TEST_MIN
Test problems for minimization

TEST_MIN is a MATLAB library which defines problems involving the minimization of a scalar function of a scalar argument.

TEST_MIN can be useful for testing algorithms that attempt to minimize a scalar function of a scalar argument. Each problem has an index number, and there are a corresponding set of routines, with names beginning with the index number, to:

• evaluate f(x);
• evaluate f'(x);
• evaluate f"(x);
• return the title of the problem;
• return a starting point;
• return a starting search interval;
• return the exact solution;

There is also a "generic" problem interface, whose routines all begin with "P00". This allows the user to call all possible problems in a single simple loop, by passing the desired index number through the generic interface.

The functions can be invoked by an index number, and include:

1. f(x) = ( x - 2 )^2 + 1;
   a PNG image;
2. f(x) = x^2 + exp ( -x );
   a PNG image;
3. f(x) = x^4 + 2x^2 + x + 3;
   a PNG image;
4. f(x) = exp ( x ) + 0.01 / x;
   a PNG image;
5. f(x) = exp ( x ) - 2 * x + 0.01 / x - 0.000001 / x^2;
   a PNG image;
6. f(x) = 2 - x;
   a PNG image;
7. f(x) = ( x + sin ( x ) ) * exp ( -x^2 );
   a PNG image;
8. f(x) = 3 * x^2 + 1 + ( log ( ( x - pi )^2 ) ) / pi^4;
   a PNG image;
9. f(x) = x^2 - 10 sin ( x^2 - 3x + 2);
   a PNG image;
10. f(x) = cos(x)+5*cos(1.6*x)-2*cos(2*x)+5*cos(4.5*x)+7*cos(9*x);
    a PNG image;
11. f(x) = 1+|3x-1|;
    a PNG image;

Licensing:

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

Languages:

TEST_MIN 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:

BRENT, a MATLAB 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.

NMS, a FORTRAN90 library which includes a routine called FMIN which seeks the minimizer of a scalar function of a scalar argument.

Reference:

1. Isabel Beichl, Dianne O'Leary, Francis Sullivan,
   Monte Carlo Minimization and Counting: One, Two, Too Many,
   Computing in Science and Engineering,
   Volume 9, Number 1, January/February 2007.
2. Richard Brent,
   Algorithms for Minimization without Derivatives,
   Dover, 2002,
   ISBN: 0-486-41998-3,
   LC: QA402.5.B74.
3. David Kahaner, Cleve Moler, Steven Nash,
   Numerical Methods and Software,
   Prentice Hall, 1989,
   ISBN: 0-13-627258-4,
   LC: TA345.K34.
4. Arnold Krommer, Christoph Ueberhuber,
   Numerical Integration on Advanced Computer Systems,
   Springer, 1994,
   ISBN: 3540584102,
   LC: QA299.3.K76.
5. Dianne O'Leary,
   Scientific Computing with Case Studies,
   SIAM, 2008,
   ISBN13: 978-0-898716-66-5,
   LC: QA401.O44.
6. LE Scales,
   Introduction to Non-Linear Optimization,
   Springer, 1985,
   ISBN: 0-387-91252-5,
   LC: QA402.5.S33.

Source Code:

