scholarly journals Robust Linear Programming with Norm Uncertainty

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Lei Wang ◽  
Hong Luo
Keyword(s):  
Linear Programming ◽  
Convex Optimization ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Optimization Problem ◽  
Convex Optimization Problem ◽  
Robust Solution ◽  
Normal Distributions ◽  
Robust Counterpart ◽  
Uncertainty Set

We consider the linear programming problem with uncertainty set described byp,w-norm. We suggest that the robust counterpart of this problem is equivalent to a computationally convex optimization problem. We provide probabilistic guarantees on the feasibility of an optimal robust solution when the uncertain coefficients obey independent and identically distributed normal distributions.

scholarly journals A multi-user multi-operator computing pricing method for Internet of things based on bi-level optimization

2020 ◽  
Vol 16 (1) ◽  
pp. 155014771990011
Author(s):  
Wei Jiang ◽  
Huiqiang Wang ◽  
Bingyang Li ◽  
Haibin Lv ◽  
Qingchuan Meng
Keyword(s):  
Decision Making ◽  
Linear Programming ◽  
Internet Of Things ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Optimization Problem ◽  
Time Data ◽  
Integer Linear Programming Problem ◽  
User Tasks ◽  
Computing Capacity

The Internet of mobile things is a promising paradigm that generates, stores, and processes amount of real-time data to render rich services for mobile users. Along with the increase of mobile devices in the field of Internet of things, more and more intelligent applications, such as face recognition and virtual reality, have emerged. These applications typically consume large amounts of computing and energy resources. However, due to the physical size limitations of Internet of things terminals, their computing capacity and power are limited, where users’ needs for application processing delay and power consumption cannot be met. Therefore, the concept of edge cloud computing has been proposed, which enhances the computing capacity of Internet of things terminals by offloading user tasks to edge servers for computation. When there are multiple operators, it is important to understand how users choose an operator to perform computation and how operators can reasonably price the computing capacity to meet their own interests. Therefore, we study the computation pricing and user decision-making problems of Internet of things under multi-user and multi-operator scenarios. The problem is divided into three phases and modeled as a two-level optimization problem. While an operator’s goal is to minimize the loss of his interests, the user’s goal is to minimize the computation cost (energy consumption and price). First, since the lower-level user decision-making problem is an integer linear programming problem, we transform it into an equivalent continuous linear programming problem by relaxation. Second, we transform the bi-level optimization problem into an equivalent single-level optimization problem by substituting the lower problem’s Karush–Kuhn–Tucker conditions into an upper problem. Finally, we use a spatial branch and bound algorithm to solve the problem. Experimental results show that the proposed algorithm can effectively maintain the benefits of both operators and users in the field of Internet of things.

scholarly journals Convex projection and convex multi-objective optimization

2021 ◽  
Author(s):  
Gabriela Kováčová ◽  
Birgit Rudloff
Keyword(s):  
Linear Programming ◽  
Approximate Solution ◽  
Convex Optimization ◽  
Optimization Problem ◽  
Approximate Solutions ◽  
Convex Optimization Problem ◽  
Multi Objective ◽  
Convex Case ◽  
Projection Problem ◽  
Polyhedral Projection

AbstractIn this paper we consider a problem, called convex projection, of projecting a convex set onto a subspace. We will show that to a convex projection one can assign a particular multi-objective convex optimization problem, such that the solution to that problem also solves the convex projection (and vice versa), which is analogous to the result in the polyhedral convex case considered in Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016). In practice, however, one can only compute approximate solutions in the (bounded or self-bounded) convex case, which solve the problem up to a given error tolerance. We will show that for approximate solutions a similar connection can be proven, but the tolerance level needs to be adjusted. That is, an approximate solution of the convex projection solves the multi-objective problem only with an increased error. Similarly, an approximate solution of the multi-objective problem solves the convex projection with an increased error. In both cases the tolerance is increased proportionally to a multiplier. These multipliers are deduced and shown to be sharp. These results allow to compute approximate solutions to a convex projection problem by computing approximate solutions to the corresponding multi-objective convex optimization problem, for which algorithms exist in the bounded case. For completeness, we will also investigate the potential generalization of the following result to the convex case. In Löhne and Weißing (Math Methods Oper Res 84(2):411–426, 2016), it has been shown for the polyhedral case, how to construct a polyhedral projection associated to any given vector linear program and how to relate their solutions. This in turn yields an equivalence between polyhedral projection, multi-objective linear programming and vector linear programming. We will show that only some parts of this result can be generalized to the convex case, and discuss the limitations.

