Matrix Perturbation and Optimal Partition Invariancy in Linear Optimization

2015 ◽  
Vol 32 (03) ◽  
pp. 1550013 ◽  
Author(s):  
Alireza Ghaffari-Hadigheh ◽  
Nayyer Mehanfar
Keyword(s):  
Linear Optimization ◽  
Optimal Solution ◽  
Singular Values ◽  
Coefficient Matrix ◽  
Optimal Partition ◽  
Matrix Perturbation ◽  
Optimal Value ◽  
Special Perturbation ◽  
Transition Points ◽  
The Given

Understanding the effect of variation of the coefficient matrix in linear optimization problem on the optimal solution and the optimal value function has its own importance in practice. However, most of the published results are on the effect of this variation when the current optimal solution is a basic one. There is only a study of the problem for special perturbation on the coefficient matrix, when the given optimal solution is strictly complementary and the optimal partition (in some sense) is known. Here, we consider an arbitrary direction for perturbation of the coefficient matrix and present an effective method based on generalized inverse and singular values to detect invariancy intervals and corresponding transition points.

Linear Optimization in Uncertain Environment: Sensitivity of the Solution to the Belief Degree

2021 ◽  
pp. 2150005
Author(s):  
Alireza Ghaffari-Hadigheh
Keyword(s):  
Uncertainty Theory ◽  
Optimization Problem ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Parametric Programming ◽  
Point Of View ◽  
Linear Programs ◽  
Optimal Partition ◽  
Linear Optimization Problem ◽  
Uncertain Environment

Uncertainty theory has been initiated in 2007 by Liu, as an axiomatically developed notion, which considers the uncertainty on data as a belief degree on the domain expert’s opinion. Uncertain linear optimization is devised to model linear programs in an uncertain environment. In this paper, we investigate the relation between uncertain linear optimization and parametric programming. It is denoted that the problem can be converted to parametric linear optimization problem, at which belief degrees play the role of parameters, and parametric linear optimization with its rich literature provides insightful interpretations. In a point of view, a strictly complementary optimal solution of problem is known for the belief degree [Formula: see text], as well as the associated optimal partition. One may be interested in knowing the region of belief degrees (parameters) where this optimal partition remains invariant for all parameter values (belief degrees) in this region. We consider the linear optimization problem with uncertain rim data, i.e., the right-hand side and the objective function data. The known results in the literature are translated to the language of uncertainty theory, and managerial interpretations are provided. The methodology is illustrated via concrete examples.

scholarly journals Approaching the Optimal Solution of the Maximal α-quasi-clique Local Community Problem

Electronics ◽  
2020 ◽  
Vol 9 (9) ◽  
pp. 1438
Author(s):  
Patricia Conde-Cespedes
Keyword(s):  
Graph Theory ◽  
Community Detection ◽  
Local Community ◽  
Optimal Solution ◽  
Local Communities ◽  
Optimal Value ◽  
Interesting Task ◽  
Local Community Detection ◽  
Networks Analysis ◽  
Node Problem

Complex networks analysis (CNA) has attracted so much attention in the last few years. An interesting task in CNA complex network analysis is community detection. In this paper, we focus on Local Community Detection, which is the problem of detecting the community of a given node of interest in the whole network. Moreover, we study the problem of finding local communities of high density, known as α-quasi-cliques in graph theory (for high values of α in the interval ]0,1[). Unfortunately, the higher α is, the smaller the communities become. This led to the maximal α-quasi-clique community of a given node problem, which is, the problem of finding local communities that are α-quasi-cliques of maximal size. This problem is NP-hard, then, to approach the optimal solution, some heuristics exist. When α is high (>0.5) the diameter of a maximal α-quasi-clique is at most 2. Based on this property, we propose an algorithm to calculate an upper bound to approach the optimal solution. We evaluate our method in real networks and conclude that, in most cases, the bound is very accurate. Furthermore, for a real small network, the optimal value is exactly achieved in more than 80% of cases.

scholarly journals Optimal Control and Positional Controllability in a One-Sector Economy

Games ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 11
Author(s):  
Nikolai Grigorenko ◽  
Lilia Luk’yanova
Keyword(s):  
Optimal Control ◽  
Production Capacity ◽  
Quality Criterion ◽  
Zero Order ◽  
Suboptimal Control ◽  
Time Interval ◽  
Bounded Control ◽  
Optimal Value ◽  
Current Production ◽  
The Given

