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implement full dimensional polytope computational examples
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‎examples/full_dimensional_polytope/CMakeLists.txt‎

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# VolEsti (volume computation and sampling library)
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# Copyright (c) 2012-2025 Vissarion Fisikopoulos
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# Copyright (c) 2018-2025 Apostolos Chalkis
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# Licensed under GNU LGPL.3, see LICENCE file
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project( VolEsti )
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CMAKE_MINIMUM_REQUIRED(VERSION 3.11)
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set(CMAKE_ALLOW_LOOSE_LOOP_CONSTRUCTS true)
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# Locate Intel MKL root (in case it is enabled)
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if (APPLE)
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set(MKLROOT /opt/intel/oneapi/mkl/latest)
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elseif(UNIX)
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set(MKLROOT /opt/intel/oneapi/mkl/latest)
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endif()
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if(COMMAND cmake_policy)
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cmake_policy(SET CMP0003 NEW)
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endif(COMMAND cmake_policy)
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option(DISABLE_NLP_ORACLES "Disable non-linear oracles (used in collocation)" ON)
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option(BUILTIN_EIGEN "Use eigen from ../../external" OFF)
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option(USE_MKL "Use MKL library to build eigen" OFF)
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if(DISABLE_NLP_ORACLES)
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add_definitions(-DDISABLE_NLP_ORACLES)
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else()
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find_library(IFOPT NAMES libifopt_core.so PATHS /usr/local/lib)
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find_library(IFOPT_IPOPT NAMES libifopt_ipopt.so PATHS /usr/local/lib)
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find_library(GMP NAMES libgmp.so PATHS /usr/lib/x86_64-linux-gnu /usr/lib/i386-linux-gnu)
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find_library(MPSOLVE NAMES libmps.so PATHS /usr/local/lib)
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find_library(PTHREAD NAMES libpthread.so PATHS /usr/lib/x86_64-linux-gnu /usr/lib/i386-linux-gnu)
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find_library(FFTW3 NAMES libfftw3.so.3 PATHS /usr/lib/x86_64-linux-gnu /usr/lib/i386-linux-gnu)
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if (NOT IFOPT)
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message(FATAL_ERROR "This program requires the ifopt library, and will not be compiled.")
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elseif (NOT GMP)
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message(FATAL_ERROR "This program requires the gmp library, and will not be compiled.")
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elseif (NOT MPSOLVE)
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message(FATAL_ERROR "This program requires the mpsolve library, and will not be compiled.")
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elseif (NOT FFTW3)
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message(FATAL_ERROR "This program requires the fftw3 library, and will not be compiled.")
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else()
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message(STATUS "Library ifopt found: ${IFOPT}")
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message(STATUS "Library gmp found: ${GMP}")
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message(STATUS "Library mpsolve found: ${MPSOLVE}")
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message(STATUS "Library fftw3 found:" ${FFTW3})
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endif(NOT IFOPT)
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endif(DISABLE_NLP_ORACLES)
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# Include CMake modules for dependencies
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include("../../external/cmake-files/Eigen.cmake")
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GetEigen()
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include("../../external/cmake-files/Boost.cmake")
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GetBoost()
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include("../../external/cmake-files/LPSolve.cmake")
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GetLPSolve()
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include("../../external/cmake-files/SuiteSparse.cmake")
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GetSuiteSparse()
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set(CMAKE_EXPORT_COMPILE_COMMANDS "ON")
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if (USE_MKL)
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find_library(BLAS NAMES libblas.so libblas.dylib PATHS /usr/local/Cellar/lapack/3.9.1_1/lib /usr/lib/x86_64-linux-gnu /usr/lib/i386-linux-gnu /usr/local/Cellar/openblas/0.3.15_1/lib /usr/lib)
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find_library(GFORTRAN NAME libgfortran.dylib PATHS /usr/local/Cellar/gcc/10.2.0_4/lib/gcc/10)
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find_library(LAPACK NAME liblapack.dylib PATHS /usr/lib)
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find_library(OPENMP NAME libiomp5.dylib PATHS /opt/intel/oneapi/compiler/2021.1.1/mac/compiler/lib)
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include_directories (BEFORE ${MKLROOT}/include)
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set(PROJECT_LIBS ${BLAS_LIBRARIES} ${LAPACK_LIBRARIES} ${GFORTRAN_LIBRARIES})
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set(MKL_LINK "-L${MKLROOT}/lib -Wl,-rpath,${MKLROOT}/lib -lmkl_intel_ilp64 -lmkl_sequential -lmkl_core -lpthread -lm -ldl")
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add_definitions(-DEIGEN_USE_MKL_ALL)
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else()
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set(MKL_LINK "")
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endif(USE_MKL)
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# Include directories
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include_directories (BEFORE ../../external)
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include_directories (BEFORE ../../external/minimum_ellipsoid)
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include_directories (BEFORE ../../include/)
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# for Eigen
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if (${CMAKE_VERSION} VERSION_LESS "3.12.0")
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add_compile_options(-D "EIGEN_NO_DEBUG")
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else ()
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add_compile_definitions("EIGEN_NO_DEBUG")
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endif ()
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# Compiler flags
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add_definitions(${CMAKE_CXX_FLAGS} "-std=c++17") # enable C++17 standard
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add_definitions(${CMAKE_CXX_FLAGS} "-O3") # optimization
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add_definitions(${CXX_COVERAGE_COMPILE_FLAGS} "-lm")
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add_definitions(${CXX_COVERAGE_COMPILE_FLAGS} "-ldl")
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add_definitions(${CXX_COVERAGE_COMPILE_FLAGS} "-DBOOST_NO_AUTO_PTR")
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# Build the example executable
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add_executable(full_dimensional_example full_dimensional_example.cpp)
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# Link libraries (including SuiteSparse for the full_dimensional_polytope function)
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target_link_libraries(full_dimensional_example
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lp_solve
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${SUITESPARSE_LIBRARIES}
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${MKL_LINK}
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)

