The problem of scheduling irregular polygons with vertices on a circle is considered under different objectives. The optimization criteria are depending on the distances between adjacent vertices on the circle. This polygon scheduling problem is a continuous optimization problem, which is NP-hard in the strong sense. It has been reformulated as a discrete optimization problem \(\min_{s\in S}c(s)\), where each solution \(s\in S\) represents a continuous set of schedules. The objective function value \(c(s)\) of a solution \(s\in S\) is the optimal value of a convex optimization problem with linear constraints. For maximizing the minimum weighted distance, minimizing the maximum weighted distance, and minimizing the sum of weighted distances the problems of calculating the value \(c(s)\) reduce to linear programming problems. The corresponding dual problems are special network flow problems which can be solved by well-known methods. Since the cardinality of the set \(S\) is exponential, it is practicable for small problems to use these results for an enumerative solution method. Based on the discrete formulation of the polygon scheduling problem the author has developed two neighbourhood structures. They can be used to adapt local search heuristics to the polygon scheduling problem. The solution methods for the polygon scheduling problem can also be used for a more general problem where polygons on different circles have to be scheduled simultaneously. This generalization of the polygon scheduling problem has an application in the area of railway scheduling.