An Approximate Quasi-Newton Bundle-Type Method for Nonsmooth Optimization

and Applied Analysis 3 rate of convergence under some additional assumptions, and it should be noted that we only use the approximate values of the objective function and its subgradients which makes the algorithm easier to implement. Some notations are listed below for presenting the algorithm. (i) ∂f(x) = {ξ ∈ Rn | f(z) ≥ f(x) + ξT(z − x), ∀z ∈ R}, the subdifferential of f at x, and each such ξ is called a subgradient of f at x. (ii) ∂ ε f(x) = {η ∈ Rn | f(z) ≥ f(x) + ηT(z − x) − ε}, the ε-subdifferential off at x, and each such η is called an ε-subgradient of f at x. (iii) p(x) = argmin z∈R n{f(z)+(2λ)−1 ‖z−x‖2}, the unique minimizer of (2). (iv) G(x) = λ−1(x − p(x)), the gradient of F at x. This paper is organized as follows: in Section 2, to approximate the unique minimizer p(x) of (2), we introduce the bundle idea, which uses approximate values of the objective function and its subgradients. The approximate quasiNewton bundle-type algorithm is presented in Section 3. In the last section, we prove the global convergence and, under additional assumptions, Q-superlinear convergence of the proposed algorithm. 2. The Approximation of p(x) Let x = xk and s = z − x, where xk is the current iterate point of AQNBT algorithm presented in Section 3, then (13) has the form F (xk) = min s∈R n {f (xk + s) + (2λ) −1 ‖s‖ 2} . (12) Nowwe consider approximatingf(x+s) by using the bundle idea. Suppose we have a bundle Jk generated sequentially starting from xk and possibly a subset of the previous set used to generate x.The bundle includes the data (zi, fi, ga(zi, ε i )), i ∈ J, where zi ∈ Rn, fi ∈ R, and ga(zi, ε i ) ∈ Rn satisfy f (zi) ≥ fi ≥ f (zi) − ε i , f (z) ≥ f i + ⟨ga (zi, ε i ) , z − zi⟩ , ∀z ∈ Rn. (13) Suppose that the elements in Jk can be arranged according to the order of their entering the bundle. Without loss of generality we may suppose Jk = {1, . . . , j}. ε i is updated by the rule ε i+1 = γε i , 0 < γ < 1, i ∈ J. The condition (13) means ga(zi, ε i ) ∈ ∂ ε i f(z), i ∈ J. By using the data in the bundle we construct a polyhedral function f a (xk + s) defined by