Pill capsule minimum-area problem

A two-variable constrained optimization problem that minimizes capsule surface area at fixed volume.

See Optimization Problem Catalog for the optimization family index.

Quick Facts

Field	Value
Problem ID	pill_capsule_min_area
Problem Family	optimization
Implementation	design_research_problems.problems.optimization._pill:PillCapsuleMinArea
Capabilities	baseline-solver, bounded-variables, equality-constraint, statement-markdown
Study Suitability	none
Tags	optimization, continuous, nonlinear, single-objective

Taxonomy

Formulation

nonlinear_program

Convexity

not_guaranteed

Design Variable Type

continuous

Is Dynamic

no

Orientation

mathematical

Feasibility Ratio Hint

0.5

Objective Mode

single

Constraint Nature

hard

Bounds Summary

radius and length bounded on [0, 1]

Tags

optimization, continuous, nonlinear, single-objective

Statement

Minimize the surface area of a capsule-shaped pill while satisfying a fixed volume requirement. The design variables are the capsule radius and cylindrical section length. Both variables are bounded on [0, 1].

This seed problem is a compact constrained optimization benchmark for design research workflows and solver experiments.

Problem Shape

Field	Value
Design Variable Count	2
Bound Summary	radius and length bounded on [0, 1]
Total Constraint Count	1
Equality Constraint Count	1
Inequality Constraint Count	0

Variable Bounds

Variable	Lower Bound	Upper Bound
x[0]	0	1
x[1]	0	1

Manifest Parameters

Key	Value
required_volume	1e-06

Library Interface

• generate_initial_solution(seed=None)

• objective(x)

• evaluate(x)

• solve(initial_solution=None, seed=None, maxiter=200)