It is seen that GPOPS-Ⅱ finds a suboptimal solution when used as a direct transcription delayed optimal control problem solver but that it is often able to produce a good solution of the optimal control problem when used as a delayed boundary value solver of the necessary conditions.

2021, 11(2): 307-320 doi: 10.3934/naco.2020027 +[Abstract](1432) +[HTML](471) +[PDF](380.17KB)
Abstract:

A singularly perturbed linear time-dependent controlled system with a point-wise nonsmall (of order of \begin{document}$1$\end{document}) delay in the input (the control variable) is considered. Sufficient conditions of the complete Euclidean space controllability for this system, robust with respect to the parameter of singular perturbation, are derived. This derivation is based on an asymptotic analysis of the controllability matrix for the considered system and on such an analysis of the determinant of this matrix. However, this derivation does not use a slow-fast decomposition of the considered system. The theoretical result is illustrated by an example.

2021, 11(2): 321-332 doi: 10.3934/naco.2020028 +[Abstract](1598) +[HTML](491) +[PDF](918.83KB)
Abstract:

In a grid network, the nodes could be traversed either horizontally or vertically. The constrained shortest Hamiltonian path goes over the nodes between a source node and a destination node, and it is constrained to traverse some nodes at least once while others could be traversed several times. There are various applications of the problem, especially in routing problems. It is an NP-complete problem, and the well-known Bellman-Held-Karp algorithm could solve the shortest Hamiltonian circuit problem within \begin{document}${\rm O(}{{\rm 2}}^{{\rm n}}{{\rm n}}^{{\rm 2}}{\rm )}$\end{document} time complexity; however, the shortest Hamiltonian path problem is more complicated. So, a metaheuristic algorithm based on ant colony optimization is applied to obtain the optimal solution. The proposed method applies the rooted shortest path tree structure since in the optimal solution the paths between the restricted nodes are the shortest paths. Then, the shortest path tree is obtained by at most \begin{document}${\rm O(}{{\rm n}}^{{\rm 3}}{\rm )}$\end{document} time complexity at any iteration and the ants begin to improve the solution and the optimal solution is constructed in a reasonable time. The algorithm is verified by some numerical examples and the ant colony parameters are tuned by design of experiment method, and the optimal setting for different size of networks are determined.

2021 CiteScore: 1.9