# test_opt

test_opt, a MATLAB code which defines test problems for the scalar function optimization problem.

The scalar function optimization problem is to find a value for the N-dimensional vector X which minimizes the value of the given scalar function F(X). The function F(X) is not usually defined as the sum of squares of other functions. The minimum function value is not guaranteed to be zero.

Any system of M nonlinear functions in N unknowns can be turned into a scalar optimization problem. One way to do this is to define the functional F(X) to be the sum of the squares of the original nonlinear functions. The minimizer of F will then minimize the sum of the squares of the residuals. Since this process involves squaring, it can be less accurate than dealing directly with the original nonlinear functions: that is to say, the derived optimization problem may be more convenient to solve, but might provide less accurate results than applying a nonlinear solver to the original system.

If a function F(X) is differentiable, then at an optimum, the gradient vector must vanish. Thus, it is also possible to start with an optimization problem involving F(X) and turn it into a problem in which we seek a zero of the nonlinear functions represented by the gradient of F. Of course, the gradient must be zero at a mininum, but the converse does not hold; thus unless we know more about F, it is not safe to try to replace the optimization problem by a nonlinear function solution.

For each test problem, routines are provided to evaluate the function, gradient vector, and hessian matrix. Routines are also provided to indicate the number of variables, the problem title, a suitable starting point, and a minimizing solution, if known.

The functions defined include:

1. The Fletcher-Powell helical valley function,
N = 3.
2. The Biggs EXP6 function,
N = 6.
3. The Gaussian function,
N = 3.
4. The Powell badly scaled function,
N = 2.
5. The Box 3-dimensional function,
N = 3.
6. The variably dimensioned function,
1 <= N.
7. The Watson function,
2 <= N.
8. The penalty function #1,
1 <= N.
9. The penalty function #2,
1 <= N.
10. The Brown badly scaled function,
N = 2.
11. The Brown and Dennis function,
N = 4.
12. The Gulf R&D function,
N = 3.
13. The trigonometric function,
1 <= N.
14. The extended Rosenbrock parabolic valley function,
1 <= N.
15. The extended Powell singular quartic function,
4 <= N.
16. The Beale function,
N = 2.
17. The Wood function,
N = 4.
18. The Chebyquad function,
1 <= N.
19. Leon's cubic valley function,
N = 2.
20. Gregory and Karney's Tridiagonal Matrix Function,
1 <= N.
21. The Hilbert function,
1 <= N.
22. The De Jong Function F1,
N = 3.
23. The De Jong Function F2,
N = 2.
24. The De Jong Function F3 (discontinuous),
N = 5.
25. The De Jong Function F4 (Gaussian noise),
N = 30.
26. The De Jong Function F5,
N = 2.
27. The Schaffer Function F6,
N = 2.
28. The Schaffer Function F7,
N = 2.
29. The Goldstein Price Polynomial,
N = 2.
30. The Branin RCOS Function,
N = 2.
31. The Shekel SQRN5 Function,
N = 4.
32. The Shekel SQRN7 Function,
N = 4.
33. The Shekel SQRN10 Function,
N = 4.
34. The Six-Hump Camel-Back Polynomial,
N = 2.
35. The Shubert Function,
N = 2.
36. The Stuckman Function,
N = 2.
37. The Easom Function,
N = 2.
38. The Bohachevsky Function #1,
N = 2.
39. The Bohachevsky Function #2,
N = 2.
40. The Bohachevsky Function #3,
N = 2.
41. The Colville Polynomial,
N = 4.
42. The Powell 3D function,
N = 3.
43. The Himmelblau function,
N = 2.

### Licensing:

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

### Languages:

test_opt is available in a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

asa047, a MATLAB code which minimizes a scalar function of several variables using the Nelder-Mead algorithm.

compass_search, a MATLAB code which seeks the minimizer of a scalar function of several variables using compass search, a direct search algorithm that does not use derivatives.

nelder_mead, a MATLAB code which minimizes a scalar function of several variables using the Nelder-Mead algorithm.

polynomials, a MATLAB code which defines multivariate polynomials over rectangular domains, for which certain information is to be determined, such as the maximum and minimum values.

praxis, a MATLAB code which minimizes a scalar function of several variables, without requiring derivative information, by Richard Brent.

test_opt_con, a MATLAB code which defines test problems for the minimization of a scalar function of several variables, with the search constrained to lie within a specified hyper-rectangle.

test_optimization, a MATLAB code which defines test problems for the minimization of a scalar function of several variables, as described by Molga and Smutnicki.

