On positive definiteness of Powell symmetric Broyden (H-version) update for unconstrained optimization

In this paper, we propose to search for the Powell Symmetric Broyden (PSB) update modification, according to the quasi-Newton’s ( H_ĸ+1у_ĸ=s_ĸ ) method, regarding the unconstrained optimization problem. Therefore ( β − PSB ) is the proposed new method of updating. PSB of the second order update classes, which solves the problem of unconstrained optimization, note that the update does not preserve the positive property of the inverse definite of the following Hessian matrix. In this our proposed method, guarantees of the positive property of the following Hessian matrix inverse definite are achieved by performing an update of the s_k vector, (i.e.) Subtraction is made between the next gradient and the current gradient of the target function, which must be differentiable and continuous twice. After that we confirmed the existence of the update. Then the numerical results are presented and then the proposed method is compared with the original update method ƤЅΒ in light of the standard problems.