ACM TOMS Algorithm 1028: VTMOP: Solver for Blackbox Multiobjective Optimization Problems

VTMOP is a Fortran 2008 package containing a robust, portable solver and a flexible framework for solving MOPs. Designed for efficiency and scalability to an arbitrary number of objectives, VTMOP attempts to generate uniformly spaced points on a (possibly nonconvex) Pareto front with minimal cost function evaluations. The driver subroutine is VTMOP_SOLVE, which can be run both serially and with parallel function evaluations. Minimal subsets of dependencies such as VTDIRECT, QNSTOP, SHEPPACK, DELAUNAYSPARSE, SLATEC, LAPACK, and BLAS are also provided. Comments at the top of each subroutine document their usage, and examples demonstrating the driver's usage are given in src/sample.f90.

A Python interface for VTMOP is also available through libEnsemble:

https://github.com/Libensemble/libe-community-examples

Usage Summary

VTMOP features two distinctly different user-interfaces, with varying levels of flexibility.

The first and simplest user-interface is the driver subroutine VTMOP_SOLVE, which is included in the module VTMOP_LIB. VTMOP_SOLVE accepts problem dimensions, simple bound constraints, an objective function, some (optional) parameter settings, and (optionally) any user-defined surrogate models and optimization routines. The default setting is to perform an adaptive search using the VTDIRECT95 software package, fit LSHEP surrogate models, and optimize the surrogate models using direct search with the polling strategy generalized pattern search (GPS).

In order to use VTMOP_SOLVE, the blackbox multiobjective cost function must be available as a Fortran 2008 subroutine. For information on how to achieve this, including cases where F is an ISO C/C++ function and when F is a command line executable, see src/OBJ_FUNC_README. Additional information on VTMOP_SOLVE is provided in the comments around each subroutine definition, in the file vtmop.f90.

The second interface is the return-to-caller interface, which allows advanced users to perform function evaluations in batches, in an independent environment. The return-to-caller interface is contained in the module VTMOP_MOD, and contains four driver subroutines (below). Additional information on these subroutines is provided in the code documentation, in the file vtmop.f90.

• VTMOP_INIT initializes an instance of the data type VTMOP_TYPE for passing data between subroutines.
• VTMOP_LTR identifies the most isolated point on the Pareto front, constructs a local trust region (LTR) about that point, and returns a region of the design for the user to explore independently.
• VTMOP_OPT fits several surrogate models to the current dataset, and uses these models to propose a set of candidate points in the LTR, for evaluation by the user.
• VTMOP_FINALIZE post processes the dataset gathered throughout the iterations of the algorithm, returns the nondominated and efficient point sets, and frees all internal memory.

VTMOP also includes a checkpointing system, a detailed error handling system, and maintains a database of all function evaluations ever taken, which can be retrieved using optional output arrays.

For more detailed information on usage, see the corresponding sections in the src/USERS file.

Source Code and Package Layout

The directory src contains all source code and documentation, exactly as it appeared in ACM TOMS Algorithm 1028.

The physical organization of the src directory is as follows:

• The file depend_graph.txt contains a diagram of the dependency tree for the VTMOP package and its sample main program.
• The file vtmop.f90 is the main Fortran 2008 file containing the VTMOP_MOD and VTMOP_LIB libraries containing worker and driver interfaces, respectively.
• The file delsparse.f90 contains the module and driver subroutines for DELAUNAYSPARSE (ACM TOMS Algorithm 1012).
• The file linear_shepard.f90 is a Fortran 95 module for computing the LSHEP surrogate model (ACM TOMS Algorithm 905).
• The file qnstop.f90 contains the LATINDESIGN function from QNSTOP (ACM TOMS Algorithm 1007).
• The file sVTdirect.f90 contains a serial implementation of the Fortran 95 algorithm VTdirect95 (ACM TOMS Algorithm 897).
• The file bVTdirect.f90 contains a slight modification to sVTdirect.f90, which allows for usage in VTMOP_SOLVE's parallel paradigm.
• The file shared_modules.f90 contains modules and subroutines that are used by VTdirect95, as well as the module REAL_PRECISION, which is used for approximately 64 bit arithmetic.
• The file slatec.f contains the subroutine DWNNLS and its dependencies from the SLATEC library. This library has been slightly modified to comply with the modern Fortran standards. Additionally, legacy implementations of the BLAS subroutines DROTM and DTROMG have been included under different names to avoid dependency issues.
• The files lapack.f and blas.f contain all LAPACK and BLAS subroutines that are referenced (both directly and indirectly) in VTMOP or its dependencies.
• vtmop_func.f90 contains the module VTMOP_FUNC_MOD, which contains several multiobjective test problems.
• samples.f90 contains sample code for building and running the test problems in vtmop_func.f90 and checks the serial installation for correctness.
• samplep.f90 contains sample code for building and running the test problems in vtmop_func.f90 and checks the parallel installation for correctness.
• cl_objfunc.f90 contains sample code for implementing a command line executable as a Fortran subroutine, while matching the interface expected by VTMOP_SOLVE.
• A sample GNU Makefile is also provided.

Building and Running

From here on, the files samples.f90 and samplep.f90 will be referred to collectively as sample{s|p}.f90.

To build the sample code and test the installation, use the following command, where $(F90) is a Fortran 2008 compiler and $(OPTS) contains the compiler options (including the option to build with OpenMP).

If BLAS and LAPACK exist on your system, then $(LIBS) should contain flags to link those libraries and blas.f and lapack.f can be removed from the following command; otherwise, $(LIBS) can be ignored.

$(F90) $(OPTS) $(LIBS) shared_modules.f90 blas.f lapack.f slatec.f \
    qnstop.f90 sVTdirect.f90 bVTdirect.f90 delsparse.f90           \
    linear_shepard.f90 vtmop.f90 vtmop_func.f90 sample{s|p}.f90    \
    -o sample{s|p}

To test the installation, use

./sample{s|p}

Before running the parallel driver "samplep", set the following environment variables:

• export OMP_NESTED=TRUE
• export OMP_NUM_THREADS=T1,T2

where T1 = MAX(NUMBER OF PROCESSORS, NUMBER OF OBJECTIVES) and T2 = CEILING(NUMBER OF PROCESSORS / T1). For the sample code provided, the number of objectives is three.

This code has been tested with the GNU 5.4.0 (and newer) and the Intel 17.0.4 Fortran compilers. Other modern Fortran compilers may not offer full support for the Fortran 2008 standard. If using a different compiler, first check whether it supports

• passing internal procedures as actual arguments
• usage of the IEEE_ARITHMETIC intrinsic module.

Citations and Additional References

If you use VTMOP as part of a published work, please cite the following publication:

@article{chang2022algorithm,
  author={Chang, Tyler H. and Watson, Layne T. and Larson, Jeffrey and Neveu, Nicole and Thacker, William I. and Deshpande, Shubhangi and Lux, Thomas C. H.},
  year={2022},
  title={Algorithm {1028}: {VTMOP}: {S}olver for blackbox multiobjective optimization problems},
  journal={ACM Transactions on Mathematical Software},
  volume={48},
  number={3},
  articleno={36},
  pages={1-34},
  doi={10.1145/3529258}
}

Contacts and Contributors

For further inquiries, contact

• Tyler Chang, [email protected]
• Layne Watson, [email protected]

For a full list of contributors, including contributions to the source code, theoretical contributions, and proof-reading/writing documentation, see the src/CONTRIBUTORS document.

Special thanks to the editors and anonymous reviewers of ACM TOMS, for their helpful comments and suggestions.