### Reference:

1. Evelyn Beale,
On an Iterative Method for Finding a Local Minimum of a Function of More than One Variable,
Technical Report 25,
Statistical Techniques Research Group,
Princeton University, 1958.
2. Richard Brent,
Algorithms for Minimization without Derivatives,
Dover, 2002,
ISBN: 0-486-41998-3,
LC: QA402.5.B74.
3. John Dennis, David Gay, Phuong Vu,
A new nonlinear equations test problem,
Technical Report 83-16,
Mathematical Sciences Department,
Rice University, 1983.
4. John Dennis, Robert Schnabel,
Numerical Methods for Unconstrained Optimization and Nonlinear Equations,
SIAM, 1996,
ISBN13: 978-0-898713-64-0,
LC: QA402.5.D44.
5. Noel deVilliers, David Glasser,
A continuation method for nonlinear regression,
SIAM Journal on Numerical Analysis,
Volume 18, 1981, pages 1139-1154.
6. Chris Fraley,
Solution of nonlinear least-squares problems,
Technical Report STAN-CS-1165,
Computer Science Department,
Stanford University, 1987.
7. Chris Fraley,
Software performance on nonlinear least-squares problems,
Technical Report SOL 88-17,
Systems Optimization Laboratory,
Department of Operations Research,
Stanford University, 1988.
8. David Himmelblau,
Applied Nonlinear Programming,
McGraw Hill, 1972,
ISBN13: 978-0070289215,
LC: T57.8.H55.
9. A Leon,
A Comparison of Eight Known Optimizing Procedures,
in Recent Advances in Optimization Techniques,
edited by Abraham Lavi, Thomas Vogl,
Wiley, 1966.
10. JJ McKeown,
Specialized versus general-purpose algorithms for functions that are sums of squared terms,
Mathematical Programming,
Volume 9, 1975, pages 57-68.
11. JJ McKeown,
On algorithms for sums of squares problems,
in Towards Global Optimization,
edited by L Dixon, Gabor Szego,
North-Holland, 1975, pages 229-257.
12. Zbigniew Michalewicz,
Genetic Algorithms + Data Structures = Evolution Programs,
Third Edition,
Springer, 1996,
ISBN: 3-540-60676-9,
LC: QA76.618.M53.
13. Jorge More, Burton Garbow, Kenneth Hillstrom,
Testing unconstrained optimization software,
ACM Transactions on Mathematical Software,
Volume 7, Number 1, March 1981, pages 17-41.
14. Jorge More, Burton Garbow, Kenneth Hillstrom,
Algorithm 566: FORTRAN Subroutines for Testing unconstrained optimization software,
ACM Transactions on Mathematical Software,
Volume 7, Number 1, March 1981, pages 136-140.
15. Michael Powell,
An Efficient Method for Finding the Minimum of a Function of Several Variables Without Calculating Derivatives,
Computer Journal,
Volume 7, Number 2, 1964, pages 155-162.
16. Douglas Salane,
A continuation approach for solving large residual nonlinear least squares problems,
SIAM Journal on Scientific and Statistical Computing,
Volume 8, 1987, pages 655-671.

