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A secant equation (quasi-Newton) has one of the most important rule to find an optimal solution in nonlinear optimization. Curvature information must satisfy the usual secant equation to ensure positive definiteness of the Hessian approximation. In this work, we present a new diagonal updating to improve the Hessian approximation with a modifying weak secant equation for the diagonal quasi-Newton (DQN) method. The gradient and function evaluation are utilized to obtain a new weak secant equation and achieve a higher order accuracy in curvature information in the proposed method. Modified DQN methods based on the modified weak secant equation are globally convergent. Extended numerical results indicate the advantages of modified DQN methods over the usual ones and some classical conjugate gradient methods. [ABSTRACT FROM AUTHOR]

In this work, we present a new class of diagonal quasi-Newton methods for solving large-scale unconstrained optimization problems. The methods are derived by means of variational principle under the generalized Frobenius norm. We show global convergence of our methods under the standard line search with Armijo condition. Numerical results are carried out in standard test problems and clearly indicate vast superiority over some classical conjugate gradient methods. [ABSTRACT FROM PUBLISHER]

We present a new gradient method that uses scaling and extra updating within the diagonal updating for solving unconstrained optimization problem. The new method is in the frame of Barzilai and Borwein (BB) method, except that the Hessian matrix is approximated by a diagonal matrix rather than the multiple of identity matrix in the BB method. The main idea is to design a new diagonal updating scheme that incorporates scaling to instantly reduce the large eigenvalues of diagonal approximation and otherwise employs extra updates to increase small eigenvalues. These approaches give us a rapid control in the eigenvalues of the updating matrix and thus improve stepwise convergence. We show that our method is globally convergent. The effectiveness of the method is evaluated by means of numerical comparison with the BB method and its variant. [ABSTRACT FROM AUTHOR]

Diagonal quasi-Newton (DQN) methods are a class of quasi-Newton methods which alter the standard quasi-Newton updates of approximations to the Hessian or its inverse to diagonal updating matrices. Most often, the updating formulae for this class of methods are derived by the variational approach. A major drawback under this approach is that the derived diagonal matrix may suffer from the loss of positive definiteness and thus it may not be appropriate for use within a descent-gradient algorithm. Previous strategies to overcome this difficulty concentrated on skipping or restarting the non-descent steps. Doing so would abandon the second derivative information that is found on the previous step and consequently, the speed of convergence is usually slower than it would be without these remedies. Hence the present paper intends to propose a simple yet effective remedy to overcome the difficulty that gives arise non-positive-definite updating matrices in the variational based DQN methods. To this end we find that by incorporating an appropriate scaling for the diagonal updating, it improves step-wise convergence while avoiding non-positive definiteness of the updates. Finally, the new DQN method is tested for computational efficiency and stability on numerous test functions, and the numerical results indicate clear superiority over the current methods. [ABSTRACT FROM AUTHOR]

This paper focuses on developing diagonal gradient-type methods that employ accumulative approach in multistep diagonal updating to determine a better Hessian approximation in each step. The interpolating curve is used to derive a generalization of the weak secant equation, which will carry the information of the local Hessian. The new parameterization of the interpolating curve in variable space is obtained by utilizing accumulative approach via a norm weighting defined by two positive definite weighting matrices. We also note that the storage needed for all computation of the proposed method is just On. Numerical results show that the proposed algorithm is efficient and superior by comparison with some other gradient-type methods. [ABSTRACT FROM AUTHOR]

This paper presents the study of multi-objective optimization of a pharmaceutical portfolio when both cost and return values are uncertain. Decision makers in the pharmaceutical industry encounter several challenges in deciding the optimal selection of drug projects for their portfolio since they have to consider several key aspects such as a long product-development process split into multiple phases, high cost and low probability of success. Additionally, the optimization often involves more than a single objective (goal) with a non-deterministic nature. The aim of the study is to develop a stochastic multi-objective approach in the frame of chance-constrained goal programming. The application of the results of this study allows pharmaceutical decision makers to handle two goals simultaneously, where one objective is to achieve a target return and another is to keep the cost within a finite annual budget. Finally, the numerical results for portfolio optimization are presented and discussed. [ABSTRACT FROM AUTHOR]