scholarly journals Output-Space Branch-and-Bound Reduction Algorithm for a Class of Linear Multiplicative Programs

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 315
Author(s):  
Bo Zhang ◽  
Yuelin Gao ◽  
Xia Liu ◽  
Xiaoli Huang
Keyword(s):  
Linear Programming ◽  
Branch And Bound ◽  
Numerical Experiments ◽  
Dimensional Space ◽  
Equivalence Problem ◽  
Reduction Technique ◽  
Reduction Algorithm ◽  
Advantages And Disadvantages ◽  
Multiplicative Programs ◽  
Output Space

In this paper, a new relaxation bounding method is proposed for a class of linear multiplicative programs. Although the 2 p − 1 variable is introduced in the construction of equivalence problem, the branch process of the algorithm is only carried out in p − dimensional space. In addition, a super-rectangular reduction technique is also given to greatly improve the convergence rate. Furthermore, we construct an output-space branch-and-bound reduction algorithm based on solving a series of linear programming sub-problems, and prove the convergence and computational complexity of the algorithm. Finally, to verify the feasibility and effectiveness of the algorithm, we carried out a series of numerical experiments and analyzed the advantages and disadvantages of the algorithm by numerical results.

scholarly journals An Effective Computational Algorithm for the Global Solution of a Class of Linear Fractional Programming

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
XiaoLi Huang ◽  
YueLin Gao ◽  
Bo Zhang ◽  
Xia Liu
Keyword(s):  
Objective Function ◽  
Global Solution ◽  
Branch And Bound ◽  
Fractional Programming ◽  
Numerical Experiments ◽  
Computational Algorithm ◽  
Decision Variable ◽  
Linear Fractional Programming ◽  
Algorithm Framework ◽  
Output Space

For the minimization of the sum of linear fractions on polyhedra, it is likewise a class of linear fractional programming (LFP). In this paper, we mainly propose a new linear relaxation technique and combine the branch-and-bound algorithm framework to solve the LFP globally. It is worthwhile to mention that the branching operation of the algorithm occurs in the relatively small output space of the dimension rather than the space where the decision variable is located. When the number of linear fractions in the objective function is much lower than the dimension of the decision variable, the performance of the algorithm is better. After that, we also explain the effectiveness, feasibility, and other performances of the algorithm through numerical experiments.

scholarly journals Sound Localization for Ad-Hoc Microphone Arrays

Energies ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3446
Author(s):  
Muhammad Usman Liaquat ◽  
Hafiz Suliman Munawar ◽  
Amna Rahman ◽  
Zakria Qadir ◽  
Abbas Z. Kouzani ◽  
...  
Keyword(s):  
Sound Localization ◽  
Sound Source ◽  
Ad Hoc ◽  
Dimensional Space ◽  
Computation Time ◽  
Location Estimation ◽  
Microphone Arrays ◽  
Sound Signal ◽  
Experimental Arrangement ◽  
Advantages And Disadvantages

Sound localization is a field of signal processing that deals with identifying the origin of a detected sound signal. This involves determining the direction and distance of the source of the sound. Some useful applications of this phenomenon exists in speech enhancement, communication, radars and in the medical field as well. The experimental arrangement requires the use of microphone arrays which record the sound signal. Some methods involve using ad-hoc arrays of microphones because of their demonstrated advantages over other arrays. In this research project, the existing sound localization methods have been explored to analyze the advantages and disadvantages of each method. A novel sound localization routine has been formulated which uses both the direction of arrival (DOA) of the sound signal along with the location estimation in three-dimensional space to precisely locate a sound source. The experimental arrangement consists of four microphones and a single sound source. Previously, sound source has been localized using six or more microphones. The precision of sound localization has been demonstrated to increase with the use of more microphones. In this research, however, we minimized the use of microphones to reduce the complexity of the algorithm and the computation time as well. The method results in novelty in the field of sound source localization by using less resources and providing results that are at par with the more complex methods requiring more microphones and additional tools to locate the sound source. The average accuracy of the system is found to be 96.77% with an error factor of 3.8%.

scholarly journals How Can I Grab That?

i-com ◽  
2020 ◽  
Vol 19 (2) ◽  
pp. 67-85
Author(s):  
Matthias Weise ◽  
Raphael Zender ◽  
Ulrike Lucke
Keyword(s):  
Virtual Reality ◽  
User Experience ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Advantages And Disadvantages ◽  
Multiple Dimensions ◽  
Application Developers ◽  
Three Dimensional Space

AbstractThe selection and manipulation of objects in Virtual Reality face application developers with a substantial challenge as they need to ensure a seamless interaction in three-dimensional space. Assessing the advantages and disadvantages of selection and manipulation techniques in specific scenarios and regarding usability and user experience is a mandatory task to find suitable forms of interaction. In this article, we take a look at the most common issues arising in the interaction with objects in VR. We present a taxonomy allowing the classification of techniques regarding multiple dimensions. The issues are then associated with these dimensions. Furthermore, we analyze the results of a study comparing multiple selection techniques and present a tool allowing developers of VR applications to search for appropriate selection and manipulation techniques and to get scenario dependent suggestions based on the data of the executed study.