### Source Code:

• p00_f.m, evaluates the objective function for any problem.
• p00_g.m, evaluates the gradient for any problem.
• p00_gdif.m, approximates the gradient via finite differences.
• p00_h.m, evaluates the Hessian for any problem.
• p00_hdif.m, approximates the Hessian via finite differences.
• p00_problem_num.m, returns the number of problems available.
• p00_n.m, returns the number of variables for any problem.
• 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 01.
• p01_g.m, evaluates the gradient for problem 01.
• p01_h.m, evaluates the Hessian for problem 01.
• p01_n.m, returns the number of variables for problem 01.
• p01_sol.m, returns the solution for problem 01.
• p01_start.m, returns a starting point for optimization for problem 01.
• p01_th.m, returns the value of the TH parameter for problem 01.
• p01_title.m, returns a title for problem 01.
• p02_f.m, evaluates the objective function for problem 02.
• p02_g.m, evaluates the gradient for problem 02.
• p02_h.m, evaluates the Hessian for problem 02.
• p02_n.m, returns the number of variables for problem 02.
• p02_sol.m, returns the solution for problem 02.
• p02_start.m, returns a starting point for optimization for problem 02.
• p02_title.m, returns a title for problem 02.
• p03_f.m, evaluates the objective function for problem 03.
• p03_g.m, evaluates the gradient for problem 03.
• p03_h.m, evaluates the Hessian for problem 03.
• p03_n.m, returns the number of variables for problem 03.
• p03_sol.m, returns the solution for problem 03.
• p03_start.m, returns a starting point for optimization for problem 03.
• p03_title.m, returns a title for problem 03.
• p04_f.m, evaluates the objective function for problem 04.
• p04_g.m, evaluates the gradient for problem 04.
• p04_h.m, evaluates the Hessian for problem 04.
• p04_n.m, returns the number of variables for problem 04.
• p04_sol.m, returns the solution for problem 04.
• p04_start.m, returns a starting point for optimization for problem 04.
• p04_title.m, returns a title for problem 04.
• p05_f.m, evaluates the objective function for problem 05.
• p05_g.m, evaluates the gradient for problem 05.
• p05_h.m, evaluates the Hessian for problem 05.
• p05_n.m, returns the number of variables for problem 05.
• p05_sol.m, returns the solution for problem 05.
• p05_start.m, returns a starting point for optimization for problem 05.
• p05_title.m, returns a title for problem 05.
• p06_f.m, evaluates the objective function for problem 06.
• p06_g.m, evaluates the gradient for problem 06.
• p06_h.m, evaluates the Hessian for problem 06.
• p06_n.m, returns the number of variables for problem 06.
• p06_sol.m, returns the solution for problem 06.
• p06_start.m, returns a starting point for optimization for problem 06.
• p06_title.m, returns a title for problem 06.
• p07_f.m, evaluates the objective function for problem 07.
• p07_g.m, evaluates the gradient for problem 07.
• p07_h.m, evaluates the Hessian for problem 07.
• p07_n.m, returns the number of variables for problem 07.
• p07_sol.m, returns the solution for problem 07.
• p07_start.m, returns a starting point for optimization for problem 07.
• p07_title.m, returns a title for problem 07.
• p08_f.m, evaluates the objective function for problem 08.
• p08_g.m, evaluates the gradient for problem 08.
• p08_h.m, evaluates the Hessian for problem 08.
• p08_n.m, returns the number of variables for problem 08.
• p08_sol.m, returns the solution for problem 08.
• p08_start.m, returns a starting point for optimization for problem 08.
• p08_title.m, returns a title for problem 08.
• p09_f.m, evaluates the objective function for problem 09.
• p09_g.m, evaluates the gradient for problem 09.
• p09_h.m, evaluates the Hessian for problem 09.
• p09_n.m, returns the number of variables for problem 09.
• p09_sol.m, returns the solution for problem 09.
• p09_start.m, returns a starting point for optimization for problem 09.
• p09_title.m, returns a title for problem 09.
• p10_f.m, evaluates the objective function for problem 10.
• p10_g.m, evaluates the gradient for problem 10.
• p10_h.m, evaluates the Hessian for problem 10.
• p10_n.m, returns the number of variables 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_g.m, evaluates the gradient for problem 11.
• p11_h.m, evaluates the Hessian for problem 11.
