A parallel Bernstein algorithm for global optimization based on the implicit Bernstein form

Abstract

In this paper, we first present a serial Bernstein algorithm for polynomial global optimization based on the Implicit Bernstein Form (IBF) (Smith in J Glob Optim 43:445–458, 2009). The serial Bernstein algorithm based on IBF needs less computations and memory than the conventional Bernstein algorithm and its variants. To accelerate further the Bernstein algorithm based on the IBF, we next propose a parallel version for GPU computing using Compute Unified Device Architecture. With the parallel version, the exponential time-complexity of the serial algorithm reduces to linear time-complexity. We compare the performance of both the versions on a set of 12 test problems, and find that the parallel version is up to 26 times faster and takes 96% less time than the serial one. Based on these findings, we suggest the use of the parallel version of the Bernstein algorithm based on IBF in polynomial global optimization.

Appendix A

In the following, we list the polynomials p and the initial domains x, the abbreviated and full names, and the dimensionality of the problems.

• 1. Reim5: The 5-dimensional system of Reimer, l = 5 (Verschelde 2001)

  $$\begin{aligned} & p(x) = - 1 + 2x_{1}^{6} - 2x_{2}^{6} + 2x_{3}^{6} - 2x_{4}^{6} + 2x_{5}^{6} \\ & {\mathbf{x}}_{1} = {\mathbf{x}}_{2} = {\mathbf{x}}_{3} = {\mathbf{x}}_{4} = {\mathbf{x}}_{5} = \left[ { - 1, \, 1} \right] \\ \end{aligned}$$

• 2. Zakharov 5: The 5-dimensional Zakharov function, l = 5 (Salhi and Queen 2004)

  $$\begin{aligned} & p(x) = \left( {\sum\limits_{i = 1}^{l} {x_{i}^{2} } } \right) + \left( {\sum\limits_{i = 1}^{l} {0.5*i*x_{i} } } \right)^{2} + \left( {\sum\limits_{i = 1}^{l} {0.5*i*x_{i} } } \right)^{4} \\ & {\mathbf{x}}_{1} = {\mathbf{x}}_{2} = \cdots = {\mathbf{x}}_{5} = [ - 5, \, 10] \\ \end{aligned}$$

• 3. Reim 6: The 6-dimensioanl system of Reimer, l = 6 (Verschelde 2001)

  $$\begin{aligned} p(x) = - 1 + 2x_{1}^{7} - 2x_{2}^{7} + 2x_{3}^{7} - 2x_{4}^{7} + 2x_{5}^{7} - 2x_{6}^{7} \\ & {\mathbf{x}}_{1} = {\mathbf{x}}_{2} = \cdots = {\mathbf{x}}_{6} = [ - 5, \, 5] \\ \end{aligned}$$

• 4. Kear 6: The 6-dimensional extended Kearfott function l = 6 (Vrahatis et al. 1997)

  $$\begin{aligned} & p(x) = \left[ {\sum\limits_{i = 1}^{(l - 1)} {} (x_{i}^{2} - x_{i + 1} )^{2} } \right] + (x_{l}^{2} - x_{1} )^{2} \\ & {\mathbf{x}}_{1} = {\mathbf{x}}_{2} = \cdots = {\mathbf{x}}_{6} = [ - 2, \, 2] \\ \end{aligned}$$

• 5. PowerSum 6: A power sum function l = 6 (Salhi and Queen 2004)

  $$\begin{aligned} & p(x) = \sum\limits_{i = 1}^{L} {x_{i}^{g} } ,\quad {\text{where}},\;g = 8 \\ & {\mathbf{x}}_{1} = {\mathbf{x}}_{2} = \cdots = {\mathbf{x}}_{6} = [ - 5, \, 5] \\ \end{aligned}$$

• 6. Mag 7: A problem of magnetism in physics, l = 7 (Verschelde 2001)

  $$\begin{aligned} & p(x) = x_{1}^{2} + 2x_{2}^{2} + 2x_{2}^{3} + 2x_{4}^{2} + 2x_{5}^{2} + 2x_{6}^{2} + 2x_{7}^{2} - x_{1} \\ & {\mathbf{x}}_{1} = {\mathbf{x}}_{2} = \cdots = {\mathbf{x}}_{7} = [ - 5, \, 5] \\ \end{aligned}$$

• 7. Heart 8: Heart-dipole problem, l = 8 (Verschelde 2001)

  $$\begin{aligned} & p(x) = - x_{1} x_{6}^{3} + 3x_{1} x_{6} x_{7}^{2} - x_{3} x_{7}^{2} + 3x_{3} x_{7} x_{6}^{2} - x_{2} x_{5}^{3} + 3x_{2} x_{5} x_{8}^{2} - x_{4} x_{8}^{3} + 3x_{4} x_{8} x_{5}^{2} - 0.9563453 \\ & {\mathbf{x}}_{1} = \, \left[ { - 0.1, \, 0.4} \right],\,{\mathbf{x}}_{2} = \left[ {0.4, \, 1} \right],\,{\mathbf{x}}_{3} = \left[ { - 0.7, \, 0.4} \right],\,{\mathbf{x}}_{4} = \left[ { - 0.7, \, 0.4} \right],\,{\mathbf{x}}_{5} = \left[ {0.1, \, 0.2} \right],\,{\mathbf{x}}_{6} = \left[ {0.1, \, 0.2} \right], \\ & {\mathbf{x}}_{7} = \, \left[ { - 0.3, \, 1.1} \right],{\mathbf{x}}_{8} = \left[ { - 1.1, \, - 0.3} \right] \\ \end{aligned}$$

• 8. Holzmann 8: The 8-dimensional Holzmann function, l = 8 (Himmelblau and Yetes 1972)

  $$p(x) = \sum\limits_{i = 1}^{l} {i*x_{i}^{4} } ,\;{\mathbf{x}}_{1} = {\mathbf{x}}_{2} = \cdots = {\mathbf{x}}_{8} = [ - 10,\,10]$$

• 9. Quad 10: A 10 dimensional quadratic function, l = 10 (Vrahatis et al. 1997)

  $$\begin{aligned} & p(x) = x_{1}^{2} + x_{2}^{2} + \cdots + x_{10}^{2} - r,\;{\text{for}}\;r = - 2 \\ & {\mathbf{x}}_{1} = {\mathbf{x}}_{2} = \cdots = {\mathbf{x}}_{10} = [ - 1,\,1] \\ \end{aligned}$$

• 10. Rosen-6: The 6-dimensional generalized Rosenbrock’s function, l = 6 (Salhi and Queen 2004)

  $$\begin{aligned} & p(x) = \sum\limits_{i = 1}^{(l - 1)} {} \left( {100(x_{i}^{2} - x_{i + 1} )^{2} + (1 - x_{i} )^{2} } \right) \\ & {\mathbf{x}}_{1} = {\mathbf{x}}_{2} = \cdots {\mathbf{x}}_{6} = [ - 2,\;2] \\ \end{aligned}$$

• 11. Kear 10: The 10-dimensional extended Kearfott function l = 10 (Vrahatis et al. 1997)

  Same as problem 4, but with l = 10

• 12. Quad 13: A 13 dimensional quadratic function, l = 13 (Vrahatis et al. 1997)

  Same as problem 9, but with l = 13.