Convex programming: the minimum covering circle case

Abstract

In convex programming, the primal minimum covering circle case is associated with a complex nonlinear constraint. This article introduces the definition of a convex function and applies the Wolfe duality to derive the equivalent Wolfe dual problem with a single linear constraint. According to the duality theorem and the geometric characterization, the Wolfe dual problem is of equivalence basically to the primal problem with a unique feasible solution. Meanwhile, since the decision variable is optimal, the Wolfe dual problem satisfies the Karush-Kuhn-Tucker conditions in terms of Lagrange conditions. Subsequently, this article proposes a simplistic two-point covering circle case as a visualized approach to furthermore discuss the principle of convex programming in the graphic demonstration. Nevertheless, the article utilizes the Wolfe objective function and the constraint for the multi-point covering circle problem to implement the minimum covering circle and sphere across 50 randomly generated points, briefly discussing its applications.