• p11_n.m, returns the number of variables 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_th.m, returns the value of the TH parameter for problem 11.
• p11_title.m, returns a title for problem 11.
• p12_f.m, evaluates the objective function for problem 12.
• p12_g.m, evaluates the gradient for problem 12.
• p12_h.m, evaluates the Hessian for problem 12.
• p12_n.m, returns the number of variables for problem 12.
• p12_sol.m, returns the solution for problem 12.
• p12_start.m, returns a starting point for optimization for problem 12.
• p12_title.m, returns a title for problem 12.
• p13_f.m, evaluates the objective function for problem 13.
• p13_g.m, evaluates the gradient for problem 13.
• p13_h.m, evaluates the Hessian for problem 13.
• p13_n.m, returns the number of variables for problem 13.
• p13_sol.m, returns the solution for problem 13.
• p13_start.m, returns a starting point for optimization for problem 13.
• p13_title.m, returns a title for problem 13.
• p14_f.m, evaluates the objective function for problem 14.
• p14_g.m, evaluates the gradient for problem 14.
• p14_h.m, evaluates the Hessian for problem 14.
• p14_n.m, returns the number of variables for problem 14.
• p14_sol.m, returns the solution for problem 14.
• p14_start.m, returns a starting point for optimization for problem 14.
• p14_title.m, returns a title for problem 14.
• p15_f.m, evaluates the objective function for problem 15.
• p15_g.m, evaluates the gradient for problem 15.
• p15_h.m, evaluates the Hessian for problem 15.
• p15_n.m, returns the number of variables for problem 15.
• p15_sol.m, returns the solution for problem 15.
• p15_start.m, returns a starting point for optimization for problem 15.
• p15_title.m, returns a title for problem 15.
• p16_f.m, evaluates the objective function for problem 16.
• p16_g.m, evaluates the gradient for problem 16.
• p16_h.m, evaluates the Hessian for problem 16.
• p16_n.m, returns the number of variables for problem 16.
• p16_sol.m, returns the solution for problem 16.
• p16_start.m, returns a starting point for optimization for problem 16.
• p16_title.m, returns a title for problem 16.
• p17_f.m, evaluates the objective function for problem 17.
• p17_g.m, evaluates the gradient for problem 17.
• p17_h.m, evaluates the Hessian for problem 17.
• p17_n.m, returns the number of variables for problem 17.
• p17_sol.m, returns the solution for problem 17.
• p17_start.m, returns a starting point for optimization for problem 17.
• p17_title.m, returns a title for problem 17.
• p18_f.m, evaluates the objective function for problem 18.
• p18_g.m, evaluates the gradient for problem 18.
• p18_h.m, evaluates the Hessian for problem 18.
• p18_n.m, returns the number of variables for problem 18.
• p18_sol.m, returns the solution for problem 18.
• p18_start.m, returns a starting point for optimization for problem 18.
• p18_title.m, returns a title for problem 18.
• p19_f.m, evaluates the objective function for problem 19.
• p19_g.m, evaluates the gradient for problem 19.
• p19_h.m, evaluates the Hessian for problem 19.
• p19_n.m, returns the number of variables for problem 19.
• p19_sol.m, returns the solution for problem 19.
• p19_start.m, returns a starting point for optimization for problem 19.
• p19_title.m, returns a title for problem 19.
• p20_f.m, evaluates the objective function for problem 20.
• p20_g.m, evaluates the gradient for problem 20.
• p20_h.m, evaluates the Hessian for problem 20.
• p20_n.m, returns the number of variables for problem 20.
• p20_sol.m, returns the solution for problem 20.
• p20_start.m, returns a starting point for optimization for problem 20.
• p20_title.m, returns a title for problem 20.
• p21_f.m, evaluates the objective function for problem 21.
• p21_g.m, evaluates the gradient for problem 21.
• p21_h.m, evaluates the Hessian for problem 21.
• p21_n.m, returns the number of variables for problem 21.
• p21_sol.m, returns the solution for problem 21.
• p21_start.m, returns a starting point for optimization for problem 21.
• p21_title.m, returns a title for problem 21.
• p22_f.m, evaluates the objective function for problem 22.
• p22_g.m, evaluates the gradient for problem 22.
• p22_h.m, evaluates the Hessian for problem 22.
• p22_n.m, returns the number of variables for problem 22.
• p22_sol.m, returns the solution for problem 22.
