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Numerical Algebra, Control & Optimization

March 2022 , Volume 12 , Issue 1

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Shu-Cherng Fang, Ruey-Lin Sheu and Tamaki Tanaka
2022, 12(1): ⅰ-ⅱ doi: 10.3934/naco.2021058 +[Abstract](5128) +[HTML](31) +[PDF](287.86KB)
Axiomatic results and dynamic processes for two weighted indexes under fuzzy transferable-utility behavior
Yu-Hsien Liao
2022, 12(1): 1-14 doi: 10.3934/naco.2021047 +[Abstract](86) +[HTML](36) +[PDF](389.46KB)

By considering the supreme-utilities and the weights simultaneously under fuzzy behavior, we propose two indexes on fuzzy transferable-utility games. In order to present the rationality for these two indexes, we define extended reductions to offer several axiomatic results and dynamics processes. Based on different consideration, we also adopt excess functions to propose alternative formulations and related dynamic processes for these two indexes respectively.

Variable fixing method by weighted average for the continuous quadratic knapsack problem
Hsin-Min Sun and Yu-Juan Sun
2022, 12(1): 15-29 doi: 10.3934/naco.2021048 +[Abstract](107) +[HTML](30) +[PDF](458.01KB)

We analyze the method of solving the separable convex continuous quadratic knapsack problem by weighted average from the viewpoint of variable fixing. It is shown that this method, considered as a variant of the variable fixing algorithms, and Kiwiel's variable fixing method generate the same iterates. We further improve the algorithm based on the analysis regarding the semismooth Newton method. Computational results are given and comparisons are made among the state-of-the-art algorithms. Experiments show that our algorithm has significantly good performance; it behaves very much like an \begin{document}$ O(n) $\end{document} algorithm with a very small constant.

A crowdsourced dynamic repositioning strategy for public bike sharing systems
I-Lin Wang and Chen-Tai Hou
2022, 12(1): 31-46 doi: 10.3934/naco.2021049 +[Abstract](84) +[HTML](37) +[PDF](781.03KB)

Public bike sharing systems have become the most popular shared economy application in transportation. The convenience of this system depends on the availability of bikes and empty racks. One of the major challenges in operating a bike sharing system is the repositioning of bikes between rental sites to maintain sufficient bike inventory in each station at all times. Most systems hire trucks to conduct dynamic repositioning of bikes among rental sites. We have analyzed a commonly used repositioning scheme and have demonstrated its ineffectiveness. To realize a higher quality of service, we proposed a crowdsourced dynamic repositioning strategy: first, we analyzed the historical rental data via the random forest algorithm and identified important factors for demand forecasting. Second, considering 30-minute periods, we calculated the optimal bike inventory via integer programming for each rental site in each time period with a sufficient crowd for repositioning bikes. Then, we proposed a minimum cost network flow model in a time-space network for calculating the optimal voluntary rider flows for each period based on the current bike inventory, which is adjusted according to the forecasted demands. The results of computational experiments on real-world data demonstrate that our crowdsourced repositioning strategy may reduce unmet rental demands by more than 30% during rush hours compared to conventional trucks.

Levenberg-Marquardt method for absolute value equation associated with second-order cone
Xin-He Miao, Kai Yao, Ching-Yu Yang and Jein-Shan Chen
2022, 12(1): 47-61 doi: 10.3934/naco.2021050 +[Abstract](81) +[HTML](32) +[PDF](483.7KB)

In this paper, we suggest the Levenberg-Marquardt method with Armijo line search for solving absolute value equations associated with the second-order cone (SOCAVE for short), which is a generalization of the standard absolute value equation frequently discussed in the literature during the past decade. We analyze the convergence of the proposed algorithm. For numerical reports, we not only show the efficiency of the proposed method, but also present numerical comparison with smoothing Newton method. It indicates that the proposed algorithm could also be a good choice for solving the SOCAVE.

Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems
Do Sang Kim, Nguyen Ngoc Hai and Bui Van Dinh
2022, 12(1): 63-78 doi: 10.3934/naco.2021051 +[Abstract](104) +[HTML](35) +[PDF](478.64KB)

In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given in the literature. A numerical example is also provided to illustrate the proposed algorithms.

Measuring efficiency of a recycling production system with imprecise data
Cheng-Feng Hu, Hsiao-Fan Wang and Tingyang Liu
2022, 12(1): 79-91 doi: 10.3934/naco.2021052 +[Abstract](100) +[HTML](36) +[PDF](483.82KB)

Resources scarcity and environmental degradation have made sustainable resource utilization and environmental protection worldwide. A circular economy system considers economic production activities as closed-loop feedback cycles in which resources are used sustainably and cyclically. Improving the eco-efficiency of the circular economy system has both theoretical value and practical meaning. In this work, the efficiency measurement model of the circular economy system with imprecise data based on network data envelopment analysis is proposed. The two-level mathematical programming approach is employed for measuring the system and process efficiencies. The lower and upper bounds of the efficiencies scores are calculated by transformed conventional one-level linear programs so that the existing solution methods can be applied. The proposed method is applied to assess the circular economy system of EU countries. Our results show that most countries have large difference among fuzzy efficiencies between the production efficiency and recycling efficiency stages, which reveals the source that causes the low efficiency of the circular economy system.

Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators
Jutamas Kerdkaew, Rabian Wangkeeree and Rattanaporn Wangkeeree
2022, 12(1): 93-107 doi: 10.3934/naco.2021053 +[Abstract](159) +[HTML](33) +[PDF](418.58KB)

In this paper, a robust optimization problem, which features a maximum function of continuously differentiable functions as its objective function, is investigated. Some new conditions for a robust KKT point, which is a robust feasible solution that satisfies the robust KKT condition, to be a global robust optimal solution of the uncertain optimization problem, which may have many local robust optimal solutions that are not global, are established. The obtained conditions make use of underestimators, which were first introduced by Jayakumar and Srisatkunarajah [1,2] of the Lagrangian associated with the problem at the robust KKT point. Furthermore, we also investigate the Wolfe type robust duality between the smooth uncertain optimization problem and its uncertain dual problem by proving the sufficient conditions for a weak duality and a strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. The results on robust duality theorems are established in terms of underestimators. Additionally, to illustrate or support this study, some examples are presented.

Optimal pre-sale policy for deteriorating items
Lianxia Zhao, Hui Qiao and Qi An
2022, 12(1): 109-120 doi: 10.3934/naco.2021054 +[Abstract](83) +[HTML](45) +[PDF](757.04KB)

Pre-sale policy is a frequently-used sales approach for deteriorating products, e.g, fruits, vegetables, seafood, etc. In this paper, we consider an EOQ inventory model under pre-sale policy for deteriorating products, in which the demand of pre-sale period depends on price and pre-sale horizon, and the demand of spot-sale period depends on the price and stock level. Optimal pricing decisions and economic order quantity are also provided. We compare pre-sale model with a benchmark inventory model in which all the products are sold in spot-sale period. Theoretical results are derived to show the existence and uniqueness of the optimal solution. Numerical experiments are carried out to to illustrate the theoretical results. And sensitivity analysis is conducted to identify conditions under which the pre-sale policy is better off than the spot-sale only policy.

Optimality and duality for complex multi-objective programming
Tone-Yau Huang and Tamaki Tanaka
2022, 12(1): 121-134 doi: 10.3934/naco.2021055 +[Abstract](93) +[HTML](29) +[PDF](416.8KB)

We consider a complex multi-objective programming problem (CMP). In order to establish the optimality conditions of problem (CMP), we introduce several properties of optimal efficient solutions and scalarization techniques. Furthermore, a certain parametric dual model is discussed, and their duality theorems are proved.

An active set solver for constrained $ H_\infty $ optimal control problems with state and input constraints
Canghua Jiang, Dongming Zhang, Chi Yuan and Kok Ley Teo
2022, 12(1): 135-157 doi: 10.3934/naco.2021056 +[Abstract](82) +[HTML](26) +[PDF](514.89KB)

This paper proposes an active set solver for \begin{document}$ H_\infty $\end{document} min-max optimal control problems involving linear discrete-time systems with linearly constrained states, controls and additive disturbances. The proposed solver combines Riccati recursion with dynamic programming. To deal with possible degeneracy (i.e. violations of the linear independence constraint qualification), constraint transformations are introduced that allow the surplus equality constraints on the state at each stage to be moved to the previous stage together with their Lagrange multipliers. In this way, degeneracy for a feasible active set can be determined by checking whether there exists an equality constraint on the initial state over the prediction horizon. For situations when the active set is degenerate and all active constraints indexed by it are non-redundant, a vertex exploration strategy is developed to seek a non-degenerate active set. If the sampled state resides in a robust control invariant set and certain second-order sufficient conditions are satisfied at each stage, then a bounded \begin{document}$ l_2 $\end{document} gain from the disturbance to controlled output can be guaranteed for the closed-loop system under some standard assumptions. Theoretical analysis and numerical simulations show that the computational complexity per iteration of the proposed solver depends linearly on the prediction horizon.

Distributionally Robust Optimization: A review on theory and applications
Fengming Lin, Xiaolei Fang and Zheming Gao
2022, 12(1): 159-212 doi: 10.3934/naco.2021057 +[Abstract](184) +[HTML](65) +[PDF](1079.22KB)

In this paper, we survey the primary research on the theory and applications of distributionally robust optimization (DRO). We start with reviewing the modeling power and computational attractiveness of DRO approaches, induced by the ambiguity sets structure and tractable robust counterpart reformulations. Next, we summarize the efficient solution methods, out-of-sample performance guarantee, and convergence analysis. Then, we illustrate some applications of DRO in machine learning and operations research, and finally, we discuss the future research directions.

2020 CiteScore: 1.6




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