A cutting plane method for solving minimax problems in the complex plane

Reemtsen, Rembert

doi:10.1007/bf02139477

Cited by 20 publications

(3 citation statements)

References 35 publications

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“…At this time, the standard method for solving smooth unconstrained optimization problems cannot be directly applied, so the theory and solving methods of minimax problem are studied. At present, it mainly includes sub-gradient algorithm [1] , cutting plane algorithm [2] , bundle method [3] , random method [11] , non-derivative method [12] and so on. There are also many ways to solve minimax problems by smoothing methods [13][14][15] .…”

Section: Introductionmentioning

confidence: 99%

Gradient descent method for unconstrained minimax problem based on max(0,x) aggregate smoothing

Zhang,

Li,

2024

Fourth International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2024)

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Based on the aggregate smoothing of max(0,x), a smooth gradient descent method for unconstrained minimax problem is proposed. Firstly, the unconstrained minimax problem is transformed into an equivalent non-smooth problem containing max(0,x) function. The unconstrained smooth optimization problem is obtained by the aggregate smoothing approximation technique. Through the numerical experiments of three examples, compared with two typical algorithms, the proposed algorithm shows good competitiveness.

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Section: Introductionmentioning

confidence: 99%

Gradient descent method for unconstrained minimax problem based on max(0,x) aggregate smoothing

Zhang,

Li,

2024

Fourth International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2024)

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“…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]).…”

Section: Introductionmentioning

confidence: 99%

An Active Set Smoothing Method for Solving Unconstrained Minimax Problems

Zhou

2020

Mathematical Problems in Engineering

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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.

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“…Taking into account the value of minimax problems, many methods are proposed for solving problem (1). For example, in [3,4], the minimax optimization problem is viewed as an unconstrained nonsmooth optimization problem, which can be solved by the general methods, such as subgradient methods, bundle methods, and cutting plane methods. The other type of methods which solves the problem (1) is so called smoothing methods, whose approach is to transform the minimax problem (1) into an equivalent smooth constrained nonlinear programming problem as follows: min s.t.…”

Section: Introductionmentioning

confidence: 99%

Improved Filter-SQP Algorithm with Active Set for Constrained Minimax Problems

Luo

Wang

2014

Journal of Applied Mathematics

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An improved filter-SQP algorithm with active set for constrained finite minimax problems is proposed. Firstly, an active constraint subset is obtained by a pivoting operation procedure. Then, a new quadratic programming (QP) subproblem is constructed based on the active constraint subset. The main search directiondkis obtained by solving this (QP) subproblem which is feasible at per iteration point and need not to consider the penalty function by using the filter technique. Under some suitable conditions, the global convergence of our algorithm is established. Finally, some numerical results are reported to show the effectiveness of the proposed algorithm.

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A cutting plane method for solving minimax problems in the complex plane

Cited by 20 publications

References 35 publications

Gradient descent method for unconstrained minimax problem based on max(0,x) aggregate smoothing

Gradient descent method for unconstrained minimax problem based on max(0,x) aggregate smoothing

An Active Set Smoothing Method for Solving Unconstrained Minimax Problems

Improved Filter-SQP Algorithm with Active Set for Constrained Minimax Problems

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