Some Unconstrained Optimization Methods

Although it is a very old theme, unconstrained optimization is an area which is always actual for many scientists. Today, the results of unconstrained optimization are applied in different branches of science, as well as generally in practice. Here, we present the line search techniques. Further, in this chapter we consider some unconstrained optimization methods. We try to present these methods but also to present some contemporary results in this area

A comparison on classical-hybrid conjugate gradient method under exact line search

One of the popular approaches in modifying the Conjugate Gradient (CG) Method is hybridization. In this paper, a new hybrid CG is introduced and its performance is compared to the classical CG method which are Rivaie-Mustafa-Ismail-Leong (RMIL) and Syarafina-Mustafa-Rivaie (SMR) methods. The proposed hybrid CG is evaluated as a convex combination of RMIL and SMR method. Their performance are analyzed under the exact line search. The comparison performance showed that the hybrid CG is promising and has outperformed the classical CG of RMIL and SMR in terms of the number of iterations and central processing unit per time

Diagonal preconditioned conjugate gradient algorithm for unconstrained optimization

The nonlinear conjugate gradient (CG) methods have widely been used in solving unconstrained optimization problems. They are well-suited for large-scale optimization problems due to their low memory requirements and least computational costs. In this paper, a new diagonal preconditioned conjugate gradient (PRECG) algorithm is designed, and this is motivated by the fact that a pre-conditioner can greatly enhance the performance of the CG method. Under mild conditions, it is shown that the algorithm is globally convergent for strongly convex functions. Numerical results are presented to show that the new diagonal PRECG method works better than the standard CG method

Improved Diagonal Hessian Approximations for Large-Scale Unconstrained Optimization

We consider some diagonal quasi-Newton methods for solving large-scale unconstrained optimization problems. A simple and effective approach for diagonal quasi-Newton algorithms is presented by proposing new updates of diagonal entries of the Hessian. Moreover, we suggest employing an extra BFGS update of the diagonal updating matrix and use its diagonal again. Numerical experiments on a collection of standard test problems show, in particular, that the proposed diagonal quasi-Newton methods perform substantially better than certain available diagonal methods

Riemannian Conjugate Gradient Methods: General Framework and Specific Algorithms with Convergence Analyses

Conjugate gradient methods are important first-order optimization algorithms both in Euclidean spaces and on Riemannian manifolds. However, while various types of conjugate gradient methods have been studied in Euclidean spaces, there are relatively fewer studies for those on Riemannian manifolds (i.e., Riemannian conjugate gradient methods). This paper proposes a novel general framework that unifies existing Riemannian conjugate gradient methods such as the ones that utilize a vector transport or inverse retraction. The proposed framework also develops other methods that have not been covered in previous studies. Furthermore, conditions for the convergence of a class of algorithms in the proposed framework are clarified. Moreover, the global convergence properties of several specific types of algorithms are extensively analyzed. The analysis provides the theoretical results for some algorithms in a more general setting than the existing studies and new developments for other algorithms. Numerical experiments are performed to confirm the validity of the theoretical results. The experimental results are used to compare the performances of several specific algorithms in the proposed framework

Two Modified Three-Term Type Conjugate Gradient Methods and Their Global Convergence for Unconstrained Optimization

Two modified three-term type conjugate gradient algorithms which satisfy both the descent condition and the Dai-Liao type conjugacy condition are presented for unconstrained optimization. The first algorithm is a modification of the Hager and Zhang type algorithm in such a way that the search direction is descent and satisfies Dai-Liao’s type conjugacy condition. The second simple three-term type conjugate gradient method can generate sufficient decent directions at every iteration; moreover, this property is independent of the steplength line search. Also, the algorithms could be considered as a modification of the MBFGS method, but with different zk. Under some mild conditions, the given methods are global convergence, which is independent of the Wolfe line search for general functions. The numerical experiments show that the proposed methods are very robust and efficient