Nonlinear Integer Programming and Discrete Optimization

1983 ◽  
Vol 105 (2) ◽  
pp. 160-164 ◽  
Author(s):  
Omprakash K. Gupta ◽  
A. Ravindran
Keyword(s):  
Integer Programming ◽  
Branch And Bound ◽  
Discrete Optimization ◽  
Optimization Problems ◽  
Computer Code ◽  
Nonlinear Integer Programming ◽  
Mixed Integer ◽  
Test Problems ◽  
Branch And Bound Method ◽  
Discrete Optimization Problems

Branch and bound has been widely recognized as an effective method in solving linear integer programming problems. This paper presents a study on the feasibility of the branch and bound method in solving general nonlinear mixed integer programming and discrete optimization problems. First, a description of the branch and bound method as applied to the integer case is given. Next, a computer code BBNLMIP is developed to carry out an experimental study on 22 test problems. The numerical results indicate the effect of the problem parameters such as number of integer variables and constraints. Finally, a method for extending the branch and bound principle to solve nonlinear discrete optimization problems is described.

scholarly journals On the Classical Version of the Branch and Bound Method

2021 ◽  
pp. 21-44
Author(s):  
Boris Melnikov ◽  
◽  
Elena Melnikova ◽  
Keyword(s):  
Traveling Salesman Problem ◽  
Branch And Bound ◽  
Discrete Optimization ◽  
Optimization Problems ◽  
Traveling Salesman ◽  
Branch And Bound Method ◽  
Main Subject ◽  
Classical Version ◽  
The Traveling Salesman Problem ◽  
Discrete Optimization Problems

In the computer literature, a lot of problems are described that can be called discrete optimization problems: from encrypting information on the Internet (including creating programs for digital cryptocurrencies) before searching for “interests” groups in social networks. Often, these problems are very difficult to solve on a computer, hence they are called “intractable”. More precisely, the possible approaches to quickly solving these problems are difficult to solve (to describe algorithms, to program); the brute force solution, as a rule, is programmed simply, but the corresponding program works much slower. Almost every one of these intractable problems can be called a mathematical model. At the same time, both the model itself and the algorithms designed to solve it are often created for one subject area, but they can also be used in many other areas. An example of such a model is the traveling salesman problem. The peculiarity of the problem is that, given the relative simplicity of its formulation, finding the optimal solution (the optimal route). This problem is very difficult and belongs to the so-called class of NP-complete problems. Moreover, according to the existing classification, the traveling salesman problem is an example of an optimization problem that is an example of the most complex subclass of this class. However, the main subject of the paper is not the problem, but the method of its soluti- on, i.e. the branch and bound method. It consists of several related heuristics, and in the monographs, such a multi-heuristic branch and bound method was apparently not previously noted: the developers of algorithms and programs should have understood this themselves. At the same time, the method itself can be applied (with minor changes) to many other discrete optimization problems. So, the classical version of branch and bound method is the main subject of this paper, but also important is the second subject, i.e. the traveling salesman problem, also in the classical formulation. The paper deals with the application of the branch and bound method in solving the traveling salesman problem, and about this application, we can also use the word “classical”. However, in addition to the classic version of this implementation, we consider some new heuristics, related to the need to develop real-time algorithms.

scholarly journals Investigation of the estimates of the length of parallel alignment of the vertices of the graph

2018 ◽  
Author(s):  
V. A. Turchina ◽  
K. D. Karavaev
Keyword(s):  
Exact Solution ◽  
Branch And Bound ◽  
Discrete Optimization ◽  
Optimization Problems ◽  
Fixed Number ◽  
Minimum Length ◽  
Branch And Bound Method ◽  
Optimal Sequencing ◽  
Time Expenditures ◽  
The Impact

A number of practical tasks require minimizing the human and material resources that are involved in tasks or time expenditures. A special place in this class of problems is occupied by theoretical problems that have a broad practical application, which belong to a class of discrete optimization problems. When minimizing time expenditures in such problems the question of determining the optimal sequencing of execution of a finite set of works (tasks, operations, projects, etc.) is raised. This sequencing can be linear, circular or parallel. The latter is considered by the authors. This article is devoted to the analysis of one of the problems of discrete optimization, which belongs to the class of problems of the scheduling theory, and, taking into account its specificity, can be considered as an optimization graph problem. Specifically, in terms of the theory of graphs, the problem of finding a parallel sequencing of vertices of a given graph of minimum length, in which at each place there is no more than a given fixed number of vertices, is under consideration. Since this problem is NP-hard, its exact solution can be found by using one of the methods that implements state search scheme. The authors investigated the impact of the accuracy of the estimation of the length of optimal sequencing on the rate of finding the solution by using one of the most common methods, namely the branch and bound method. As a result, an improved lower-bound estimate of time expenditures was obtained and an upper-bound estimate was proposed. The latter was used to justify the relationship of the problem under consideration with the inverse one. Also, on the basis of the computational experiment results were obtained that refuted the a priori consideration about the impact of the accuracy of the estimation on the rate of finding the exact solution by using the branch and bound method

Optimal Distributed Generator Allocation in Active Distribution Network Considering Reconfiguration

2014 ◽  
Vol 672-674 ◽  
pp. 1117-1122
Author(s):  
Yu De Yang ◽  
Hong Bo Xie
Keyword(s):  
Integer Programming ◽  
Mixed Integer Programming ◽  
Branch And Bound ◽  
Distribution Network ◽  
Mixed Integer ◽  
Branch And Bound Method ◽  
Allocation Model ◽  
Distributed Generator ◽  
Active Distribution Network ◽  
Planning And Operation

For the structure of active distribution network can flexibly adjust, an optimal distributed generator allocation model considering reconfiguration is proposed. This model is implemented in the GAMS software and solved using SBB solver based on branch and bound method, due to it is a 0-1 nonlinear mixed integer programming. IEEE 33 standard example is used to simulate for this model. Results show that reconfiguration can increase the capacity of absorbing DG, providing thoughts and guidance for confirming the capacity of DG, and for planning and operation of active distribution network.

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