A New Algorithm for Solving Quadratic Programming Problems subject to Addition-Min Fuzzy Relation Inequalities

This paper investigates a quadratic programming problem subject to fuzzy relation inequalities with the addition-min composition operator. Three different approaches for solving the problem are introduced and compared through numerical experiments. The first one is converting the problem into an ordinary quadratic programming problem with linear constraints. The second one is smoothing the constraints with polynomial functions and approximating the problem by ordinary smooth nonlinear programming. However, when problem scale is large, both of the two approaches become very time-consuming in practical computation. Inspired by the active-set method for quadratic programming problems, a new iterative algorithm is developed in this paper. At each iteration, a quadratic programming subproblem is generated and solved according to the active set of the current iteration point. Based on the Karuch-Kuhn-Tucker condition for nonlinear programming, the optimality condition of the considered problem is given, which guarantees the constructed algorithm terminates at an optimal solution. Numerical results show that all these three algorithms can solve the problem efficiently when the problem scale is small. While the new algorithm has obvious superiority in terms of running time and computation accuracy for large-scale problems.