Abstract
For some practical problems, the exact computation of the function and (sub)gradient values may be difficult. In this paper, a proximal-projection bundle method for minimizing convex nonsmooth optimization problems with on-demand accuracy oracles is proposed. Our method essentially generalizes the work of Kiwiel (SIAM J Optim, 17: 1015-1034, 2006) from exact and inexact oracles to various oracles, including exact, inexact, partially inexact, asymptotically exact and partially asymptotically exact oracles. At each iteration, a proximal subproblem is solved to generate a linear model of the objective function, and then a projection subproblem is solved to obtain a trial point. Finally, global convergence of the algorithm is established under different types of inexactness.
Highlights
IntroductionWe focus on solving problems of the form f∗ := min f (u), u∈C (1.1). where C ⊆ Rn is a nonempty closed convex set, and f : Rn → R is a convex function but not necessarily differentiable
In this paper, we focus on solving problems of the form f∗ := min f (u), u∈C (1.1)where C ⊆ Rn is a nonempty closed convex set, and f : Rn → R is a convex function but not necessarily differentiable.It is well known that bundle methods are among the most efficient methods for solving nonsmooth optimization problems
By combining the on-demand accuracy approach of [3] with the proximal-projection bundle method of [7], we proposed a proximal-projection bundle method with on-demand accuracy oracles for solving problem (1.1)
Summary
We focus on solving problems of the form f∗ := min f (u), u∈C (1.1). where C ⊆ Rn is a nonempty closed convex set, and f : Rn → R is a convex function but not necessarily differentiable. Kiwiel [4] proposed a bundle method with a partially inexact oracle which becomes exact when an objective target level for a descent step is reached, and applied it to solve generalized assignment problems. Oliveira et al [5] proposed inexact bundle methods for solving two-stage stochastic programming. Oliveira and Sagastizabal [3] developed level bundle methods for oracles with on-demand accuracy for solving problems of the form (1.1). Wolf et al [16] presented a computational study for oracles of ondemand accuracy in solving two-stage stochastic linear programming problems. The subdifferential of f at u ∈ Rn is denoted by ∂f (u) := {g : f (y) ≥ f (u) + ⟨g, y − u⟩, ∀y ∈ Rn}, and each element g ∈ ∂f (u) is called a subgradient
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