# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

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## Numerical Algebra, Control & Optimization

March 2020 , Volume 10 , Issue 1

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2020, 10(1): 1-11 doi: 10.3934/naco.2019028 +[Abstract](806) +[HTML](366) +[PDF](1250.65KB)
Abstract:

As a generalization of the fuzzy set and intuitionistic fuzzy set, the neutrosophic set (NS) have been developed to represent uncertain, imprecise, incomplete and inconsistent information existing in the real world. Now the interval neutrosophic set (INS) which is an expansion of the neutrosophic set have been proposed exactly to address issues with a set of numbers in the real unit interval, not just one specific number. After definition of concepts and operations, INS is applied to image segmentation. Images are converted to the INS domain, which is described using three membership interval sets: T, I and F. Then, in order to increase the contrast between membership and evaluate the indeterminacy, a fuzzy intensification for each element in the interval set is made and a score function in the INS is defined. Finally, the proposed method is employed to perform image segmentation using the traditional k-means clustering. The experimental results on a variety of images demonstrate that the proposed approach can segment different sorts of images. Especially, it can segment "clean" images and images with various levels of noise.

2020, 10(1): 13-21 doi: 10.3934/naco.2019029 +[Abstract](608) +[HTML](338) +[PDF](408.02KB)
Abstract:

Optimal correction of an infeasible system of absolute value equations (AVEs), leads into a nonconvex and nonsmooth fractional problem. Using Dinkelbach's approach, this problem can be reformulated to form a single variable equation. In this paper, first, we have smoothed the equation by considering four important and famous smoothing functions (see[2,23]) and thus, to solve it, a smoothing-type algorithm based on the Difference of Convex (DC) algorithm-Newtown methods is proposed. Finally, the randomly generated AVEs were compared to find the best smoothing function.

2020, 10(1): 23-38 doi: 10.3934/naco.2019030 +[Abstract](621) +[HTML](385) +[PDF](1725.98KB)
Abstract:

In this work we propose a novel numerical method for a finite-dimensional optimization problem arising from the discretization of an infinite-dimensional constrained optimization problem, called an obstacle problem, with interval coefficients. In this method, the two different ways of characterizing the optimal solutions, i.e., minimizing the mid-point and one end-point (the worst-case scenario) or the mid-point and the width of the objective interval, are formulated as a single constrained multi-objective minimization problem and the KKT conditions of the optimization problem defining the Pareto optimal solution to the multi-objective problem are of the form of a Linear Complementarity Problem (LCP) which is shown to have a unique solution. The LCP is the approximated by a non-linear equation using an interior penalty approach. We prove that the penalty equation is uniquely solvable and its solution converges to that of LCP as the penalty constant approaches to zero. Numerical results are presented to demonstrate the usefulness of the numerical method proposed.

2020, 10(1): 39-41 doi: 10.3934/naco.2019031 +[Abstract](681) +[HTML](374) +[PDF](275.11KB)
Abstract:

An optimal control problem is presented that exhibited unexpected initial guess dependence when being solved with direct transcription methods. This note presents that example and the cautionary tale it provides.

2020, 10(1): 45-74 doi: 10.3934/naco.2019032 +[Abstract](684) +[HTML](342) +[PDF](708.66KB)
Abstract:

A deteriorating inventory model with imperfect product and variable demand is formulated in this paper. A time-dependent deterioration factor is considered because the rate of deterioration is highly hinging on time. We introduce imperfect quality of production which leads to imperfect items in our proposed model. The retailer adopts inspection policy to pick over the perfect items from imperfect. Type Ⅰ and Type Ⅱ, both type of errors are included and the retailer invest some capital to improve the production process quality of the supplier. There is also a penalty cost for the retailer if they deliver some defective items by mistake. Sometime, there is a high amount of demand and, consequently, we assume shortages and partial backorder in our formulated model. The retailer adopts the trade-credit policy for his customers in order to promote market competition. The main objective of the paper is to show that the total cost is globally minimized and we have aimed at reducing the total cycle length, defectiveness of the system and the optimal order size by maximizing the total profit of the system. Then, we present three theorems and prove them to find an easy solution procedure to reduce the total cost of a system. The results are discussed with the help of numerical examples to approve the proposed model. A sensitivity analysis of the optimal solutions for the parameters is also provided. The paper ends with the conclusions and an outlook to possible future studies.

2020, 10(1): 75-92 doi: 10.3934/naco.2019034 +[Abstract](862) +[HTML](303) +[PDF](408.61KB)
Abstract:

We study a new class of mixed equilibrium problem, in short MEP, under weakly relaxed \begin{document}$\alpha$\end{document}-monotonicity in Banach spaces. This class of problems extends and generalizes some related fundamental results such as mixed variational-like inequalities, variational inequalities, and classical equilibrium problems as special cases. Existence and uniqueness of the solution to the problem is established. Auxiliary principle technique is used to obtain an iterative algorithm. Solvability of the auxiliary problem is established in the paper and finally the convergence of the iterates to the exact solution is proved. As applications of the approach developed in this paper, we study the existence and algorithmic approach for a general class of nonlinear mixed variational-like inequalities. The results obtained in this paper are interesting and improve considerably many existing results in literature.

2020, 10(1): 93-106 doi: 10.3934/naco.2019035 +[Abstract](675) +[HTML](289) +[PDF](405.03KB)
Abstract:

In this paper, we define a V-KT-pseudoinvex multidimensional vector control problem. More precisely, we introduce a new condition on the functionals which are involved in a multidimensional multiobjective (vector) control problem and we prove that a V-KT-pseudoinvex multidimensional vector control problem is characterized so that all Kuhn-Tucker points are efficient solutions. Also, the theoretical results derived in this paper are illustrated with an application.

2020, 10(1): 107-125 doi: 10.3934/naco.2019036 +[Abstract](720) +[HTML](308) +[PDF](473.59KB)
Abstract:

This paper has two objectives. The first one is to propose a new vector quasiequilibrium problem where the ordering relation is defined via an improvement set \begin{document}$D$\end{document}, and its weak version, also their Minty-type dual problems and the corresponding set-valued cases. These models provide unified frameworks to deal with well-known exact and approximate vector quasiequilibrium problems with vector-valued or set-valued mappings. The second one is to study solution stability in the sense of Hölder continuity of the unique solution to parametric unified (resp. weak) vector quasiequilibrium problems, by employing the Gerstewitz scalarization techniques. In particular, we deduce a new stability result for the typical vector optimization problem related with (resp. weak) \begin{document}$D$\end{document}-optimality, by considering perturbations of both the objective function and the feasible set.