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Pill capsule minimum-area problem
=================================

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

See :doc:`../optimization` for the optimization family index.

Quick Facts
-----------

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   * - 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
-------------

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   * - 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
---------------

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   :widths: 25 37 38

   * - Variable
     - Lower Bound
     - Upper Bound
   * - ``x[0]``
     - 0
     - 1
   * - ``x[1]``
     - 0
     - 1

Manifest Parameters
-------------------

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   * - 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)``