Numerical Experiments in Using Voronoi Cell Finite Elements for Topology Optimization

2009 ◽  
Author(s):  
James M. Gibert ◽  
Georges M. Fadel
Keyword(s):  
Topology Optimization ◽  
Isotropic Material ◽  
Numerical Experiments ◽  
Voronoi Cell ◽  
Material Model ◽  
Topological Optimization ◽  
Advantages And Disadvantages ◽  
Compliance Minimization ◽  
Design Variables ◽  
Standard Linear

This paper provides two separate methodologies for implementing the Voronoi Cell Finite Element Method (VCFEM) in topological optimization. Both exploit two characteristics of VCFEM. The first approach utilizes the property that a hole or inclusion can be placed in the element: the design variables for the topology optimization are sizes of the hole. In the second approach, we note that VCFEM may mesh the design domain as n sided polygons. We restrict our attention to hexagonal meshes of the domain while applying Solid Isotropic Material Penalization (SIMP) material model. Researchers have shown that hexagonal meshes are not subject to the checker boarding problem commonly associated with standard linear quad and triangle elements. We present several examples to illustrate the efficacy of the methods in compliance minimization as well as discuss the advantages and disadvantages of each method.

scholarly journals A Global Optimization Algorithm for Sum of Linear Ratios Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yuelin Gao ◽  
Siqiao Jin
Keyword(s):  
Global Optimization ◽  
Lower Bound ◽  
Programming Problem ◽  
Branch And Bound ◽  
Numerical Experiments ◽  
Original Problem ◽  
Bilinear Programming ◽  
Global Optimization Algorithm ◽  
Product Function ◽  
Optimal Value

We equivalently transform the sum of linear ratios programming problem into bilinear programming problem, then by using the linear characteristics of convex envelope and concave envelope of double variables product function, linear relaxation programming of the bilinear programming problem is given, which can determine the lower bound of the optimal value of original problem. Therefore, a branch and bound algorithm for solving sum of linear ratios programming problem is put forward, and the convergence of the algorithm is proved. Numerical experiments are reported to show the effectiveness of the proposed algorithm.

scholarly journals Penjadwalan Integer Linear Programming pada Penjadwalan Produksi Tipe Flowshop dan Program Optimasi Waktu dengan Metode Branch and Bound

2021 ◽  
Vol 3 (1) ◽  
pp. 44-51
Author(s):  
Ismawati Khotimah ◽  
Hagni Wijayanti ◽  
Sri Setyaningsih
Keyword(s):  
Linear Programming ◽  
Branch And Bound ◽  
Integer Linear Programming

Penjadwalan merupakan pengalokasian sumber daya yang tersedia untuk menyelesaikan sejumlah pekerjaan dengan mempertimbangkan batasan yang ada. Hal yang cukup penting dalam perusahaan adalah menentukan penjadwalan yang optimal agar kegiatan produksi dapat berjalan dengan lancar, efisien, dan sistematis. PT Unitex merupakan perusahaan yang bergerak dalam bidang tekstil. Perusahaan ini menerapkan penjadwalan yang kurang efisien sehingga menyebabkan permasalahan seperti keterlambatan waktu dalam penyelesaian produksi akibat terlalu banyaknya permintaan konsumen yang tidak menentu. Perusahaan melakukan penjadwalan hanya diperkirakan saja dan tidak menggunakan metode ilmiah. Metode Branch and Bound dapat diterapkan pada permasalahan tersebut karena dapat menentukan penjadwalan yang tepat dengan waktu optimal. Metode Branch and Bound adalah metode pencarian dalam menentukan solusi optimal pada penjadwalan dengan menentukan nilai batas atas dan nilai  batas bawah untuk menghasilkan nilai makespan dari tiap job yang dikerjakan. Penjadwalan menggunakan Metode Branch and Bound menghasilkan urutan penjadwalan produksi dengan job 1-5-3-6-7-2-4 dengan makespan sebesar 17290.73 menit. Nilai ini lebih kecil dibandingkan hasil makespan perusahaan yaitu 19278.13 menit. Hal tersebut meminimumkan makespan sebesar 10.31%.

scholarly journals Verifying a Solver for Linear Mixed Integer Arithmetic in Isabelle/HOL

2020 ◽  
pp. 233-250
Author(s):  
Ralph Bottesch ◽  
Max W. Haslbeck ◽  
Alban Reynaud ◽  
René Thiemann
Keyword(s):  
Linear Programming ◽  
Branch And Bound ◽  
Decision Procedure ◽  
Simplex Algorithm ◽  
Upper Bounds ◽  
Mixed Integer ◽  
Small Solution ◽  
Code Generator ◽  
Integer Arithmetic ◽  
Termination Proofs

