A Two-Stage Sequential Approximation Method for Non-Linear Discrete-Variable Optimization

1995 ◽  
Author(s):  
Yeh-Liang Hsu ◽  
Tzyh-Li Sun ◽  
Li-Hwang Leu
Keyword(s):  
Linear Programming ◽  
Approximation Method ◽  
Optimization Problems ◽  
Control Factor ◽  
Discrete Variable ◽  
Two Stage ◽  
Second Stage ◽  
Non Linear ◽  
Sequential Approximation ◽  
Discrete Variable Optimization

Abstract A two-stage sequential approximation method is developed for non-linear discrete-variable optimization. The concept of this technique is similar to that of sequential linear programming (SLP), only in each iteration, the linear programming subproblem in the first stage is modified into a discrete programming subproblem in the second stage in order to solve for a discrete solution. SLP is often impractical when applied to engineering optimization problems with implicit constraints, because of the difficulties in choosing proper move limits. For this reason, in the second stage a “boundary control factor” is introduced to augment the function of move limits. Several mechanical design optimization problems are presented to demonstrate this algorithm.

scholarly journals Two-stage multi-tasking transform framework for large-scale many-objective optimization problems

2021 ◽  
Author(s):  
Lu Chen ◽  
Handing Wang ◽  
Wenping Ma
Keyword(s):  
Large Scale ◽  
Original Problem ◽  
Optimization Problems ◽  
Reference Points ◽  
Optimization Strategy ◽  
Two Stage ◽  
Decision Space ◽  
Second Stage ◽  
Decision Variables ◽  
Objective Space

AbstractReal-world optimization applications in complex systems always contain multiple factors to be optimized, which can be formulated as multi-objective optimization problems. These problems have been solved by many evolutionary algorithms like MOEA/D, NSGA-III, and KnEA. However, when the numbers of decision variables and objectives increase, the computation costs of those mentioned algorithms will be unaffordable. To reduce such high computation cost on large-scale many-objective optimization problems, we proposed a two-stage framework. The first stage of the proposed algorithm combines with a multi-tasking optimization strategy and a bi-directional search strategy, where the original problem is reformulated as a multi-tasking optimization problem in the decision space to enhance the convergence. To improve the diversity, in the second stage, the proposed algorithm applies multi-tasking optimization to a number of sub-problems based on reference points in the objective space. In this paper, to show the effectiveness of the proposed algorithm, we test the algorithm on the DTLZ and LSMOP problems and compare it with existing algorithms, and it outperforms other compared algorithms in most cases and shows disadvantage on both convergence and diversity.

scholarly journals Dynamic Programming as a Method of Spline Approximation in the CAD Systems of Linear Constructions

2019 ◽  
Vol 7 (3) ◽  
pp. 77-88 ◽  
Author(s):  
D. A. Karpov ◽  
V. I. Struchenkov
Keyword(s):  
Dynamic Programming ◽  
Linear Programming ◽  
Spline Approximation ◽  
Initial Approximation ◽  
Longitudinal Profile ◽  
Non Linear Programming ◽  
Second Stage ◽  
Linear Programming Methods ◽  
Non Linear ◽  
Three Stages

Under study is a problem of the line structure routing of roads, railways and other linear constructions. Designing a trace plan and longitudinal profile are considered as non-linear programming tasks. Since the number of elements of the plan and the longitudinal profile is not known, the problem is solved in three stages. First, a search is performed for a polyline consisting of short elements. On the second stage it is used to determine the initial approximation of the desired line, which is optimized at the last stage. The required line consists of a given type elements and it is a spline with a number of features:- In contrast to the polynomial elements considered in the theory of splines, when designing roads unknown spline is a sequence of elements: straight, clothoid, circle, clothoid, straight and so on.- In this task, the spline does not have to be a single-valued function.- The parameters of the elements of the desired spline must satisfy the constraints in the form of inequalities.These features of the task do not allow the use of non-linear programming methods to solve it. Converting a broken line to a spline is carried out using dynamic programming. For this purpose a special formalization of this task is proposed. A new algorithm of dynamic programming is given. The result is used as an initial approximation to optimize the parameters of the spline using a previously developed non-linear programming program.

