Abstract
In this paper, we propose a partial bundle method for a convex constrained minimax problem where the objective function is expressed as maximum of finitely many convex (not necessarily differentiable) functions. To avoid complete evaluation of all component functions of the objective, a partial cutting-planes model is adopted instead of the traditional one. Based on the proximal-projection idea, at each iteration, an unconstrained proximal subproblem is solved first to generate an aggregate linear model of the objective function, and then another subproblem based on this model is solved to obtain a trial point. Moreover, a new descent test criterion is proposed, which can not only simplify the presentation of the algorithm, but also improve the numerical performance significantly. An explicit upper bound for the number of bundle resets is also derived. Global convergence of the algorithm is established, and some preliminary numerical results show that our method is very encouraging.
Highlights
We consider the following convex constrained minimax problem min {f (x) : x ∈ C}, (1)where f (x) = max{fi(x), i ∈ I} with I = {1, . . . , m}, and the component functions fi (i ∈ I) : Rn → R are convex but not necessarily differentiable, and C is a nonempty closed convex set in Rn.Minimax problems are a typical and special class of nonsmooth optimization problems, which aims to make an “optimal” decision under the “worst” cost
Where f (x) = max{fi(x), i ∈ I} with I = {1, . . . , m}, and the component functions fi (i ∈ I) : Rn → R are convex but not necessarily differentiable, and C is a nonempty closed convex set in Rn
Project supported by the National Natural Science Foundation (11761013, 11771383) and Guangxi Natural Science Foundation (2013GXNSFAA019013, 2014GXNSFFA118001, 2016GXNSFDA380019) of China
Summary
We consider the following convex constrained minimax problem min {f (x) : x ∈ C}, (1). Where f (x) = max{fi(x), i ∈ I} with I = {1, . M}, and the component functions fi (i ∈ I) : Rn → R are convex but not necessarily differentiable, and C is a nonempty closed convex set in Rn. Minimax problems are a typical and special class of nonsmooth optimization problems, which aims to make an “optimal” decision under the “worst” cost. Partial bundle method, proximal-projection, descent test criterion, global convergence. Project supported by the National Natural Science Foundation (11761013, 11771383) and Guangxi Natural Science Foundation (2013GXNSFAA019013, 2014GXNSFFA118001, 2016GXNSFDA380019) of China. This work was essentially done while the first author was visiting The University of New South Wales, Australia
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