[Paper Review] Minimal time functions and the smallest intersecting ball problem generated by unbounded dynamics

Nguyen Mau Nam, C.I Villalobos+1 more|arXiv (Cornell University)|Dec 24, 2011

Facility Location and Emergency Management16 references3 citations

TL;DR

This paper introduces and analyzes minimal time functions generated by unbounded dynamics, extending the classical smallest intersecting ball problem to convex target sets in normed spaces. By leveraging subdifferential calculus and the subgradient method, the authors derive explicit formulas for subgradients and prove convergence of the algorithm, demonstrating its effectiveness through numerical examples with optimal solutions converging to values like 9.10208.

ABSTRACT

The smallest enclosing circle problem introduced in the 19th century by J. J. Sylvester [20] aks for the circle of smallest radius enclosing a given set of finite points in the plane. An extension of the smallest enclosing circle problem called the smallest intersecting ball problem was considered in [17,18]: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that intersects all of the sets. In this paper we initiate the study of minimal time functions generated by unbounded dynamics and discuss their applications to extensions of the smallest intersecting ball problem. This approach continues our effort in applying convex and nonsmooth analysis to the well-established field of facility location.

Motivation & Objective

• Extend the smallest intersecting ball problem to unbounded dynamics in normed spaces, generalizing classical facility location models.
• Investigate minimal time functions with unbounded, closed, convex dynamics F, where the origin is not necessarily an interior point.
• Develop generalized differentiation tools—particularly subdifferential formulas—for minimal time functions under unbounded dynamics.
• Apply these tools to solve the extended smallest intersecting ball problem via nonsmooth optimization techniques.
• Establish convergence of the subgradient method for the maximal minimal time function under convexity and natural assumptions.

Proposed method

• Model the smallest intersecting ball problem as minimizing the maximal minimal time function: $\mathcal{T}(x) = \max_{i=1,\dots,m} \mathcal{T}^F_{\Omega_i}(x)$, where $\mathcal{T}^F_Q(x) = \inf\{t \geq 0 : (x + tF) \cap Q \neq \emptyset\}$.
• Use asymptotic cone theory to characterize the subdifferential of $\mathcal{T}^F_Q$ when $F$ is unbounded, linking it to the normal cone of $Q$ and the recession cone $F_\infty$.
• Derive explicit subgradient formulas: $\partial \mathcal{T}^F_Q(x) = \{x^* \in X^* : \langle x^*, v \rangle \leq \rho_F(v) \text{ for all } v \in X\}$, where $\rho_F$ is the Minkowski gauge of $F$.
• Apply the subgradient method: $x_{k+1} = x_k - \alpha_k x_k^*$, with $x_k^* \in \partial \mathcal{T}(x_k)$, and $\alpha_k = 1/k$ for convergence.
• Compute projections onto the constraint set $\Omega$ (e.g., a ball) explicitly to ensure feasibility in each iteration.
• Use the max-function subdifferential rule: $\partial \max_i f_i(x) = \text{conv} \bigcup_{i \in I(x)} \partial f_i(x)$, where $I(x) = \{i : f_i(x) = \max_j f_j(x)\}$.

Experimental results

Research questions

• RQ1 How can minimal time functions be generalized to unbounded dynamics, and what are their key analytical properties?
• RQ2 What is the subdifferential structure of the minimal time function $\mathcal{T}^F_Q(x)$ when $F$ is an unbounded, closed, convex set?
• RQ3 How does the asymptotic cone $F_\infty$ influence the subdifferential of the minimal time function?
• RQ4 Can the subgradient method be effectively applied to the smallest intersecting ball problem with unbounded dynamics and convex target sets?
• RQ5 What convergence guarantees and numerical performance can be expected for the subgradient algorithm in this setting?

Key findings

• The subdifferential of the minimal time function $\mathcal{T}^F_Q(x)$ is characterized via the Minkowski gauge $\rho_F$ and the normal cone to $Q$, providing a complete analytical tool for nonsmooth optimization.
• For unbounded $F$, the subdifferential formula $\partial \mathcal{T}^F_Q(x) = \{x^* \in X^* : \langle x^*, v \rangle \leq \rho_F(v) \text{ for all } v \in X\}$ holds under natural assumptions.
• The subgradient method with $\alpha_k = 1/k$ converges to a minimizer of $\mathcal{T}(x)$, as proven in Theorem 5.2.
• In Example 5.1, the algorithm converges to $\bar{x} \approx (-3.10208, -3.10208)$ with optimal value $\overline{V} \approx 9.10208$ after 500,000 iterations.
• In Example 5.2 with 8 square targets and a circular constraint set, the algorithm converges to the same optimal solution and value, confirming robustness.
• The method effectively handles non-singleton target sets (e.g., squares) and unbounded dynamics, extending classical smallest enclosing circle results to broader facility location models.