A modified Broyden family algorithm with global convergence under a weak Wolfe-Powell line search for unconstrained nonc
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A modified Broyden family algorithm with global convergence under a weak Wolfe‑Powell line search for unconstrained nonconvex problems Gonglin Yuan1 · Zhan Wang1 · Pengyuan Li1 Received: 20 December 2019 / Revised: 30 August 2020 / Accepted: 22 September 2020 © Istituto di Informatica e Telematica (IIT) 2020
Abstract The Quasi-Newton method is one of the most effective methods using the first derivative for solving all unconstrained optimization problems. The Broyden family method plays an important role among the quasi-Newton algorithms. However, the study of the convergence of the classical Broyden family method is still not enough. While in the special case, BFGS method, there have been abundant achievements. Yuan et al. (Appl Math Model. 47:811–825, (2017)) presented a modified weak Wolfe-Powell line search and obtained the convergence of BFGS method for general functions under this line search. Motivated by their works, a new modified weak Wolfe-Powell line search technique is proposed for unconstrained problems. We assume that the objective function is nonconvex and the global convergence of the restricted Broyden family method is established. Preliminary numerical results including the classical optimization problems and the Muskingum model show that the presented algorithm is promising. Keywords Broyden family method · Weak Wolfe line search · Global convergence · Nonconvex optimization problems
This work was supported by the National Natural Science Foundation of China (Grant No. 11661009), the High Level Innovation Teams and Excellent Scholars Program in Guangxi institutions of higher education (Grant No. [2019]52), the Guangxi Natural Science Key Fund (No. 2017GXNSFDA198046), and the Special Funds for Local Science and Technology Development Guided by the Central Government (No. ZY20198003). * Gonglin Yuan [email protected]; [email protected] Zhan Wang [email protected] Pengyuan Li [email protected] 1
College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, People’s Republic of China
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Mathematics Subject Classification 90C26 · 90C53
1 Introduction Considering the following model:
min f (x),
(1.1)
x∈ℜn
the following iterative formula is often used to solve (1.1):
xk+1 = xk + 𝛼k dk , where xk is the kth iteration point, 𝛼k is the steplength, and dk is the search direction designated by
dk = −B−1 gk , k where gk = g(xk ) is the gradient of f(x) at xk , and the symmetric positive definite matrix Bk ∈ ℜn×n is generated by the Broyden family iterative formula of the quasiNewton method. The sequence Bk satisfies the quasi-Newton equation
Bk+1 sk = yk , where yk = gk+1 − gk , sk = xk+1 − xk . Bk is updated by the formula of the Broyden family
Bk+1 = Bk −
Bk sk sTk Bk sTk Bk sk
+
yk yTk yTk sk
+ 𝜔(sTk Bk sk )vk vTk ,
(1.2)
where 𝜔 ∈ ℜ is a scalar and
vk =
yk yTk sk
−
Bk sk . T sk Bk sk
Obviously, we can obtain two popular update formulas: one is the BFGS formula for 𝜔 = 0 , and the other is the
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