# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

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## Numerical Algebra, Control & Optimization

June 2019 , Volume 9 , Issue 2

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2019, 9(2): 113-132 doi: 10.3934/naco.2019009 +[Abstract](2172) +[HTML](391) +[PDF](446.47KB)
Abstract:

In today's competitive environment, businesses are searching for tools and instruments, using which they can reduce their costs as much as possible in order to increase profits. The supply chain as a process which requires comprehensive management can be a great help in this regard for companies. The majority of approaches evaluated in supply chain primarily deal with logistic and material flows and neglect a lot of financial dimensions. This is while the financial flow in the supply chain can play an effective role in improving and optimizing the chain and contribute heavily to the profitability of the business. This paper deals with the financial flow of the supply chain model along with the material flow. It indicates that while optimizing the financial flow will provide the maximum profit for the plant, through simultaneous modeling of both these flows, better results can be reached. On the other hand, optimizing the financial flow allows the financial factors to be also considered in the model, which helps the business reach higher profits and better management of financial processes, which in turn shifts the business towards a modern industrial unit.

2019, 9(2): 133-145 doi: 10.3934/naco.2019010 +[Abstract](1923) +[HTML](311) +[PDF](345.44KB)
Abstract:

In the paper, a vector optimization problem with twice differentiable functions and cone constraints is considered. The second order modified objective function method is used for solving such a multiobjective programming problem. In this method, for the considered twice differentiable multi-criteria optimization problem, its associated second order vector optimization problem with the modified objective function is constructed at the given arbitrary feasible solution. Then, the equivalence between the sets of (weakly) efficient solutions in the original twice differentiable vector optimization problem with cone constraints and its associated modified vector optimization problem is established. Further, the relationship between an (weakly) efficient solution in the original vector optimization problem and a saddle-point of the second order Lagrange function defined for the modified vector optimization problem is also analyzed.

2019, 9(2): 147-156 doi: 10.3934/naco.2019011 +[Abstract](1676) +[HTML](382) +[PDF](344.27KB)
Abstract:

In this paper, we analyze a feasible predictor-corrector linear programming variant of Mehrotra's algorithm. The analysis is done in the negative infinity neighborhood of the central path. We demonstrate the theoretical efficiency of this algorithm by showing its polynomial complexity. The complexity result establishes an improvement of factor \begin{document}$n^3$\end{document} in the theoretical complexity of an earlier presented variant in [2], which is a huge improvement. We examine the performance of our algorithm by comparing its implementation results to solve some NETLIB problems with the algorithm presented in [2].

2019, 9(2): 157-171 doi: 10.3934/naco.2019012 +[Abstract](1501) +[HTML](347) +[PDF](418.23KB)
Abstract:

In the paper we apply a Parlett–Kahan's "twice is enough" type algorithm to conjugate directions. We give a lower bound of the digits of precision of the conjugate directions, too.

2019, 9(2): 173-186 doi: 10.3934/naco.2019013 +[Abstract](1946) +[HTML](383) +[PDF](487.85KB)
Abstract:

To implement the balancing based model reduction of large-scale dynamical systems we need to compute the low-rank (controllability and observability) Gramian factors by solving Lyapunov equations. In recent time, Rational Krylov Subspace Method (RKSM) is considered as one of the efficient methods for solving the Lyapunov equations of large-scale sparse dynamical systems. The method is well established for solving the Lyapunov equations of the standard or generalized state space systems. In this paper, we develop algorithms for solving the Lyapunov equations for large-sparse structured descriptor system of index-1. The resulting algorithm is applied for the balancing based model reduction of large sparse power system model. Numerical results are presented to show the efficiency and capability of the proposed algorithm.

2019, 9(2): 187-209 doi: 10.3934/naco.2019014 +[Abstract](2078) +[HTML](359) +[PDF](1116.99KB)
Abstract:

Product portfolio optimization (PPO) is a strategic decision for many organizations. There are several technical methods for facilitating this decision. According to the reviewed studies, the implementation of the robust optimization approach and the invasive weed optimization (IWO) algorithm is the research gap in this field. The contribution of this paper is the development of the PPO problem with the help of the robust optimization approach and the multi-objective IWO algorithm. Considering the profit margin uncertainty in real-world investment decisions, the robust optimization approach is used to address this issue. To illustrate the real-world applicability of the model, it is implemented for dairy products of Pegah Golpayegan Company in Iran. The numerical results obtained from the IWO algorithm demonstrate the effectiveness of the proposed algorithm in tracing out the efficiency frontier of the product portfolio. The average risk of efficient frontier solutions in the deterministic model is about 0.4 and for the robust counterpart formulation is at least 0.5 per product. The efficient frontier solutions obtained from robust counterpart formulation demonstrate a more realistic risk level than the deterministic model. The comparisons between CPLEX, IWO and genetic algorithm (GA) shows that the performance of the IWO algorithm is much better than the older algorithms and can be considered as an alternative to algorithms, such as GA in product portfolio optimization problems.

2019, 9(2): 211-224 doi: 10.3934/naco.2019015 +[Abstract](1745) +[HTML](406) +[PDF](375.08KB)
Abstract:

Based on the matrix hard thresholding method, a homotopy method is proposed for solving the matrix rank minimization problem. This method iteratively solves a series of regularization subproblems, whose solutions are given in closed form by the matrix hard thresholding operator. Under some mild assumptions, convergence of the proposed method is proved. The proposed method does not depend on a prior knowledge of exact rank value. Numerical experiments demonstrate that the proposed homotopy method weakens the affection of the choice of the regularization parameter, and is more efficient and effective than the existing sate-of-the-art methods.

2019, 9(2): 225-256 doi: 10.3934/naco.2019016 +[Abstract](1747) +[HTML](299) +[PDF](1616.0KB)
Abstract:

A main objective of this work is to assess the feasibility of space missions to a new population of near Earth asteroids which temporarily orbit Earth, called temporarily captured orbiter. We design rendezvous missions to a large random sample from a database of over 16,000 simulated temporarily captured orbiters using an indirect method based on the maximum principle. The main contribution of this paper is the development of techniques to overcome the difficulty in initializing the algorithm with the construction of the so-called cloud of extremals.