Solving Interval Quadratic Programming Problems by Using the Numerical Method and Swarm Algorithms

In this paper, we present a new approach which is based on using numerical solutions and swarm algorithms (SAs) to solve the interval quadratic programming problem (IQPP). We use numerical solutions for SA to improve its performance. Our approach replaced all intervals in IQPP by additional variables. This new form is called the modified quadratic programming problem (MQPP). The Karush-Kuhn-Tucker (KKT) conditions for MQPP are obtained and solved by the numerical method to get solutions. These solutions are functions in the additional variables. Also, they provide the boundaries of the basic variables which are used as a start point for SAs. Chaotic particle swarm optimization (CPSO) and chaotic firefly algorithm (CFA) are presented. In addition, we use the solution of dual MQPP to improve the behavior and as a stopping criterion for SAs. Finally, the comparison and relations between numerical solutions and SAs are shown in some well-known examples.