A model of production funds acquisition, which includes two differential links of the zero order and two series-connected inertial links, is considered in a one-sector economy. Zero-order differential links correspond to the equations of the Ramsey model. These equations contain scalar bounded control, which determines the distribution of the available funds into two parts: investment and consumption. Two series-connected inertial links describe the dynamics of the changes in the volume of the actual production at the current production capacity. For the considered control system, the problem is posed to maximize the average consumption value over a given time interval. The properties of optimal control are analytically established using the Pontryagin maximum principle. The cases are highlighted when such control is a bang-bang, as well as the cases when, along with bang-bang (non-singular) portions, control can contain a singular arc. At the same time, concatenation of singular and non-singular portions is carried out using chattering. A bang-bang suboptimal control is presented, which is close to the optimal one according to the given quality criterion. A positional terminal control is proposed for the first approximation when a suboptimal control with a given deviation of the objective function from the optimal value is numerically found. The obtained results are confirmed by the corresponding numerical calculations.

scholarly journals Data-based polyhedron model for optimization of engineering structures involving uncertainties

2021 ◽  
Vol 2 ◽  
Author(s):  
Zhiping Qiu ◽  
Han Wu ◽  
Isaac Elishakoff ◽  
Dongliang Liu
Keyword(s):  
Linear Programming ◽  
Convex Polyhedron ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Composite Plates ◽  
Convex Polyhedra ◽  
Chebyshev Inequality ◽  
Engineering Structures ◽  
Load Bearing ◽  
Polyhedron Model

Abstract This paper studies the data-based polyhedron model and its application in uncertain linear optimization of engineering structures, especially in the absence of information either on probabilistic properties or about membership functions in the fussy sets-based approach, in which situation it is more appropriate to quantify the uncertainties by convex polyhedra. Firstly, we introduce the uncertainty quantification method of the convex polyhedron approach and the model modification method by Chebyshev inequality. Secondly, the characteristics of the optimal solution of convex polyhedron linear programming are investigated. Then the vertex solution of convex polyhedron linear programming is presented and proven. Next, the application of convex polyhedron linear programming in the static load-bearing capacity problem is introduced. Finally, the effectiveness of the vertex solution is verified by an example of the plane truss bearing problem, and the efficiency is verified by a load-bearing problem of stiffened composite plates.

scholarly journals Sensor Location Problem for Network Traffic Flow Derivation Based on Turning Ratios at Intersection

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Minhua Shao ◽  
Lijun Sun ◽  
Xianzhi Shao
Keyword(s):  
Network Flow ◽  
Location Problem ◽  
Optimal Solution ◽  
Coefficient Matrix ◽  
Sensor Location ◽  
The Road ◽  
Minimum Number ◽  
Flow Conservation ◽  
The Greedy Algorithm ◽  
Counting Points

The sensor location problem (SLP) discussed in this paper is to find the minimum number and optimum locations of the flow counting points in the road network so that the traffic flows over the whole network can be inferred uniquely. Flow conservation system at intersections is formulated firstly using the turning ratios as the prior information. Then the coefficient matrix of the flow conservation system is proved to be nonsingular. Based on that, the minimal number of counting points is determined to be the total number of exclusive incoming roads and dummy roads, which are added to the network to represent the trips generated on real roads. So the task of SLP model based on turning ratios is just to determine the optimal sensor locations. The following analysis in this paper shows that placing sensors on all the exclusive incoming roads and dummy roads can always generate a unique network flow vector for any network topology. After that, a detection set composed of only real roads is proven to exist from the view of feasibility in reality. Finally, considering the roads importance and cost of the sensors, a weighted SLP model is formulated to find the optimal detection set. The greedy algorithm is proven to be able to provide the optimal solution for the proposed weighted SLP model.

scholarly journals A Novel Approach for Solving Nonsmooth Optimization Problems with Application to Nonsmooth Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Hamid Reza Erfanian ◽  
M. H. Noori Skandari ◽  
A. V. Kamyad
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Piecewise Linear Function ◽  
Nonsmooth Equations ◽  
Smooth Functions ◽  
Generalized Derivative ◽  
Novel Approach

