A Feasible MBFGS Method For Linearly Constrained Optimization

Optimization problems have a wide range of practical background in agriculture,national defense, transportation, finance, energy, telecommunications and other areas.Quasi-Newton methods are very welcome in the solution of middle size optimizationproblems. In a quasi-Newton method, the positive definiteness of the quasi-Newtonmatrix plays an important role in the global convergence of the method. BFGS methodis regarded as one of the most e?cient Quasi-Newton methods. Under appropriate con-ditions, the method is globally and superlinearly convergent. However, when appliedto solve a nonconvex minimization problem, the standard method may not be globallyconvergent. Recently, Li and Fukushima made a modification to the standard BFGSmethod and proposed a modified BFGS method– MBFGS method. It has been provedthat the MBFGS method is globally and superlinearly convergent even when the ob-jective function is not convex. An attractive property of the MBFGS method is thatthe quasi-Newton matrices generated by the method are always positive definite. Thisproperty does not rely the line search used.In this thesis, we extend the MBFGS method of unconstrained optimization todevelop a feasible MBFGS method for solving linearly constrained optimization prob-lems. We propose the method by the use of the idea of the feasible direction methodand the MBFGS method. We first develop a feasible MBFGS method for solving linearequality constrained problem. We then extend the idea to develop a feasible MBFGSmethod for solving non-negative constrained problem and general linearly constrainedproblems. We show that under appropriate conditions, the proposed method is globallyand superlinearly convergent. We also adopt the nonmonotone line search techniqueintroduced by Grippo et. al. to develop a nonmonotone feasible MBFGS method forsolving linear equality constrained problems. At last, we do some numerical experi-ments to test the proposed method. The results show that the proposed method ispractically e?cient.As the proposed method generates feasible points, we use the objective functionas the merit function. Consequently, the method can avoid the Maratos e?ect.