POLYNOMIALS
Polynomials for Global Optimization Tests

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

Polynomials include

• butcher;
• camel;
• camera;
• caprasse;
• cyclic5;
• cyclic7;
• cyclic8;
• goldstein_price;
• hairer;
• heart, heart dipole;
• himmelblau;
• hunecke;
• kearfott;
• lv3, adaptive Lotka-Volterra 3 system;
• lv4, adaptive Lotka-Volterra 4 system;
• magnetism6;
• magnetism7;
• quadratic;
• rd, 3 variable reaction diffusion;
• reimer5;
• reimer6;
• rosenbrock;
• schwefel;
• smith1;
• smith2;
• virasoro;
• wright;
• zakharov;

Licensing:

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

Languages:

POLYNOMIALS is available in a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

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

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.

COMPASS_SEARCH, a MATLAB library 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 program which minimizes a scalar function of several variables using the Nelder-Mead algorithm.

polynomials_test

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

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

Reference:

1. Cesar Munoz, Anthony Narkawicz,
   Formalization of Bernstein polynomials and applications to global optimization,
   Journal of Automated Reasoning,
   Volume 51, Number 2, 2013, pages 151-196.
2. Sashwati Ray, PSV Nataraj,
   An efficient algorithm for range computation of polynomials using the Bernstein form,
   Journal of Global Optimization,
   Volume 45, 2009, pages 403-426.
3. Andrew Smith,
   Fast construction of constant bound functions for sparse polynomials,
   Journal of Global Optimization,
   Volume 43, 2009, pages 445-458.
4. Jan Verschelde,
   PHCPACK: A general-purpose solver for polynomial systems by homotopy continuation,
   ACM Transactions on Mathematical Software,
   Volume 25, Number 2, June 1999, pages 251-276.

Source Code:

• butcher_b.m, bounds for the butcher polynomial.
• butcher_f.m, function for the butcher polynomial.
• butcher_m.m, problem size for the butcher polynomial.
• butcher_max_fun.m, butcher polynomial function for fmincon maximization.
• butcher_min_fun.m, butcher polynomial function for fmincon minimization.
• camel_b.m, bounds for the camel polynomial.
• camel_f.m, function for the camel polynomial.
• camel_m.m, problem size for the camel polynomial.
• camera_b.m, bounds for the camera polynomial.
• camera_f.m, function for the camera polynomial.
• camera_m.m, problem size for the camera polynomial.
• caprasse_b.m, bounds for the caprasse polynomial.
• caprasse_f.m, function for the caprasse polynomial.
• caprasse_m.m, problem size for the caprasse polynomial.
• cyclic5_b.m, bounds for the cyclic5 polynomial.
• cyclic5_f.m, function for the cyclic5 polynomial.
• cyclic5_m.m, problem size for the cyclic5 polynomial.
• cyclic7_b.m, bounds for the cyclic7 polynomial.
• cyclic7_f.m, function for the cyclic7 polynomial.
• cyclic7_m.m, problem size for the cyclic7 polynomial.
• cyclic8_b.m, bounds for the cyclic8 polynomial.
• cyclic8_f.m, function for the cyclic8 polynomial.
• cyclic8_m.m, problem size for the cyclic8 polynomial.
• goldstein_price_b.m, bounds for the goldstein_price polynomial.
• goldstein_price_f.m, function for the goldstein_price polynomial.
• goldstein_price_m.m, problem size for the goldstein_price polynomial.
• hairer_b.m, bounds for the hairer polynomial.
• hairer_f.m, function for the hairer polynomial.
• hairer_m.m, problem size for the hairer polynomial.
• heart_b.m, bounds for the heart dipole polynomial.
• heart_f.m, function for the heart dipole polynomial.
• heart_m.m, problem size for the heart dipole polynomial.
• himmelblau_b.m, bounds for the himmelblau polynomial.
• himmelblau_f.m, function for the himmelblau polynomial.
• himmelblau_m.m, problem size for the himmelblau polynomial.
• hunecke_b.m, bounds for the hunecke polynomial.
• hunecke_f.m, function for the hunecke polynomial.
• hunecke_m.m, problem size for the hunecke polynomial.
• kearfott_b.m, bounds for the kearfott polynomial.
• kearfott_f.m, function for the kearfott polynomial.
• kearfott_m.m, problem size for the kearfott polynomial.
• lv3_b.m, bounds for the adaptive Lotka-Volterra 3 polynomial.
• lv3_f.m, function for the adaptive Lotka-Volterra 3 polynomial.
• lv3_m.m, problem size for the adaptive Lotka-Volterra 3 polynomial.
• lv4_b.m, bounds for the adaptive Lotka-Volterra 4 polynomial.
• lv4_f.m, function for the adaptive Lotka-Volterra 4 polynomial.
• lv4_m.m, problem size for the adaptive Lotka-Volterra 4 polynomial.
• magnetism6_b.m, bounds for the magnetism6 polynomial.
• magnetism6_f.m, function for the magnetism6 polynomial.
• magnetism6_m.m, problem size for the magnetism6 polynomial.
• magnetism7_b.m, bounds for the magnetism7 polynomial.
• magnetism7_f.m, function for the magnetism7 polynomial.
• magnetism7_m.m, problem size for the magnetism7 polynomial.
• quadratic_b.m, bounds for the quadratic polynomial.
• quadratic_f.m, function for the quadratic polynomial.
• quadratic_m.m, problem size for the quadratic polynomial.
• r9mat_uniform_abvec.m, returns a matrix of N samples in an M dimensional space with separate limits for each dimensions.
• rd_b.m, bounds for the 3 variable reaction diffusion polynomial.
• rd_f.m, function for the 3 variable reaction diffusion polynomial.
• rd_m.m, problem size for the 3 variable reaction diffusion polynomial.
• reimer5_b.m, bounds for the reimer5 polynomial.
• reimer5_f.m, function for the reimer5 polynomial.
• reimer5_m.m, problem size for the reimer5 polynomial.
• reimer6_b.m, bounds for the reimer6 polynomial.
• reimer6_f.m, function for the reimer6 polynomial.
• reimer6_m.m, problem size for the reimer6 polynomial.
• rosenbrock_b.m, bounds for the rosenbrock polynomial.
• rosenbrock_f.m, function for the rosenbrock polynomial.
• rosenbrock_m.m, problem size for the rosenbrock polynomial.
• schwefel_b.m, bounds for the schwefel polynomial.
• schwefel_f.m, function for the schwefel polynomial.
• schwefel_m.m, problem size for the schwefel polynomial.
• smith1_b.m, bounds for the smith1 polynomial.
• smith1_f.m, function for the smith1 polynomial.
• smith1_m.m, problem size for the smith1 polynomial.
• smith2_b.m, bounds for the smith2 polynomial.
• smith2_f.m, function for the smith2 polynomial.
• smith2_m.m, problem size for the smith2 polynomial.
• timestamp.m, prints the current YMDHMS date as a time stamp.
• virasoro_b.m, bounds for the virasoro polynomial.
• virasoro_f.m, function for the virasoro polynomial.
• virasoro_m.m, problem size for the virasoro polynomial.
• wright_b.m, bounds for the wright polynomial.
• wright_f.m, function for the wright polynomial.
• wright_m.m, problem size for the wright polynomial.
• zakharov_b.m, bounds for the zakharov polynomial.
• zakharov_f.m, function for the zakharov polynomial.
• zakharov_m.m, problem size for the zakharov polynomial.