TEST_OPT_CON
Test Functions for Scalar Optimization
Constrained to a Hyper-Rectangle

TEST_OPT_CON is a C++ library which defines a set of constrained global optimization problems.

A typical constrained global optimization problem presents an M-dimensional hyper-rectangle bounded by A(1:M) <= X(1:M) <= B(1:M), and a scalar-valued function F(X). The task is to find a point X within the hyper-rectangle at which the function takes its minimum value.

This task is impossible, mathematically and in general. However, the problems that can be solved mathematically are often not the ones encountered in real life. Thus, it is useful to try to solve an impossible problem, since an approximate answer to such a problem can be all we can hope for or need.

The functions defined include:

1. NM1: Niederreiter-McCurley function #1, M = 4;
2. NM2: Niederreiter-McCurley function #2, M = 4;
3. NM3: Niederreiter-McCurley function #3, M = 4;
4. NM4: Niederreiter-McCurley function #4, M = 4;
5. NM5: Niederreiter-McCurley function #5, M = 4;
6. NM6: Niederreiter-McCurley function #6, M = 4;
7. L02: Langerman function, M = 2;
8. L10: Langerman function, M = 10;

For each function, the library includes a routine to evaluate the function, but also routines to return the limits of the hyper-rectangle, the spatial dimension, the solution, if known, and a title for the problem. These routines have a standard set of names based on the function index. For instance, for function #3, we have the routines:

• P03_AB returns bounds for problem 3.
• P03_F returns the objective function value for problem 3.
• P03_M returns the spatial dimension for problem 3.
• P03_SOL returns known solutions for problem 3.
• P03_TITLE returns a title for problem 3.

Since the same interface is used for each function, if you wish to work with problem 6 instead, you simply change the "03" to "06" in your routine calls.

If you wish to call all of the functions, you can write a concise program to do so by using the generic interface, in which the function names use the prefix P00_, and require the specific problem index to be supplied as an extra input argument:

• P00_AB returns bounds for a problem specified by index.
• P00_F returns the objective function value for a problem specified by index.
• P00_M returns the spatial dimension for a problem specified by index.
• P00_SOL returns known solutions for a problem specified by index.
• P00_TITLE returns a title for a problem specified by index.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

TEST_OPT_CON is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

ASA047, a C++ library which minimizes a scalar function of several variables using the Nelder-Mead algorithm.

BRENT, a C++ library which contains Richard Brent's routines for finding the zero, local minimizer, or global minimizer of a scalar function of a scalar argument, without the use of derivative information.

COMPASS_SEARCH, a C++ program which minimizes a scalar function of several variables using the compass search algorithm.

DQED, a FORTRAN90 library which solves constrained least squares problems.

ENTRUST, a MATLAB program which minimizes a scalar function of several variables using trust-region methods.

FMINCON, a MATLAB library which demonstrates the use of the fmincon() function to seek the minimizer of a function f(x) subject to a constraint.

NELDER_MEAD, a MATLAB program which minimizes a scalar function of several variables using the Nelder-Mead algorithm, by Jeff Borggaard.

PRAXIS, a FORTRAN90 library which implements the principal axis method of Richard Brent for minimization of a function without the use of derivatives.

TEST_OPT, a FORTRAN90 library which defines test problems requiring the minimization of a scalar function of several variables.

TEST_OPTIMIZATION, a FORTRAN90 library which defines test problems for the minimization of a scalar function of several variables, as described by Molga and Smutnicki.

TOMS178, a C++ library which optimizes a scalar functional of multiple variables using the Hooke-Jeeves method.

TOMS611, a FORTRAN90 library which can be used to seek the minimizer of a scalar functional of multiple variables.

Reference:

1. Harald Niederreiter, Kevin McCurley,
   Optimization of functions by quasi-random search methods,
   Computing,
   Volume 22, Number 2, 1979, pages 119-123.

Source Code:

• test_opt_con.cpp, the source code.
• test_opt_con.hpp, the include file.
• test_opt_con.sh, BASH commands to compile the source code.

Examples and Tests:

• test_opt_con_prb.cpp, a sample calling program.
• test_opt_con_prb.sh, BASH commands to compile and run the sample program.
• test_opt_con_prb_output.txt, the output file.

List of Routines:

• P00_AB returns bounds for a problem specified by index.
• P00_F returns the objective function value for a problem specified by index.
• P00_M returns the spatial dimension for a problem specified by index.
• P00_PROBLEM_NUM returns the number of problems.
• P00_SOL returns known solutions for a problem specified by index.
• P00_TITLE returns a title for a problem specified by index.
• P01_AB returns bounds for problem 1.
• P01_F returns the objective function value for problem 1.
• P01_M returns the spatial dimension for problem 1.
• P01_SOL returns known solutions for problem 1.
• P01_TITLE returns a title for problem 1.
• P02_AB returns bounds for problem 2.
• P02_F returns the objective function value for problem 2.
• P02_M returns the spatial dimension for problem 2.
• P02_SOL returns known solutions for problem 2.
• P02_TITLE returns a title for problem 2.
• P03_AB returns bounds for problem 3.
• P03_F returns the objective function value for problem 3.
• P03_M returns the spatial dimension for problem 3.
• P03_SOL returns known solutions for problem 3.
• P03_TITLE returns a title for problem 3.
• P04_AB returns bounds for problem 4.
• P04_F returns the objective function value for problem 4.
• P04_M returns the spatial dimension for problem 4.
• P04_SOL returns known solutions for problem 4.
• P04_TITLE returns a title for problem 4.
• P05_AB returns bounds for problem 5.
• P05_F returns the objective function value for problem 5.
• P05_M returns the spatial dimension for problem 5.
• P05_SOL returns known solutions for problem 5.
• P05_TITLE returns a title for problem 5.
• P06_AB returns bounds for problem 6.
• P06_F returns the objective function value for problem 6.
• P06_M returns the spatial dimension for problem 6.
• P06_SOL returns known solutions for problem 6.
• P06_TITLE returns a title for problem 6.
• P07_AB returns bounds for problem 7.
• P07_F returns the objective function value for problem 5.
• P07_M returns the spatial dimension for problem 7.
• P07_SOL returns known solutions for problem 7.
• P07_TITLE returns a title for problem 7.
• P08_AB returns bounds for problem 8.
• P08_F returns the objective function value for problem 8.
• P08_M returns the spatial dimension for problem 8.
• P08_SOL returns known solutions for problem 8.
• P08_TITLE returns a title for problem 8.
• R8COL_UNIFORM_NEW fills an R8COL with scaled pseudorandom numbers.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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