• p00_f.m evaluates the function for any problem.
• p00_f1.m evaluates the first derivative for any problem.
• p00_f1_dif.m approximates the first derivative via finite differences.
• p00_f2.m evaluates the second derivative for any problem.
• p00_f2_dif.m approximates the second derivative via finite differences.
• p00_fmin.m seeks a minimizer of a scalar function of a scalar variable.
• p00_interval.m returns a bracketing interval for any problem.
• p00_prob_num.m returns the number of problems available.
• p00_sol.m returns the solution for any problem.
• p00_start.m returns a starting point for optimization for any problem.
• p00_title.m returns a title for any problem.
• p01_f.m evaluates the objective function for problem 1.
• p01_f1.m evaluates the first derivative for problem 1.
• p01_f2.m evaluates the second derivative for problem 1.
• p01_interval.m returns a starting interval for optimization for problem 1.
• p01_sol.m returns the solution for problem 1.
• p01_start.m returns a starting point for optimization for problem 1.
• p01_title.m returns a title for problem 1.
• p02_f.m evaluates the objective function for problem 2.
• p02_f1.m evaluates the first derivative for problem 2.
• p02_f2.m evaluates the second derivative for problem 2.
• p02_interval.m returns a starting interval for optimization for problem 2.
• p02_sol.m returns the solution for problem 2.
• p02_start.m returns a starting point for optimization for problem 2.
• p02_title.m returns a title for problem 2.
• p03_f.m evaluates the objective function for problem 3.
• p03_f1.m evaluates the first derivative for problem 3.
• p03_f2.m evaluates the second derivative for problem 3.
• p03_interval.m returns a starting interval for optimization for problem 3.
• p03_sol.m returns the solution for problem 3.
• p03_start.m returns a starting point for optimization for problem 3.
• p03_title.m returns a title for problem 3.
• p04_f.m evaluates the objective function for problem 4.
• p04_f1.m evaluates the first derivative for problem 4.
• p04_f2.m evaluates the second derivative for problem 4.
• p04_interval.m returns a starting interval for optimization for problem 4.
• p04_sol.m returns the solution for problem 4.
• p04_start.m returns a starting point for optimization for problem 4.
• p04_title.m returns a title for problem 4.
• p05_f.m evaluates the objective function for problem 5.
• p05_f1.m evaluates the first derivative for problem 5.
• p05_f2.m evaluates the second derivative for problem 5.
• p05_interval.m returns a starting interval for optimization for problem 5.
• p05_sol.m returns the solution for problem 5.
• p05_start.m returns a starting point for optimization for problem 5.
• p05_title.m returns a title for problem 5.
• p06_f.m evaluates the objective function for problem 6.
• p06_f1.m evaluates the first derivative for problem 6.
• p06_f2.m evaluates the second derivative for problem 6.
• p06_interval.m returns a starting interval for optimization for problem 6.
• p06_sol.m returns the solution for problem 6.
• p06_start.m returns a starting point for optimization for problem 6.
• p06_title.m returns a title for problem 6.
• p07_f.m evaluates the objective function for problem 7.
• p07_f1.m evaluates the first derivative for problem 7.
• p07_f2.m evaluates the second derivative for problem 7.
• p07_interval.m returns a starting interval for optimization for problem 7.
• p07_sol.m returns the solution for problem 7.
• p07_start.m returns a starting point for optimization for problem 7.
• p07_title.m returns a title for problem 7.
• p08_f.m evaluates the objective function for problem 8.
• p08_f1.m evaluates the first derivative for problem 8.
• p08_f2.m evaluates the second derivative for problem 8.
• p08_interval.m returns a starting interval for optimization for problem 8.
• p08_sol.m returns the solution for problem 8.
• p08_start.m returns a starting point for optimization for problem 8.
• p08_title.m returns a title for problem 8.
• p09_f.m evaluates the objective function for problem 9.
• p09_f1.m evaluates the first derivative for problem 9.
• p09_f2.m evaluates the second derivative for problem 9.
• p09_interval.m returns a starting interval for optimization for problem 9.
• p09_sol.m returns the solution for problem 9.
• p09_start.m returns a starting point for optimization for problem 9.
• p09_title.m returns a title for problem 9.
• p10_f.m evaluates the objective function for problem 10.
• p10_f1.m evaluates the first derivative for problem 10.
• p10_f2.m evaluates the second derivative for problem 10.
• p10_interval.m returns a starting interval for optimization for problem 10.
• p10_sol.m returns the solution for problem 10.
• p10_start.m returns a starting point for optimization for problem 10.
• p10_title.m returns a title for problem 10.
• p11_f.m evaluates the objective function for problem 11.
• p11_f1.m evaluates the first derivative for problem 11.
• p11_f2.m evaluates the second derivative for problem 11.
• p11_interval.m returns a starting interval for optimization for problem 11.
• p11_sol.m returns the solution for problem 11.
• p11_start.m returns a starting point for optimization for problem 11.
• p11_title.m returns a title for problem 11.
• r8_sign.m returns the sign of an R8.
• timestamp.m prints the current YMDHMS date as a time stamp.

Examples and Tests:

• test_min_test.m, calls all the tests.
• test_min_test01.m, prints the title of each problem.
• test_min_test02.m, evaluates the objective function at each starting point.
• test_min_test03.m, compares the exact and approximate first derivatives.
• test_min_test04.m, compares the exact and approximate second derivatives.
• test_min_test05.m, carries out a simple bisection method.
• test_min_test06.m, carries out a version of Brent's derivative-free minimizer.
• test_min_prb_output.txt, the output file.

You can go up one level to the MATLAB source codes.