Abstract
Conjugate gradient methods are well-known methods which are widely applied in many practical fields. CD conjugate gradient method is one of the classical types. In this paper, a modified three-term type CD conjugate gradient algorithm is proposed. Some good features are presented as follows: (i) A modified three-term type CD conjugate gradient formula is presented. (ii) The given algorithm possesses sufficient descent property and trust region property. (iii) The algorithm has global convergence with the modified weak Wolfe–Powell (MWWP) line search technique and projection technique for general function. The new algorithm has made great progress in numerical experiments. It shows that the modified three-term type CD conjugate gradient method is more competitive than the classical CD conjugate gradient method.
Highlights
Considering the problem minf(x) | x ∈ Rn, (1)where f: Rn ⟶ R is a continuously differentiable function. is kind of model is often used to solve some problems in applied mathematics, economics, engineering, and so on.Generally, the following iteration formula is used to generate the iteration point: xk+1 xk + αkdk, k 0, 1, 2, . . . , (2)where xk+1 and xk denote the iteration point and current iteration point, respectively. αk is a step-length and dk is the search direction. e search direction dk generated by Conjugated Gradient (CG) method is defined by the following formula: dk+1 − − gk+1, gk+1 +βkdk, if k 0, if k ≥ 1, (3)
Where xk+1 and xk denote the iteration point and current iteration point, respectively. αk is a step-length and dk is the search direction. e search direction dk generated by Conjugated Gradient (CG) method is defined by the following formula: dk+1
Some conclusions are obtained from this experiment: (1) from Figures 4–6, we conclude that we can calculate approximations of flood outflows by using Algorithm 1, and Algorithm 1 is effective for the nonlinear Muskingum model; (2) the final points (x1, x2 and x3) of these three algorithms are interesting, and they are competitive with the final points of similar algorithms; and (3) the final points of Algorithm 1 are different from those of the BFGS method and HIWO method, which implies that the nonlinear Muskingum model has different optimum approximation solutions
Summary
A Modified Three-Term Type CD Conjugate Gradient Algorithm for Unconstrained Optimization Problems. Conjugate gradient methods are well-known methods which are widely applied in many practical fields. CD conjugate gradient method is one of the classical types. A modified three-term type CD conjugate gradient algorithm is proposed. E new algorithm has made great progress in numerical experiments. Some good features are presented as follows: (i) A modified three-term type CD conjugate gradient formula is presented. It shows that the modified three-term type CD conjugate gradient method is more competitive than the classical CD conjugate gradient method
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