• p22_start.m, returns a starting point for optimization for problem 22.
• p22_title.m, returns a title for problem 22.
• p23_f.m, evaluates the objective function for problem 23.
• p23_g.m, evaluates the gradient for problem 23.
• p23_h.m, evaluates the Hessian for problem 23.
• p23_n.m, returns the number of variables for problem 23.
• p23_sol.m, returns the solution for problem 23.
• p23_start.m, returns a starting point for optimization for problem 23.
• p23_title.m, returns a title for problem 23.
• p24_f.m, evaluates the objective function for problem 24.
• p24_g.m, evaluates the gradient for problem 24.
• p24_h.m, evaluates the Hessian for problem 24.
• p24_n.m, returns the number of variables for problem 24.
• p24_sol.m, returns the solution for problem 24.
• p24_start.m, returns a starting point for optimization for problem 24.
• p24_title.m, returns a title for problem 24.
• p25_f.m, evaluates the objective function for problem 25.
• p25_g.m, evaluates the gradient for problem 25.
• p25_h.m, evaluates the Hessian for problem 25.
• p25_n.m, returns the number of variables for problem 25.
• p25_p_get.m, get parameters for problem 25.
• p25_p_set.m, set parameters for problem 25.
• p25_p_val.m, sets or gets parameters for problem 25.
• p25_sol.m, returns the solution for problem 25.
• p25_start.m, returns a starting point for optimization for problem 25.
• p25_title.m, returns a title for problem 25.
• p26_f.m, evaluates the objective function for problem 26.
• p26_g.m, evaluates the gradient for problem 26.
• p26_h.m, evaluates the Hessian for problem 26.
• p26_n.m, returns the number of variables for problem 26.
• p26_sol.m, returns the solution for problem 26.
• p26_start.m, returns a starting point for optimization for problem 26.
• p26_title.m, returns a title for problem 26.
• p27_f.m, evaluates the objective function for problem 27.
• p27_g.m, evaluates the gradient for problem 27.
• p27_h.m, evaluates the Hessian for problem 27.
• p27_n.m, returns the number of variables for problem 27.
• p27_sol.m, returns the solution for problem 27.
• p27_start.m, returns a starting point for optimization for problem 27.
• p27_title.m, returns a title for problem 27.
• p28_f.m, evaluates the objective function for problem 28.
• p28_g.m, evaluates the gradient for problem 28.
• p28_h.m, evaluates the Hessian for problem 28.
• p28_n.m, returns the number of variables for problem 28.
• p28_sol.m, returns the solution for problem 28.
• p28_start.m, returns a starting point for optimization for problem 28.
• p28_title.m, returns a title for problem 28.
• p29_f.m, evaluates the objective function for problem 29.
• p29_g.m, evaluates the gradient for problem 29.
• p29_h.m, evaluates the Hessian for problem 29.
• p29_n.m, returns the number of variables for problem 29.
• p29_sol.m, returns the solution for problem 29.
• p29_start.m, returns a starting point for optimization for problem 29.
• p29_title.m, returns a title for problem 29.
• p30_f.m, evaluates the objective function for problem 30.
• p30_g.m, evaluates the gradient for problem 30.
• p30_h.m, evaluates the Hessian for problem 30.
• p30_n.m, returns the number of variables for problem 30.
• p30_sol.m, returns the solution for problem 30.
• p30_start.m, returns a starting point for optimization for problem 30.
• p30_title.m, returns a title for problem 30.
• p31_f.m, evaluates the objective function for problem 31.
• p31_g.m, evaluates the gradient for problem 31.
• p31_h.m, evaluates the Hessian for problem 31.
• p31_n.m, returns the number of variables for problem 31.
• p31_sol.m, returns the solution for problem 31.
• p31_start.m, returns a starting point for optimization for problem 31.
• p31_title.m, returns a title for problem 31.
• p32_f.m, evaluates the objective function for problem 32.
• p32_g.m, evaluates the gradient for problem 32.
• p32_h.m, evaluates the Hessian for problem 32.
• p32_n.m, returns the number of variables for problem 32.
• p32_sol.m, returns the solution for problem 32.
• p32_start.m, returns a starting point for optimization for problem 32.
• p32_title.m, returns a title for problem 32.
• p33_f.m, evaluates the objective function for problem 33.
• p33_g.m, evaluates the gradient for problem 33.
• p33_h.m, evaluates the Hessian for problem 33.
