Minimum enclosing circle/ellipse 2

2026-04-24 · Source: Yet Another Math Programming Consultant · Field: Science & Research — Mathematics & Computational Sciences · Depth: Advanced, medium

Summary

This content addresses the problem of finding the minimum enclosing circle for a given set of $n$ circles of varying sizes. It begins by illustrating a dataset of 15 circles with their x, y coordinates and radii, and then introduces a preprocessing step to identify and remove circles entirely contained within another, reducing the dataset. The article presents two Nonlinear Programming (NLP) formulations for this optimization problem, including one that uses a small constant $\varepsilon = 0.0001$ to prevent gradient issues and another that removes the square root by squaring both sides, adding a lower bound constraint for the radius. Finally, it reformulates the problem as a Second-Order Conic Program (SOCP), which introduces additional variables but eliminates the need for an initial point or explicit lower bound on the radius. All three models yield identical solutions, with an optimal enclosing circle having a center at (14.296, 12.600) and a radius of 16.875.

Key takeaway

For a Research Scientist or Software Engineer working on geometric optimization problems, understanding the equivalence and nuances of NLP and SOCP formulations for the minimum enclosing circle problem is crucial. You should consider preprocessing your data to simplify the problem and evaluate both NLP (e.g., CONOPT) and SOCP (e.g., CPLEX) solvers, potentially using one's solution as a warm start for the other to achieve optimal precision.

Key insights

The smallest enclosing circle for a set of circles can be found using NLP or SOCP models.

Principles

• Preprocessing can simplify complex geometric optimization problems.
• SOCP formulations often remove the need for initial points or explicit bounds.
• NLP and SOCP can yield identical solutions for geometric optimization.

Method

The method involves preprocessing to remove contained circles, then solving an NLP or SOCP model to minimize the enclosing circle's radius, subject to constraints ensuring all remaining circles are contained.

In practice

• Use preprocessing to reduce problem complexity.
• Consider SOCP for robust geometric optimization.
• Compare NLP and SOCP solver precision for specific cases.

Topics

• Minimum Enclosing Circle
• Nonlinear Programming
• Second-Order Conic Program
• Geometric Optimization
• Data Preprocessing

Best for: Research Scientist, Software Engineer

Related on AIssential

• Minimum enclosing ellipse — Optimizing geometric shapes around data points benefits from convex formulations and good initial solutions.
• Revisiting a crazy global NLP problem — Non-convex quadratic optimization problems can be intractable even for small instances.
• Prism: Symbolic Superoptimization of Tensor Programs — Prism superoptimizes tensor programs using symbolic graphs and a two-level search for significant speedups.
• Parametric Modular Answer Set Programs Made Declarative — Parametric modular logic programs enhance first-order ASP with structured, parameterizable subprograms and intensionality.
• Uncovering Symmetry Transfer in Large Language Models via Layer-Peeled Optimization — LLM training implicitly transfers data symmetries into structured geometric configurations within model parameters.
• Newton's Method is Just Tangent Lines — Newton's Method efficiently finds function roots and optimization minima by iteratively solving local linear or parabolic approximations.

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by Yet Another Math Programming Consultant.