scholarly journals An Exact Algorithm for Bilevel 0-1 Knapsack Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Raid Mansi ◽  
Cláudio Alves ◽  
J. M. Valério de Carvalho ◽  
Saïd Hanafi
Keyword(s):  
Optimization Problem ◽  
Original Problem ◽  
Parametric Optimization ◽  
Feasible Solution ◽  
Optimal Solution ◽  
Exact Algorithm ◽  
Decision Makers ◽  
Knapsack Problems ◽  
Hierarchical Decision ◽  
Parametric Optimization Problem

We propose a new exact method for solving bilevel 0-1 knapsack problems. A bilevel problem models a hierarchical decision process that involves two decision makers called the leader and the follower. In these processes, the leader takes his decision by considering explicitly the reaction of the follower. From an optimization standpoint, these are problems in which a subset of the variables must be the optimal solution of another (parametric) optimization problem. These problems have various applications in the field of transportation and revenue management, for example. Our approach relies on different components. We describe a polynomial time procedure to solve the linear relaxation of the bilevel 0-1 knapsack problem. Using the information provided by the solutions generated by this procedure, we compute a feasible solution (and hence a lower bound) for the problem. This bound is used together with an upper bound to reduce the size of the original problem. The optimal integer solution of the original problem is computed using dynamic programming. We report on computational experiments which are compared with the results achieved with other state-of-the-art approaches. The results attest the performance of our approach.

scholarly journals Upper Semicontinuity of Solution Mappings to Parametric Generalized Vector Quasiequilibrium Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Shu-qiang Shan ◽  
Yu Han ◽  
Nan-jing Huang
Keyword(s):  
Nash Equilibria ◽  
Optimization Problem ◽  
Parametric Optimization ◽  
Upper Semicontinuity ◽  
Saddle Point Problem ◽  
Point Problem ◽  
Quasiequilibrium Problems ◽  
Parametric Variational Inequality ◽  
Parametric Optimization Problem ◽  
Vector Quasiequilibrium Problems

We establish the upper semicontinuity of solution mappings for a class of parametric generalized vector quasiequilibrium problems. As applications, we obtain the upper semicontinuity of solution mappings to several problems, such as parametric optimization problem, parametric saddle point problem, parametric Nash equilibria problem, parametric variational inequality, and parametric equilibrium problem.

Parametric Optimization for Structural Design Problems

2019 ◽  
Author(s):  
Krupakaran Ravichandran ◽  
Nafiseh Masoudi ◽  
Georges M. Fadel ◽  
Margaret M. Wiecek
Keyword(s):  
Structural Design ◽  
Optimization Problem ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
Nonlinear Constraint ◽  
Computational Performance ◽  
Design Variables ◽  
Input Parameters ◽  
Parametric Optimization Problem ◽  
Objective Function Values

Abstract Parametric Optimization is used to solve problems that have certain design variables as implicit functions of some independent input parameters. The optimal solutions and optimal objective function values are provided as functions of the input parameters for the entire parameter space of interest. Since exact solutions are available merely for parametric optimization problems that are linear or convex-quadratic, general non-convex non-linear problems require approximations. In the present work, we apply three parametric optimization algorithms to solve a case study of a benchmark structural design problem. The algorithms first approximate the nonlinear constraint(s) and then solve the optimization problem. The accuracy of their results and their computational performance are then compared to identify a suitable algorithm for structural design applications. Using the identified method, sizing optimization of a truss structure for varying load conditions such as a varying load direction is considered and solved as a parametric optimization problem to evaluate the performance of the identified algorithm. The results are also compared with non-parametric optimization to assess the accuracy of the solution and computational performance of the two methods.

scholarly journals An Improved Hybrid Encoding Cuckoo Search Algorithm for 0-1 Knapsack Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yanhong Feng ◽  
Ke Jia ◽  
Yichao He
Keyword(s):  
Optimization Problem ◽  
Feasible Solution ◽  
Search Algorithm ◽  
Cuckoo Search ◽  
Genetic Mutation ◽  
Cuckoo Search Algorithm ◽  
Knapsack Problems ◽  
Local Optimum ◽  
Transform Method ◽  
Continuous Space

Cuckoo search (CS) is a new robust swarm intelligence method that is based on the brood parasitism of some cuckoo species. In this paper, an improved hybrid encoding cuckoo search algorithm (ICS) with greedy strategy is put forward for solving 0-1 knapsack problems. First of all, for solving binary optimization problem with ICS, based on the idea of individual hybrid encoding, the cuckoo search over a continuous space is transformed into the synchronous evolution search over discrete space. Subsequently, the concept of confidence interval (CI) is introduced; hence, the new position updating is designed and genetic mutation with a small probability is introduced. The former enables the population to move towards the global best solution rapidly in every generation, and the latter can effectively prevent the ICS from trapping into the local optimum. Furthermore, the greedy transform method is used to repair the infeasible solution and optimize the feasible solution. Experiments with a large number of KP instances show the effectiveness of the proposed algorithm and its ability to achieve good quality solutions.

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