Variable fixing method by weighted average for the continuous quadratic knapsack problem

<p style='text-indent:20px;'>We analyze the method of solving the separable convex continuous quadratic knapsack problem by weighted average from the viewpoint of variable fixing. It is shown that this method, considered as a variant of the variable fixing algorithms, and Kiwiel's variable fixing method generate the same iterates. We further improve the algorithm based on the analysis regarding the semismooth Newton method. Computational results are given and comparisons are made among the state-of-the-art algorithms. Experiments show that our algorithm has significantly good performance; it behaves very much like an <inline-formula><tex-math id="M1">\begin{document}$ O(n) $\end{document}</tex-math></inline-formula> algorithm with a very small constant.</p>

A Generalized Parallel Quantum Inspired Evolutionary Algorithm Framework for Hard Subset Selection Problems

Quantum-inspired evolutionary algorithms (QIEAs) like all evolutionary algorithms (EAs) perform well on many problems but cannot perform equally better than random for all problems due to the No Free Lunch theorem. However, a framework providing near-optimal solutions on reasonably hard instances of a large variety of problems is feasible. It has an effective general strategy for easy incorporation of domain information along with effective control on the randomness in the search process to balance the exploration and exploitation. Moreover, its effective parallel implementation is desired in the current age. Such a Generalized Parallel QIEA framework designed for the solution of Subset Selection Problems is presented here. The computational performance results demonstrate its effectiveness in the solution of different large-sized hard SSPs like the Difficult Knapsack Problem, the Quadratic Knapsack Problem and the Multiple Knapsack problem. This is the first such a generalized framework and is a major step towards creating an adaptive search framework for combinatorial optimization problems.

Application of Whale Optimization Algorithm (WOA) on Quadratic Knapsack 0-1 Problem

<span lang="EN">Quadratic Knapsack Problem is a variation of the knapsack problem that aims to maximize an objective function. The objective function in this case is quadratic. While the constraints used are binary and linear capacity constraints. The Whale Optimization Algorithm is a metaheuristic algorithm that can solve this problem. Therefore, this paper aims to find out the best solution to solve the Knapsack 0-1 Quadratic Problem using the Whale Optimization Algorithm so that its effectiveness and efficiency are known. Based on the research has been done, the algorithm is said to be effective because, from each experiment, the algorithm is always converging or towards maximum profit. Also, with the right parameters, the algorithm can achieve optimal results. It is said to be efficient because getting optimal profit does not require more time and iteration. The combination of item parameters and maximum iteration dramatically Affect the total value of profit and its running time. However, the addition of item parameter combinations is faster to achieve optimal than the maximum iteration parameter.</span>

On the rectangular knapsack problem: approximation of a specific quadratic knapsack problem

In this article, we introduce the rectangular knapsack problem as a special case of the quadratic knapsack problem consisting in the maximization of the product of two separate knapsack profits subject to a cardinality constraint. We propose a polynomial time algorithm for this problem that provides a constant approximation ratio of 4.5. Our experimental results on a large number of artificially generated problem instances show that the average ratio is far from theoretical guarantee. In addition, we suggest refined versions of this approximation algorithm with the same time complexity and approximation ratio that lead to even better experimental results.