Abstract
In this paper, an active set smoothing function based on the plus function is constructed for the maximum function. The active set strategy used in the smoothing function reduces the number of gradients and Hessians evaluations of the component functions in the optimization. Combing the active set smoothing function, a simple adjustment rule for the smoothing parameters, and an unconstrained minimization method, an active set smoothing method is proposed for solving unconstrained minimax problems. The active set smoothing function is continuously differentiable, and its gradient is locally Lipschitz continuous and strongly semismooth. Under the boundedness assumption on the level set of the objective function, the convergence of the proposed method is established. Numerical experiments show that the proposed method is feasible and efficient, particularly for the minimax problems with very many component functions.
Highlights
IntroductionWe consider the following unconstrained minimax problem: minF(x) x∈Rn
In this paper, we consider the following unconstrained minimax problem: minF(x) x∈Rn max j∈Q fj(x), (1)where the component functions fj: Rn ⟶ R, j ∈ Q 1, . . . , q, are twice continuously differentiable
Many methods have been proposed for solving minimax problem (1), such as subgradient methods ([8]), bundle type methods ([9, 10]), cutting plane methods ([11]), sequential quadratic programming methods ([12,13,14]), interior point methods ([15,16,17]), conjugate gradient methods ([18]), and smoothing methods ([19,20,21,22,23,24,25,26]). e main advantage of smoothing methods is that the minimax problem is transformed into a sequence of simple, smooth, and unconstrained optimization problems, which can be solved by standard unconstrained minimization solvers
Summary
We consider the following unconstrained minimax problem: minF(x) x∈Rn. For the minimax problems with very many component functions, several active set strategies have been developed for the smoothing methods to reduce the number of gradients or Hessians evaluations of the component functions at each iteration. In this paper, based on the plus function, an active set smoothing function for the maximum function is proposed, and the smoothing function only relates to a part of component functions, whose function values are close to F(x) It is continuously differentiable, and its gradient is locally Lipschitz continuous and strongly semismooth. Combing the active set smoothing function, a geometric reduction rule for the smoothing parameters, the Armijo line search strategy, the steepest decent direction, and the Newton direction, an active set smoothing method is proposed for solving unconstrained minimax problems.
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