‎examples/full_dimensional_polytope/README.md‎

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# Full Dimensional Polytope Examples
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This directory contains examples demonstrating the `compute_full_dimensional_polytope` function from `include/preprocess/full_dimensional_polytope.hpp`.
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## Overview
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The `compute_full_dimensional_polytope` function transforms a potentially lower-dimensional polytope defined by:
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- **Equality constraints**: `Aeq * x = beq`
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- **Inequality constraints**: `A * x <= b`
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Into a full-dimensional polytope: `A_full * y <= b_full`
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The transformation provides:
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- **shift**: A point satisfying the equality constraints
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- **N**: Nullspace basis matrix with orthonormal columns
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- **Mapping**: `x = shift + N * y` (from reduced space to original space)
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## Building the Example
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From this directory:
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```bash
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mkdir build
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cd build
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cmake ..
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make
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```
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## Running the Example
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```bash
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./full_dimensional_example
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```
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The program will run through 6 examples, pausing between each for you to review the output.
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## Output Format
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For each example, the program displays:
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1. **Description**: What the example demonstrates
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2. **Input**: The equality and inequality constraints
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3. **Results**:
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- Original and reduced dimensions
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- Shift vector
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- Nullspace basis N
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- Output polytope dimensions
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- Orthonormality verification
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## Key Features Demonstrated
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- ✅ Automatic dimension reduction
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- ✅ Orthonormal nullspace basis
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- ✅ Handling of sparse constraints
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- ✅ Multiple equality constraints
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- ✅ Extreme reductions (nD → 1D)
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- ✅ Real-world polytopes (Birkhoff)
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## Mathematical Background
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Given a polytope `P = {x : Aeq*x = beq, A*x <= b}` which is lower-dimensional (rank(Aeq) > 0), the function computes:
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1. **Shift vector**: Solves `Aeq * shift = beq` using SPQR
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2. **Nullspace basis**: Computes `N` such that `Aeq * N = 0` via QR factorization
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3. **Transformed polytope**: `P' = {y : A*N*y <= b - A*shift}`
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Any point `y` in `P'` maps to a feasible point in `P` via: `x = shift + N*y`
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## Dependencies
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- Eigen3 (matrix operations)
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- SuiteSparse (SPQR for QR factorization, CHOLMOD for linear systems)
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- Boost (for polytope generators)
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- lp_solve (for polytope operations)
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## See Also
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- **Header file**: `include/preprocess/full_dimensional_polytope.hpp`
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- **Tests**: `test/full_dimensional_polytope_test.cpp`

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