Bounds for convex quadratic programming problems and some important applications

Abstract

This paper proposes bounds for the objective function of convex quadratic programming problem (QPP) in general form. The method uses just the eigenvalues of the Hessian matrix. By use of the eigenvalues of the Hessian matrix and solving two simple optimisation problems, we propose an interval which contains the optimal value of QPP. This method can be useful for complicated and large scale optimisation problems as well as for the integer quadratic programming problems and also can be used as a start interval for the other existing mathematical methods for QPP. Another application of the proposed interval is to help the decision maker in real applications to estimate the bounds of the optimal solution. Thus the method is useful from both theoretical and practical approaches. This method can also be applied to solve fractional quadratic programming problems as well as for binary and mixed integer quadratic programming problems. Sensitivity analysis for the objective function is another application of the method which will be discussed. To illustrate the method, several problems are solved.