AbstractWe implement a decision procedure for linear mixed integer arithmetic and formally verify its soundness in Isabelle/HOL. We further integrate this procedure into one application, namely into , a formally verified certifier to check untrusted termination proofs. This checking involves assertions of unsatisfiability of linear integer inequalities; previously, only a sufficient criterion for such checks was supported. To verify the soundness of the decision procedure, we first formalize the proof that every satisfiable set of linear integer inequalities also has a small solution, and give explicit upper bounds. To this end we mechanize several important theorems on linear programming, including statements on integrality and bounds. The procedure itself is then implemented as a branch-and-bound algorithm, and is available in several languages via Isabelle’s code generator. It internally relies upon an adapted version of an existing verified incremental simplex algorithm.

Three-dimensional path planning for unmanned aerial vehicle based on linear programming

Robotica ◽  
2011 ◽  
Vol 30 (5) ◽  
pp. 773-781 ◽  
Author(s):  
Yang Chen ◽  
Jianda Han ◽  
Xingang Zhao
Keyword(s):  
Linear Programming ◽  
Path Planning ◽  
Real Time ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Linear Constraints ◽  
Flying Height ◽  
Ant Colony Optimization Algorithm ◽  
Optimal Velocity ◽  
Aerial Vehicle

SUMMARYIn this paper, an approach based on linear programming (LP) is proposed for path planning in three-dimensional space, in which an aerial vehicle is requested to pursue a target while avoiding static or dynamic obstacles. This problem is very meaningful for many aerial robots, such as unmanned aerial vehicles. First, the tasks of target-pursuit and obstacle-avoidance are modelled with linear constraints in relative coordination according to LP formulation. Then, two weighted cost functions, representing the optimal velocity resolution, are integrated into the final objective function. This resolution, defined to achieve the optimal velocity, deals with the optimization of a pair of orthogonal vectors. Some constraints, such as boundaries of the vehicle velocity, acceleration, sensor range, and flying height, are considered in this method. A number of simulations, under static and dynamic environments, are carried out to validate the performance of generating optimal trajectory in real time. Compared with ant colony optimization algorithm and genetic algorithm, our method has less parameters to tune and can achieve better performance in real-time application.

A Criterion Space Branch-and-Cut Algorithm for Mixed Integer Bilinear Maximum Multiplicative Programs

2022 ◽  
Author(s):  
Vahid Mahmoodian ◽  
Iman Dayarian ◽  
Payman Ghasemi Saghand ◽  
Yu Zhang ◽  
Hadi Charkhgard
Keyword(s):  
Objective Function ◽  
Branch And Bound ◽  
Capacity Allocation ◽  
Branch And Cut ◽  
Nash Bargaining ◽  
Mixed Integer ◽  
Bargaining Solution ◽  
Reliability Optimization ◽  
Criterion Space ◽  
Multiplicative Programs

This study introduces a branch-and-bound algorithm to solve mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). This class of optimization problems arises in many applications, such as finding a Nash bargaining solution (Nash social welfare optimization), capacity allocation markets, reliability optimization, etc. The proposed algorithm applies multiobjective optimization principles to solve MIBL-MMPs exploiting a special characteristic in these problems. That is, taking each multiplicative term in the objective function as a dummy objective function, the projection of an optimal solution of MIBL-MMPs is a nondominated point in the space of dummy objectives. Moreover, several enhancements are applied and adjusted to tighten the bounds and improve the performance of the algorithm. The performance of the algorithm is investigated by 400 randomly generated sample instances of MIBL-MMPs. The obtained result is compared against the outputs of the mixed-integer second order cone programming (SOCP) solver in CPLEX and a state-of-the-art algorithm in the literature for this problem. Our analysis on this comparison shows that the proposed algorithm outperforms the fastest existing method, that is, the SOCP solver, by a factor of 6.54 on average. Summary of Contribution: The scope of this paper is defined over a class of mixed-integer programs, the so-called mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). The importance of MIBL-MMPs is highlighted by the fact that they are encountered in applications, such as Nash bargaining, capacity allocation markets, reliability optimization, etc. The mission of the paper is to introduce a novel and effective criterion space branch-and-cut algorithm to solve MIBL-MMPs by solving a finite number of single-objective mixed-integer linear programs. Starting with an initial set of primal and dual bounds, our proposed approach explores the efficient set of the multiobjective problem counterpart of the MIBL-MMP through a criterion space–based branch-and-cut paradigm and iteratively improves the bounds using a branch-and-bound scheme. The bounds are obtained using novel operations developed based on Chebyshev distance and piecewise McCormick envelopes. An extensive computational study demonstrates the efficacy of the proposed algorithm.

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