We present a new approach for solving nonsmooth optimization problems and a system of nonsmooth equations which is based on generalized derivative. For this purpose, we introduce the first order of generalized Taylor expansion of nonsmooth functions and replace it with smooth functions. In other words, nonsmooth function is approximated by a piecewise linear function based on generalized derivative. In the next step, we solve smooth linear optimization problem whose optimal solution is an approximate solution of main problem. Then, we apply the results for solving system of nonsmooth equations. Finally, for efficiency of our approach some numerical examples have been presented.

scholarly journals Distributed Constrained Stochastic Subgradient Algorithms Based on Random Projection and Asynchronous Broadcast over Networks

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Junlong Zhu ◽  
Ping Xie ◽  
Qingtao Wu ◽  
Mingchuan Zhang ◽  
Ruijuan Zheng ◽  
...  
Keyword(s):  
Information Exchange ◽  
Optimal Solution ◽  
Random Projection ◽  
Learning Rate ◽  
Time Varying ◽  
Optimal Value ◽  
Constraint Set ◽  
Time Varying Networks ◽  
Subgradient Algorithms ◽  
Asymptotic Upper Bound

We consider a distributed constrained optimization problem over a time-varying network, where each agent only knows its own cost functions and its constraint set. However, the local constraint set may not be known in advance or consists of huge number of components in some applications. To deal with such cases, we propose a distributed stochastic subgradient algorithm over time-varying networks, where the estimate of each agent projects onto its constraint set by using random projection technique and the implement of information exchange between agents by employing asynchronous broadcast communication protocol. We show that our proposed algorithm is convergent with probability 1 by choosing suitable learning rate. For constant learning rate, we obtain an error bound, which is defined as the expected distance between the estimates of agent and the optimal solution. We also establish an asymptotic upper bound between the global objective function value at the average of the estimates and the optimal value.

Combo separability criteria and lower bound on concurrence

2021 ◽  
Author(s):  
Zangi Sultan ◽  
Jiansheng Wu ◽  
Cong-Feng Qiao
Keyword(s):  
Lower Bound ◽  
Correlation Matrix ◽  
Density Operator ◽  
Singular Values ◽  
Quantum Information Theory ◽  
Separability Criteria ◽  
The Given ◽  
Ky Fan ◽  
Natural Way ◽  
Extract Information

Abstract Detection and quantification of entanglement are extremely important in quantum information theory. We can extract information by using the spectrum or singular values of the density operator. The correlation matrix norm deals with the concept of quantum entanglement in a mathematically natural way. In this work, we use Ky Fan norm of the Bloch matrix to investigate the disentanglement of quantum states. Our separability criterion not only unifies some well-known criteria but also leads to a better lower bound on concurrence. We explain with an example how the entanglement of the given state is missed by existing criteria but can be detected by our criterion. The proposed lower bound on concurrence also has advantages over some investigated bounds.

scholarly journals Mixed integer nonlinear programming (MINLP)-based bandwidth utility function on internet pricing scheme with monitoring and marginal cost

2019 ◽  
Vol 9 (2) ◽  
pp. 1240
Author(s):  
Robinson Sitepu ◽  
Fitri Maya Puspita ◽  
Elika Kurniadi ◽  
Yunita Yunita ◽  
Shintya Apriliyani
Keyword(s):  
Utility Function ◽  
Marginal Cost ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Mixed Integer ◽  
Monitoring Cost ◽  
Internet Pricing ◽  
Pricing Scheme ◽  
Linear Optimization Problems

<span>The development of the internet in this era of globalization has increased fast. The need for internet becomes unlimited. Utility functions as one of measurements in internet usage, were usually associated with a level of satisfaction of users for the use of information services used. There are three internet pricing schemes used, that are flat fee, usage based and two-part tariff schemes by using one of the utility function which is Bandwidth Diminished with Increasing Bandwidth with monitoring cost and marginal cost. Internet pricing scheme will be solved by LINGO 13.0 in form of non-linear optimization problems to get optimal solution. The optimal solution is obtained using the either usage-based pricing scheme model or two-part tariff pricing scheme model for each services offered, if the comparison is with flat-fee pricing scheme. It is the best way for provider to offer network based on usage based scheme. The results show that by applying two part tariff scheme, the providers can maximize its revenue either for homogeneous or heterogeneous consumers.</span>

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