Iterative methods for optimization can be classified into two categories: line search methods and trust region methods. In this paper, we propose a modified regularized Newton method for minimizing nonconvex functions whose Hessian matrix may be singular without line search. The proposed method is proved to converge globally if the Gradient and Hessian of the objective function are Lipschitz continuous. Moreover, we report numerical results that show that the proposed algorithm is competitive with the existing methods.

In this paper, optimization of a linear objective func-tion with fuzzy relational inequality constraints is in-vestigated where the feasible region is formed as the in-tersection of two inequality fuzzy systems and Dombi family of t-norms is considered as fuzzy composition. Dombi family of t-norms includes a parametric family of continuous strict t-norms, whose members are increas-ing functions of the parameter. This family of t-norms covers the whole spectrum of t-norms when the param-eter is changed from zero to in nity. The resolution of the feasible region of the problem is  rstly investigated when it is de ned with max-Dombi composition. Based on some theoretical results, a necessary and su cient condition and three other necessary conditions are de-rived for determining the feasibility. Moreover, in order to simplify the problem, some procedures are presented.

In this paper, we solve unConstrained optimization problem using a free line search steepest descent method. First, we propose a double parameter scaled quasi Newton formula for calculating an approximation of the Hessian matrix. The approximation obtained from this formula is a positive definite matrix that is satisfied in the standard secant relation. We also show that the largest eigen value of this matrix is not greater than the number of variables of the problem. Then, using this double parameter scaled quasi Newton formula, an explicit formula for calculating the step length in the steepest descent method is presented and therefore, this method does not require the use of approximate methods for calculating step length. The numerical results obtained from the implementation of the algorithm in MATLAB software environment are presented for some optimization problems. These results show the efficiency of the proposed method in comparison with other existing methods.

In this paper, the modeling and impedance-control of a one link flexible robot is presented. The concept of impedance control of flexible link robots is rather new and is being addressed for the first time. The control algorithm is valid for both Constrained and unConstrained maneuvers. First, equations of motion and the associated boundary conditions are derived using Hamilton's principle. A linear finite dimensional model is, then, generated in the Cartesian coordinates, using the assumed mode method and by introduction of a proper coordinate transformation. The target impedance is, then, introduced in the Cartesian coordinate system and a control law is designed to realize the proposed target impedance for a given frequency range, using the Sliding Mode Control Theory. A set of computer simulations are carried out to demonstrate the effectiveness of the proposed control law. Simulations are carried out with various contact stiffness. As the results show, when the environmental surface stiffness is smaller than, or comparable to, that of the link, the control system is able to achieve stable behavior and the link vibration diminishes rather rapidly. However, when the environmental stiffness is much greater than that of the stiffness of the link, although the robot achieves stable behavior during contact, the vibrations tend to increase.

The conjugate gradient (CG) method is an optimization technique known for its rapid convergence; it has blossomed into significant developments and applications. Numerous variations of CG methods have emerged to en-hance computational efficiency and address real-world challenges. In this work, a novel conjugate gradient method is introduced to solve nonlinear unConstrained optimization problems. Based on the combination of PRP (Polak–Ribière–Polyak), HRM (Hamoda–Rivaie–Mamat) and NMFR (new modified Fletcher–Reeves) algorithms, our method produces a descent di-rection without depending on any line search. Moreover, it enjoys global convergence under mild assumptions and is applied successfully on various standard test problems as well as image processing. The numerical results indicate that the proposed method outperforms several existing methods in terms of efficiency.

While many real-world optimization problems typically involve multiple constraints, unConstrained problems hold practical and fundamental significance. They can arise directly in specific applications or as transformed versions of Constrained optimization problems.‎ ‎Newton's method‎, ‎a notable numerical technique within the category of line search algorithms, is widely used for function optimization‎. The search direction and step length play crucial roles in this algorithm. ‎This paper introduces an algorithm aimed at enhancing the step length within the Broyden quasi-Newton process‎. ‎Additionally‎, ‎numerical examples are provided to compare the effectiveness of this new method with another approach‎.