Abstract
In this paper, we propose an improved partial bundle method for solving linearly constrained minimax problems. In order to reduce the number of component function evaluations, we utilize a partial cutting-planes model to substitute for the traditional one. At each iteration, only one quadratic programming subproblem needs to be solved to obtain a new trial point. An improved descent test criterion is introduced to simplify the algorithm. The method produces a sequence of feasible trial points, and ensures that the objective function is monotonically decreasing on the sequence of stability centers. Global convergence of the algorithm is established. Moreover, we utilize the subgradient aggregation strategy to control the size of the bundle and therefore overcome the difficulty of computation and storage. Finally, some preliminary numerical results show that the proposed method is effective.
Highlights
IntroductionIn order to reduce the number of component function evaluations, Gaudioso et al [6] proposed an incremental bundle method for solving convex unconstrained minimax problems, in which a partial cutting-planes model of the objective function f is introduced
We consider the linearly constrained minimax problem min f (x) x∈Rn (1)s.t. ⟨ai, x⟩ ≤ bi, i ∈ I = {1, . . . , p}, where the objective function f (x) = max{fj(x), j ∈ J} with J = {1, . . . , m}, the component functions fj (j ∈ J) : Rn → R are convex but not necessarily differentiable, and ai(i ∈ I) ∈ Rn, bi(i ∈ I) ∈ R
In order to reduce the number of component function evaluations, Gaudioso et al [6] proposed an incremental bundle method for solving convex unconstrained minimax problems, in which a partial cutting-planes model of the objective function f is introduced
Summary
In order to reduce the number of component function evaluations, Gaudioso et al [6] proposed an incremental bundle method for solving convex unconstrained minimax problems, in which a partial cutting-planes model of the objective function f is introduced. Jian et al [9] proposed a feasible descent bundle method for general inequality constrained minimax problems by using the partial cutting-planes model of [6]. Tang et al [22] proposed a proximal-projection partial bundle method for convex constrained minimax problems by combining the partial cutting-planes model of [6] with the proximal-projection idea of [13, 15]. We propose an improved partial bundle method for solving linearly constrained minimax problem (1). The subdifferential (in convex analysis) of a function f at any point x is denoted by ∂f (x), and each element g ∈ ∂f (x) is called a subgradient
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