Abstract
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.
Highlights
Nonlinear programming has been appeared in solving many real-world problems
By numerical methods, we tried to get all optimal solutions in the feasible region, while by swarm algorithms (SAs), we found the best objective value in the whole interval
The equations are solved as algebraic equations where the solutions can be expressed as a function of the additional variables. ese solutions are very helpful for decision maker (DM) if the optimal solution at certain values of interval coefficients is required
Summary
Nonlinear programming has been appeared in solving many real-world problems. Solving interval programming problems is a hot issue in the research area. Interval nonlinear programming problems are used in modeling and solving many real applications such as planning of waste management activities [13]. Many researchers and authors solve the interval nonlinear programming problems by different methods [15,16,17,18], but all these methods try to get the optimal solution under some specific conditions. In [8, 9], Hladık divided the problem into subclasses which can be reduced to easy problems He put a condition for solving these problems that they must be convex quadratic programming. Liu and Wang [17] presented a numerical method to interval quadratic programming. Li and Tian [18] generalized Liu and Wang’s method [17] to solve interval quadratic programming. Li and Tian [18] generalized Liu and Wang’s method [17] to solve interval quadratic programming. eir proposed method requires less computing compared with Liu and Wang’s method
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