Outer and Inner Approximation

1998 ◽  
pp. 177-222 ◽  
Author(s):  
Hoang Tuy
Keyword(s):  
Inner Approximation

scholarly journals A Model for Demand Planning in Supply Chains with Congestion Effects

Logistics ◽  
2021 ◽  
Vol 5 (1) ◽  
pp. 3
Author(s):  
Uday Venkatadri ◽  
Shentao Wang ◽  
Ashok Srinivasan
Keyword(s):  
Supply Chains ◽  
Capacity Planning ◽  
Demand Management ◽  
Capacity Limitations ◽  
Demand Planning ◽  
Clearing Functions ◽  
Order Promising ◽  
Linear Aggregate ◽  
Inner Approximation ◽  
Congestion Effects

This paper is concerned with demand planning for internal supply chains consisting of workstations, production facilities, warehouses, and transportation links. We address the issue of how to help a supplier firmly accept orders and subsequently plan to fulfill demand. We first formulate a linear aggregate planning model for demand management that incorporates elements of order promising, recipe run constraints, and capacity limitations. Using several scenarios, we discuss the use of the model in demand planning and capacity planning to help a supplier firmly respond to requests for quotations. We extend the model to incorporate congestion effects at assembly and blending nodes using clearing functions; the resulting model is nonlinear. We develop and test two algorithms to solve the nonlinear model: one based on inner approximation and the other on outer approximation.

Fixed-order ℋ∞ controller design for MIMO systems via polynomial approach

2018 ◽  
Vol 41 (7) ◽  
pp. 1985-1992 ◽  
Author(s):  
Bilal Erol ◽  
Akın Delibaşı
Keyword(s):  
Traditional Method ◽  
Controller Design ◽  
Mimo Systems ◽  
Linear Matrix ◽  
Main Difficulty ◽  
Matrix Inequalities ◽  
Central Polynomial ◽  
Fixed Order ◽  
Inner Approximation ◽  
Associated Solution

This paper presents a fixed-order [Formula: see text]∞ controller design based on linear matrix inequalities for multi-input–multi-output systems. The main difficulty in the development of a fixed-order controller design is that the associated solution set of the problem is defined in a non-convex cluster, and that makes the problem computationally intractable. The convex inner approximation is used to deal with this non-convexity. The proposed controller design approach is applied to some elegant numerical problems taken from various previous works. To show the effectiveness of the proposed method, the full-order [Formula: see text]∞ controller and fixed-order controllers are constructed for these models using the traditional method and popular toolboxes, respectively. Furthermore, in this paper, some strategies for choosing the central polynomial, which is the main conservatism of the proposed method, are discussed.

A numerical method for the approximation of reachable sets of linear control systems

2017 ◽  
Vol 36 (2) ◽  
pp. 423-441 ◽  
Author(s):  
Lizhen Shao ◽  
Fangyuan Zhao ◽  
Guangda Hu
Keyword(s):  
Numerical Method ◽  
Control Systems ◽  
Optimization Problems ◽  
Approximation Error ◽  
Outer Approximation ◽  
Reachable Sets ◽  
Linear Control ◽  
Superior Performance ◽  
Linear Control Systems ◽  
Inner Approximation

Abstract In this article, a numerical method for the approximation of reachable sets of linear control systems is discussed. First a continuous system is transformed into a discrete one with Runge–Kutta methods. Then based on Benson’s outer approximation algorithm for solving multiobjective optimization problems, we propose a variant of Benson’s algorithm to sandwich the reachable set of the discrete system with an inner approximation and an outer approximation. By specifying an approximation error, the quality of the approximations measured in Hausdorff distance can be directly controlled. Furthermore, we use an illustrative example to demonstrate the working of the algorithm. Finally, computational experiments illustrate the superior performance of our proposed algorithm compared to a recent algorithm in the literature.

Inner approximation algorithms for optimization over the weakly efficient set

1999 ◽  
Vol 32 (2) ◽  
pp. 5915-5920
Author(s):  
Syuuji Yamada ◽  
Tetsuzo Tanino ◽  
Masahiro Inuiguchi
Keyword(s):  
Approximation Algorithms ◽  
Efficient Set ◽  
Inner Approximation ◽  
Weakly Efficient Set

scholarly journals Max-Min Rate Fairness Optimization for In-Band Full-Duplex IoT Networks: User Grouping and Power Control

2020 ◽  
Author(s):  
Ngo Tan Vu Khanh
Keyword(s):  
Optimal Solution ◽  
Base Station ◽  
Wireless Channel ◽  
Full Duplex ◽  
Full Potential ◽  
Traffic Demand ◽  
User Grouping ◽  
Grouping Method ◽  
Network Interference ◽  
Inner Approximation

The skyrocketing growth in the number of Internet of Things (IoT) devices will certainly pose a huge traffic demand for fifth-generation (5G) wireless networks and beyond. In-band full-duplex (IBFD), which is theoretically expected to double the spectral efficiency of a half-duplex (HD) wireless channel and to connect more devices, has been considered as a promising technology to accelerate the development of IoT. To exploit the full potential of IBFD, the key challenge is how to handle network interference (including self-interference, co-channel interference and multiuser interference) more effectively. In this paper, we propose a simple yet efficient user grouping method, where a base station (BS) serves strong downlink users and weak uplink users and vice versa in different frequency bands, mitigating severe network interference. We aim to maximize a minimum rate among all users subject to bandwidth and power constraints, which is formulated as a highly nonconvex optimization problem. By leveraging inner approximation framework, we develop a very efficient iterative algorithm to solve this problem, which guarantees at least a local optimal solution. Numerical results are provided to show not only the benefit of using full-duplex raido at BS, but also the advantage of the proposed user grouping method.

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