An approximation algorithm for solving standard quadratic optimization problems

2020 ◽  
Vol 39 (3) ◽  
pp. 4383-4392
Author(s):  
Lunshan Gao
Keyword(s):  
Linear Programming ◽  
Approximation Method ◽  
Heuristic Algorithms ◽  
Optimization Problems ◽  
Quadratic Optimization ◽  
Approximate Algorithm ◽  
Computational Time ◽  
Standard Quadratic Optimization ◽  
The Matrix ◽  
Possibility Distributions

Standard quadratic optimization problems (StQPs) are NP-hard in computational complexity theory when the matrix is indefinite. This paper describes an approximate algorithm of finding inner optimal values of StQPs. The approximate algorithm fuzzifies variable x ∈ Rn with normalized possibility distributions and simplifies the solving of StQPs. The approximation ratio is discussed and determined. Numerical results show that (1) the new algorithm achieves higher accuracy than the semidefinite programming method and linear programming approximation method; (2) the novel algorithm consumes less than one out of fourth computational time that is consumed by linear programming approximation method; (3) the computational time of the new algorithm does not correlate with the matrix densities whereas the computational times of the branch-and-bound and heuristic algorithms do.

Finance-based scheduling using meta-heuristics: discrete versus continuous optimization problems

2015 ◽  
Vol 20 (1) ◽  
pp. 85-104 ◽  
Author(s):  
Ashraf Elazouni ◽  
Anas Alghazi ◽  
Shokri Z. Selim
Keyword(s):  
Time Management ◽  
Optimization Problems ◽  
Continuous Variable ◽  
Discrete Variable ◽  
Content Type ◽  
Project Duration ◽  
Project Completion ◽  
Management Aspect ◽  
Discrete Variable Optimization ◽  
Cost Components

Purpose – The purpose of this paper is to compare the performance of the genetic algorithm (GA), simulate annealing (SA) and shuffled frog-leaping algorithm (SFLA) in solving discrete versus continuous-variable optimization problems of the finance-based scheduling. This involves the minimization of the project duration and consequently the time-related cost components of construction contractors including overheads, finance costs and delay penalties. Design/methodology/approach – The meta-heuristics of the GA, SA and SFLA have been implemented to solve non-deterministic polynomial-time hard (NP-hard) finance-based scheduling problem employing the objective of minimizing the project duration. The traditional problem of generating unfeasible solutions in scheduling problems is adequately tackled in the implementations of the meta-heuristics in this paper. Findings – The obtained results indicated that the SA outperformed the SFLA and GA in terms of the quality of solutions as well as the computational cost based on the small-size networks of 30 activities, whereas it exhibited the least total duration based on the large-size networks of 120 and 210 activities after prolonged processing time. Research limitations/implications – From researchers’ perspective, finance-based scheduling is one of the few domain problems which can be formulated as discrete and continuous-variable optimization problems and, thus, can be used by researchers as a test bed to give more insight into the performance of new developments of meta-heuristics in solving discrete and continuous-variable optimization problems. Practical implications – Finance-based scheduling discrete-variable optimization problem is of high relevance to the practitioners, as it allows schedulers to devise finance-feasible schedules of minimum duration. The minimization of project duration is focal for the minimization of time-related cost components of construction contractors including overheads, finance costs and delay penalties. Moreover, planning for the expedient project completion is a major time-management aspect of construction contractors towards the achievement of the objective of client satisfaction through the expedient delivery of the completed project for clients to start reaping the anticipated benefits. Social implications – Planning for the expedient project completion is a major time-management aspect of construction contractors towards the achievement of the objective of client satisfaction. Originality/value – SFLA represents a relatively recent meta-heuristic that proved to be promising, based on its limited number of applications in the literature. This paper is to implement SFLA to solve the discrete-variable optimization problem of the finance-based scheduling and assess its performance by comparing its results against those of the GA and SA.

scholarly journals Robust two-stage combinatorial optimization problems under convex second-stage cost uncertainty

2021 ◽  
Author(s):  
Marc Goerigk ◽  
Adam Kasperski ◽  
Paweł Zieliński
Keyword(s):  
Combinatorial Optimization ◽  
Approximation Algorithms ◽  
Network Optimization ◽  
Optimization Problems ◽  
Two Stage ◽  
Shortest Path Problems ◽  
Combinatorial Optimization Problems ◽  
Second Stage ◽  
Uncertainty Set ◽  
Basic Network

AbstractIn this paper a class of robust two-stage combinatorial optimization problems is discussed. It is assumed that the uncertain second-stage costs are specified in the form of a convex uncertainty set, in particular polyhedral or ellipsoidal ones. It is shown that the robust two-stage versions of basic network optimization and selection problems are NP-hard, even in a very restrictive cases. Some exact and approximation algorithms for the general problem are constructed. Polynomial and approximation algorithms for the robust two-stage versions of basic problems, such as the selection and shortest path problems, are also provided.

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