• p33_n.m, returns the number of variables for problem 33.
• p33_sol.m, returns the solution for problem 33.
• p33_start.m, returns a starting point for optimization for problem 33.
• p33_title.m, returns a title for problem 33.
• p34_f.m, evaluates the objective function for problem 34.
• p34_g.m, evaluates the gradient for problem 34.
• p34_h.m, evaluates the Hessian for problem 34.
• p34_n.m, returns the number of variables for problem 34.
• p34_sol.m, returns the solution for problem 34.
• p34_start.m, returns a starting point for optimization for problem 34.
• p34_title.m, returns a title for problem 34.
• p35_f.m, evaluates the objective function for problem 35.
• p35_g.m, evaluates the gradient for problem 35.
• p35_h.m, evaluates the Hessian for problem 35.
• p35_n.m, returns the number of variables for problem 35.
• p35_sol.m, returns the solution for problem 35.
• p35_start.m, returns a starting point for optimization for problem 35.
• p35_title.m, returns a title for problem 35.
• p36_f.m, evaluates the objective function for problem 36.
• p36_g.m, evaluates the gradient for problem 36.
• p36_h.m, evaluates the Hessian for problem 36.
• p36_n.m, returns the number of variables for problem 36.
• p36_p_get.m, get parameters for problem 36.
• p36_p_init.m, initializes parameters for problem 36.
• p36_p_set.m, set parameters for problem 36.
• p36_p_val.m, sets or gets parameters for problem 36.
• p36_sol.m, returns the solution for problem 36.
• p36_start.m, returns a starting point for optimization for problem 36.
• p36_title.m, returns a title for problem 36.
• p37_f.m, evaluates the objective function for problem 37.
• p37_g.m, evaluates the gradient for problem 37.
• p37_h.m, evaluates the Hessian for problem 37.
• p37_n.m, returns the number of variables for problem 37.
• p37_sol.m, returns the solution for problem 37.
• p37_start.m, returns a starting point for optimization for problem 37.
• p37_title.m, returns a title for problem 37.
• p38_f.m, evaluates the objective function for problem 38.
• p38_g.m, evaluates the gradient for problem 38.
• p38_h.m, evaluates the Hessian for problem 38.
• p38_n.m, returns the number of variables for problem 38.
• p38_sol.m, returns the solution for problem 38.
• p38_start.m, returns a starting point for optimization for problem 38.
• p38_title.m, returns a title for problem 38.
• p39_f.m, evaluates the objective function for problem 39.
• p39_g.m, evaluates the gradient for problem 39.
• p39_h.m, evaluates the Hessian for problem 39.
• p39_n.m, returns the number of variables for problem 39.
• p39_sol.m, returns the solution for problem 39.
• p39_start.m, returns a starting point for optimization for problem 39.
• p39_title.m, returns a title for problem 39.
• p40_f.m, evaluates the objective function for problem 40.
• p40_g.m, evaluates the gradient for problem 40.
• p40_h.m, evaluates the Hessian for problem 40.
• p40_n.m, returns the number of variables for problem 40.
• p40_sol.m, returns the solution for problem 40.
• p40_start.m, returns a starting point for optimization for problem 40.
• p40_title.m, returns a title for problem 40.
• p41_f.m, evaluates the objective function for problem 41.
• p41_g.m, evaluates the gradient for problem 41.
• p41_h.m, evaluates the Hessian for problem 41.
• p41_n.m, returns the number of variables for problem 41.
• p41_sol.m, returns the solution for problem 41.
• p41_start.m, returns a starting point for optimization for problem 41.
• p41_title.m, returns a title for problem 41.
• p42_f.m, evaluates the objective function for problem 42.
• p42_g.m, evaluates the gradient for problem 42.
• p42_h.m, evaluates the Hessian for problem 42.
• p42_n.m, returns the number of variables for problem 42.
• p42_sol.m, returns the solution for problem 42.
• p42_start.m, returns a starting point for optimization for problem 42.
• p42_title.m, returns a title for problem 42.
• p43_f.m, evaluates the objective function for problem 43.
• p43_g.m, evaluates the gradient for problem 43.
• p43_h.m, evaluates the Hessian for problem 43.
• p43_n.m, returns the number of variables for problem 43.
• p43_sol.m, returns the solution for problem 43.
• p43_start.m, returns a starting point for optimization for problem 43.
• p43_title.m, returns a title for problem 43.

Last revised on 30 March 2019.