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Volume 18, Issue 2
A Modularized Algorithmic Framework for Interface Related Optimization Problems Using Characteristic Functions

Dong Wang, Shangzhi Zeng, Jin Zhang & Ning Zhang

DOI: 10.4208/nmtma.OA-2024-0082

Numer. Math. Theor. Meth. Appl., 18 (2025), pp. 437-462.

Published online: 2025-05

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  • Abstract

In this paper, we consider the algorithms and convergence for a general optimization problem, which has a wide range of applications in image segmentation, topology optimization, flow network formulation, and surface reconstruction. In particular, the problem focuses on interface related optimization problems where the interface is implicitly described by characteristic functions of the corresponding domains. Under such representation and discretization, the problem is then formulated into a discretized optimization problem where the objective function is concave with respect to characteristic functions and convex with respect to state variables. We show that under such structure, the iterative scheme based on alternative minimization can converge to a local minimizer. Extensive numerical examples are performed to support the theory.

  • Keywords

Interface problems, thresholding, characteristic function, convergence analysis.

  • AMS Subject Headings

65K05, 90C26

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-18-437, author = {Wang , DongZeng , ShangzhiZhang , Jin and Zhang , Ning}, title = {A Modularized Algorithmic Framework for Interface Related Optimization Problems Using Characteristic Functions}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2025}, volume = {18}, number = {2}, pages = {437--462}, abstract = {

In this paper, we consider the algorithms and convergence for a general optimization problem, which has a wide range of applications in image segmentation, topology optimization, flow network formulation, and surface reconstruction. In particular, the problem focuses on interface related optimization problems where the interface is implicitly described by characteristic functions of the corresponding domains. Under such representation and discretization, the problem is then formulated into a discretized optimization problem where the objective function is concave with respect to characteristic functions and convex with respect to state variables. We show that under such structure, the iterative scheme based on alternative minimization can converge to a local minimizer. Extensive numerical examples are performed to support the theory.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2024-0082}, url = {http://global-sci.org/intro/article_detail/nmtma/24071.html} }
TY - JOUR T1 - A Modularized Algorithmic Framework for Interface Related Optimization Problems Using Characteristic Functions AU - Wang , Dong AU - Zeng , Shangzhi AU - Zhang , Jin AU - Zhang , Ning JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 437 EP - 462 PY - 2025 DA - 2025/05 SN - 18 DO - http://doi.org/10.4208/nmtma.OA-2024-0082 UR - https://global-sci.org/intro/article_detail/nmtma/24071.html KW - Interface problems, thresholding, characteristic function, convergence analysis. AB -

In this paper, we consider the algorithms and convergence for a general optimization problem, which has a wide range of applications in image segmentation, topology optimization, flow network formulation, and surface reconstruction. In particular, the problem focuses on interface related optimization problems where the interface is implicitly described by characteristic functions of the corresponding domains. Under such representation and discretization, the problem is then formulated into a discretized optimization problem where the objective function is concave with respect to characteristic functions and convex with respect to state variables. We show that under such structure, the iterative scheme based on alternative minimization can converge to a local minimizer. Extensive numerical examples are performed to support the theory.

Wang , DongZeng , ShangzhiZhang , Jin and Zhang , Ning. (2025). A Modularized Algorithmic Framework for Interface Related Optimization Problems Using Characteristic Functions. Numerical Mathematics: Theory, Methods and Applications. 18 (2). 437-462. doi:10.4208/nmtma.OA-2024-0082
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