scholarly journals Numerical Method for Solving the Robust Continuous-Time Linear Programming Problems

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 435
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Linear Programming ◽  
Robust Optimization ◽  
Numerical Method ◽  
Programming Problem ◽  
Continuous Time ◽  
Linear Programming Problem ◽  
Optimal Solution ◽  
Robust Counterpart ◽  
Linear Programming Problems ◽  
Time Linear

A robust continuous-time linear programming problem is formulated and solved numerically in this paper. The data occurring in the continuous-time linear programming problem are assumed to be uncertain. In this paper, the uncertainty is treated by following the concept of robust optimization, which has been extensively studied recently. We introduce the robust counterpart of the continuous-time linear programming problem. In order to solve this robust counterpart, a discretization problem is formulated and solved to obtain the ϵ -optimal solution. The important contribution of this paper is to locate the error bound between the optimal solution and ϵ -optimal solution.

scholarly journals Robust Solutions for Uncertain Continuous-Time Linear Programming Problems with Time-Dependent Matrices

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 885
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Continuous Time ◽  
Linear Programming Problem ◽  
Computational Procedure ◽  
Time Dependent ◽  
Optimal Solutions ◽  
Robust Counterpart ◽  
Linear Programming Problems ◽  
Time Linear

The uncertainty for the continuous-time linear programming problem with time-dependent matrices is considered in this paper. In this case, the robust counterpart of the continuous-time linear programming problem is introduced. In order to solve the robust counterpart, it will be transformed into the conventional form of the continuous-time linear programming problem with time-dependent matrices. The discretization problem is formulated for the sake of numerically calculating the ϵ-optimal solutions, and a computational procedure is also designed to achieve this purpose.

scholarly journals An Efficient Context-Aware Privacy Preserving Approach for Smartphones

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Lichen Zhang ◽  
Yingshu Li ◽  
Liang Wang ◽  
Junling Lu ◽  
Peng Li ◽  
...  
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Privacy Preservation ◽  
Optimization Problem ◽  
Privacy Preserving ◽  
Context Aware ◽  
Active Defense ◽  
Temporal Correlations ◽  
Improved Performance

With the proliferation of smartphones and the usage of the smartphone apps, privacy preservation has become an important issue. The existing privacy preservation approaches for smartphones usually have less efficiency due to the absent consideration of the active defense policies and temporal correlations between contexts related to users. In this paper, through modeling the temporal correlations among contexts, we formalize the privacy preservation problem to an optimization problem and prove its correctness and the optimality through theoretical analysis. To further speed up the running time, we transform the original optimization problem to an approximate optimal problem, a linear programming problem. By resolving the linear programming problem, an efficient context-aware privacy preserving algorithm (CAPP) is designed, which adopts active defense policy and decides how to release the current context of a user to maximize the level of quality of service (QoS) of context-aware apps with privacy preservation. The conducted extensive simulations on real dataset demonstrate the improved performance of CAPP over other traditional approaches.

scholarly journals A comparative study on interval arithmetic operations with intuitionistic fuzzy numbers for solving an intuitionistic fuzzy multi–objective linear programming problem

2017 ◽  
Vol 27 (3) ◽  
pp. 563-573 ◽  
Author(s):  
Rajendran Vidhya ◽  
Rajkumar Irene Hepzibah
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Fuzzy Numbers ◽  
Optimization Method ◽  
Arithmetic Operations ◽  
Intuitionistic Fuzzy Numbers ◽  
Multi Objective ◽  
Intuitionistic Fuzzy ◽  
Interval Valued

AbstractIn a real world situation, whenever ambiguity exists in the modeling of intuitionistic fuzzy numbers (IFNs), interval valued intuitionistic fuzzy numbers (IVIFNs) are often used in order to represent a range of IFNs unstable from the most pessimistic evaluation to the most optimistic one. IVIFNs are a construction which helps us to avoid such a prohibitive complexity. This paper is focused on two types of arithmetic operations on interval valued intuitionistic fuzzy numbers (IVIFNs) to solve the interval valued intuitionistic fuzzy multi-objective linear programming problem with pentagonal intuitionistic fuzzy numbers (PIFNs) by assuming differentαandβcut values in a comparative manner. The objective functions involved in the problem are ranked by the ratio ranking method and the problem is solved by the preemptive optimization method. An illustrative example with MATLAB outputs is presented in order to clarify the potential approach.

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