  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.
