Interdependence of Methods and Representations in Design of Software for Combinatorial Optimization (original) (raw)

Methods and Models for Combinatorial Optimization Modeling by Linear Programming

2017

Linear programming models are a special class of mathematical programming models. A mathematical programming model is used to describe the characteristics of the optimal solution of an optimization problem by means of mathematical relations. Besides giving a formal description of the problem, the model constitutes the basis for the application of standard optimization algorithms (available as algebraic modeling systems and optimization software) capable of finding an optimal solution. In the following we review the elements of a mathematical programming model; as an example, we will refer to the following simple optimization problem:

Development tools for problems of combinatorial optimization

It is known that the majority of combinatorial tasks can be formulated on logic (Boolean, ternary or some other) matrices, which further might be used to solve many problems of discrete optimization, such as finding the shortest and longest path in a graph, map coloring, set partitioning, etc. These problems appear in particular in embedded control systems that deal with technological processes (such as transfer line, assembly line, etc.), traffic management and other application areas. The paper presents a formal description of matrix combinatorial problems and suggests dynamically reconfigurable (FPGA-based) parallel computational devices that allow to solve them. Synthesis of these devices can be performed with the aid of software tools developed for PC computers in Visual C++.

Formalization and Classification of Combinatorial Optimization Problems

Optimization Methods and Applications

Students' solutions of enumerative combinatorial problems may be assessed along two main dimensions: the correctness of the solution and the method of enumeration. This study looks at the second dimension with reference to the Cartesian product of two sets, and at the 'odometer' combinatorial strategy defined by English (1991). Since we are not aware of any algorithm-based methods suitable for analysing combinatorial strategies on a large-scale sample, in this study we endeavour to formalize the odometer strategy and recommend a method of algorithm-based classification of solutions according to the strategy used. In the paper (1) odometer thinking is described using a formula based on its definition, and (2) constancy and cyclicity are characterized using mathematical formulae, which are then used to describe odometer thinking in a computationally efficient manner ('odometricality'). Our hypothesis, i.e. that odometer thinking may be approximated by the odometricality index, is successfully tested on a random sample of automatically generated solutions (n=10,000) by calculating the correlation between odometricality and the formal measure of odometer thinking. Finally, we offer a method (and R script) for classifying strategy use.

OptFrame: a Computational Framework for Combinatorial Optimization Problems

sobrapo.org.br

This work presents OptFrame, a computational framework for the development of efficient heuristic based algorithms. The objective is to provide a simple C++ interface for common components of trajectory and population based metaheuristics, in order to solve combinatorial optimization problems. Since many methods are very common in literature, we provide efficient implementations for simple versions of these methods but the user can develop "smarter" versions of the methods considering problem-specific characteristics. Moreover, parallel support for both shared-memory and distributed-memory computers is provided. OptFrame has been successfully applied to model and solve some combinatorial problems, showing a good balance between flexibility and efficiency.

A Software Framework for Solving Combinatorial Optimization Tasks

Due to the major practical importance of combinatorial optimization problems, many approaches for tackling them have been developed. As the problem of intelligent solution generation can be approached with reinforcement learning techniques, we aim at presenting in this paper a programming interface for solving combinatorial optimization problems using reinforcement learning techniques. The advantages of the proposed framework are emphasized, highlighting the potential of using reinforcement learning for solving optimization tasks. An experiment for solving the bidimensional protein folding problem developed using the designed interface is also presented.

A unified modeling and solution framework for combinatorial optimization problems

Or Spektrum, 2004

Combinatorial optimization problems are often too complex to be solved within reasonable time limits by exact methods, in spite of the theoretical guarantee that such methods will ultimately obtain an optimal solution. Instead, heuristic methods, which do not offer a convergence guarantee, but which have greater flexibility to take advantage of special properties of the search space, are commonly a preferred alternative. The standard procedure is to craft a heuristic method to suit the particular characteristics of the problem at hand, exploiting to the extent possible the structure available. Such tailored methods, however, typically have limited usefulness in other problems domains.