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

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.

COVID-19: Mathematical growth vs. precautionary measures in China, KSA, and the USA

“…Until now, many evolutionary computation techniques (ECTs) have been proposed in the literature and have been successfully applied to optimization processes. Examples of PBAs models are: genetic algorithm (GA) [ 34 , 35 ], particle swarm optimization (PSO) [ [36] , [37] , [38] ], artificial bee colony (ABC) [ 39 ], bacterial foraging (BF) [ 40 ], cat swarm optimization (CSO) [ 41 ], glowworm swarm optimization (GSO) [ 42 ], firefly algorithm (FA) [ 43 ], krill herd algorithm (KHA) [ 44 ], sine cosine algorithm (SCA) [ 45 ] and grasshopper optimization algorithm (GOA) [ 46 ], salp swarm algorithm (SSA) [ 47 ], gradient-based optimizer (GBO) [ 48 ], and harris hawks optimization (HHO) [ 49 ], etc.…”

Optimization of Cutting Parameters on Surface Roughness and Productivity when Milling Wood Materials

“…To solve a constrained nonlinear optimization problem in industry, there are many methods, such as the following: quadratic programming algorithm [21], convex programming method [22], penalty function method [23], Lagrange polynomial method [24], Kuhn-Tucker conditions [25], hill climbing algorithm, artificial neural networks [26], genetic algorithm, particle swarm [27,28]. In mathematical optimization, the Kuhn-Tucker conditions are first derivative tests for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied.…”

A Hybrid Approach for Solving Systems of Nonlinear Equations Using Harris Hawks Optimization and Newton’s Method

“…Usually, the popular algorithms of the population-based approaches result from animals' lifestyles. Many population-based approaches have been proposed to solve optimization problems in the literature, for instance, Harris hawks optimization (HHO) [3], equilibrium optimization (EO) [4], particle swarm optimization (PSO) [5]- [7], Slime Mould algorithm (SMA) [8], ant lion optimization (ALO) [9], grasshopper optimization algorithm (GOA) [10], genetic algorithm (GA) [11], firefly algorithm (FA) [12], Cuckoo search algorithm (CSA) [13], artificial bee colony (ABC) [14], krill herd algorithm (KHA) [15], ant colony optimization (ACO) [16], and glowworm swarm optimization (GSO) [17]. Many other examples of optimization algorithms can be found in [18].…”