Three-term Conjugate Gradient Algorithm Research Articles

An improved three-term conjugate gradient algorithm for solving unconstrained optimization problems

In this article, we present an improved three-term conjugate gradient algorithm for large-scale unconstrained optimization. The search directions in the developed algorithm are proved to satisfy an approximate secant equation as well as the Dai-Liao’s conjugacy condition. With the standard Wolfe line search and the restart strategy, global convergence of the algorithm is established under mild conditions. By implementing the algorithm to solve 75 benchmark test problems with dimensions from 1000 to 10,000, the obtained numerical results indicate that the algorithm outperforms the state-of-the-art algorithms available in the literature. It costs less CPU time and smaller number of iterations in solving the large-scale unconstrained optimization.

A new three-term conjugate gradient algorithm for unconstrained optimization

A new three-term conjugate gradient algorithm which satisfies both the descent condition and the conjugacy condition is presented. The algorithm is obtained by minimization the one-parameter quadratic model of the objective function in which the symmetrical approximation of the Hessian matrix satisfies the general quasi-Newton equation. The search direction is obtained by symmetrization of the iteration matrix corresponding to the solution of the quadratic model minimization. Using the general quasi-Newton equation the search direction includes a parameter which is determined by the minimization of the condition number of the iteration matrix. It is proved that this direction satisfies both the conjugacy and the descent condition. The new approximation of the minimum is obtained by the general Wolfe line search using by now a standard acceleration technique. Under standard assumptions, for uniformly convex functions the global convergence of the algorithm is proved. The numerical experiments using 800 large-scale unconstrained optimization test problems show that minimization of the condition number of the iteration matrix lead us to a value of the parameter in the search direction able to define a competitive three-term conjugate gradient algorithm. Numerical comparisons of this variant of the algorithm versus known conjugate gradient algorithms ASCALCG, CONMIN, TTCG and THREECG, as well as the limited memory quasi-Newton algorithm LBFGS (m = 5) and the truncated Newton TN show that our algorithm is indeed more efficient and more robust.

An accelerated subspace minimization three-term conjugate gradient algorithm for unconstrained optimization

A three-term conjugate gradient algorithm for large-scale unconstrained optimization using subspace minimizing technique is presented. In this algorithm the search directions are computed by minimizing the quadratic approximation of the objective function in a subspace spanned by the vectors: −g k+1, s k and y k . The search direction is considered as: d k+1 = −g k+1 + a k s k + b k y k , where the scalars a k and b k are determined by minimization the affine quadratic approximate of the objective function. The step-lengths are determined by the Wolfe line search conditions. We prove that the search directions are descent and satisfy the Dai-Liao conjugacy condition. The suggested algorithm is of three-term conjugate gradient type, for which both the descent and the conjugacy conditions are guaranteed. It is shown that, for uniformly convex functions, the directions generated by the algorithm are bounded above, i.e. the algorithm is convergent. The numerical experiments, for a set of 750 unconstrained optimization test problems, show that this new algorithm substantially outperforms the known Hestenes and Stiefel, Dai and Liao, Dai and Yuan and Polak, Ribiere and Poliak conjugate gradient algorithms, as well as the limited memory quasi-Newton method L-BFGS and the discrete truncated-Newton method TN.

On three-term conjugate gradient algorithms for unconstrained optimization

This paper presents a project for three-term conjugate gradient algorithms development. The search direction of the algorithms from this class has three terms and is computed as modifications of the classical conjugate gradient algorithms to satisfy both the descent and the conjugacy conditions. An example of three-term conjugate gradient algorithm from this class, as modifications of the classical and well known Hestenes and Stiefel or of the CG_DESCENT by Hager and Zhang conjugate gradient algorithms, satisfying both the descent and the conjugacy conditions is presented. These properties are independent of the line search. Also, the algorithm could be considered as a modification of the memoryless BFGS quasi-Newton method. The new approximation of the minimum is obtained by the general Wolfe line search and using by now a standard acceleration technique developed by Andrei. The proposed three-term conjugate gradient algorithm substantially outperforms the well known Hestenes and Stiefel conjugate gradient algorithm, as well as the more elaborated CG_DESCENT algorithm. A number of 5 applications from MINPACK-2 test problems collection, with 106 variables, prove that the suggested three-term conjugate gradient algorithm is top performer versus CG_DESCENT.

A simple three-term conjugate gradient algorithm for unconstrained optimization

A simple three-term conjugate gradient algorithm which satisfies both the descent condition and the conjugacy condition is presented. This algorithm is a modification of the Hestenes and Stiefel algorithm (Hestenes and Stiefel, 1952) [10], or that of Hager and Zhang (Hager and Zhang, 2005) [23] in such a way that the search direction is descent and it satisfies the conjugacy condition. These properties are independent of the line search. Also, the algorithm could be considered as a modification of the memoryless BFGS quasi-Newton method. The new approximation of the minimum is obtained by the general Wolfe line search, now using a standard acceleration technique developed by Andrei (2009) [27]. For uniformly convex functions, under standard assumptions, the global convergence of the algorithm is proved. Numerical comparisons of the suggested three-term conjugate gradient algorithm versus six other three-term conjugate gradient algorithms, using a set of 750 unconstrained optimization problems, show that all these computational schemes have similar performances, the suggested one being slightly faster and more robust. The proposed three-term conjugate gradient algorithm substantially outperforms the well-known Hestenes and Stiefel conjugate gradient algorithm, as well as the more elaborate CG_DESCENT algorithm. Using five applications from the MINPACK-2 test problem collection (Averick et al., 1992) [25], with 106 variables, we show that the suggested three-term conjugate gradient algorithm is the top performer versus CG_DESCENT.

A three-term conjugate gradient algorithm using subspace for large-scale unconstrained optimization

A three-term conjugate gradient algorithm using subspace for large-scale unconstrained optimization