A GPU parallel Bernstein algorithm for polynomial global optimization

Abstract

We come up with a graphics processing unit (GPU) parallel Bernstein algorithm (BA) aimed at global optimization of multi-variate real polynomials (Garloff in Interval Comput 2:164–168, 1993). We first propose parallel algorithms for (a) computing the multi-index set associated with the Bernstein coefficients (BCs), (b) computing the initial set of BCs using the Matrix method (Ray and Nataraj in Reliab Comput 17(1):40–71, 2012), (c) finding the minimum BC from a given set of BCs, and (d) finding the BCs of the child patches from the parent patch. We then incorporate the above components into the proposed parallel Bernstein algorithm. All the parallel algorithms are programmed for GPU accelerating devices through compute unified device architecture. We compared performance of serial and GPU parallel BA using a test suite of 8 multivariate examples. For the test examples, the proposed parallel algorithm is found 30 times faster as compared to serial one, and needs 96% less time.

Appendix 1

Following is the list of the polynomials p used in this work, along with their starting domain x. Each test problem has given a short name (bold face), full name and the dimensionality, as like in (Dhabe and Nataraj 2017a, 2017).

• 1. Heart-8: A 8-dimensional heart-dipole test problem, l = 8 (Verschelde 2001)

  $$ p(x) = - \,x_{1} x_{6}^{3} + 3x_{1} x_{6} x_{7}^{2} - x_{3} x_{7}^{2} + 3x_{3} x_{7} x_{6}^{2} - x_{2} x_{5}^{3} + 3x_{2} x_{5} x_{8}^{2} - x_{4} x_{8}^{3} + 3x_{4} x_{8} x_{5}^{2} - 0.9563453 $$

  x₁ = [− 0.1,0.4], x₂ = [0.4,1.0], x₃ = [− 0.7,0.4], x₄ = [− 0.7,0.4], x₅ = [0.1,0.2], x₆ = [0.1,0.2], x₇ = [− 0.3,1.1], x₈ = [− 1.1, − 0.3]

• 2. Rosen-6: A 6-dimensional general function, l = 6 (Sahil and Queen 2004)

  $$ p(x) = \int\limits_{i = 1}^{(l - 1)} {} \left( {100(x_{i}^{2} - x_{i + 1} )^{2} + (1 - x_{i} )^{2} } \right) $$

  x₁ = x₂ = ··· = x₆ = [− 2.0,2.0]

• 3. Kear-6: A 6-variable extended function of Kearfott l = 6 (Vrahatis et al. 1997)

  $$ p(x) = \left[ {\int\limits_{i = 1}^{(l - 1)} {} (x_{i}^{2} - x_{i + 1} )^{2} } \right] + (x_{l}^{2} - x_{1} )^{2} $$

  x₁ = x₂ =···= x₆ = [− 2.0,2.0]

• 4. Reim-5: A 5-variable Reimer’s system, l = 5 (Verschelde 2001)

  $$ p(x) = - \,1 + 2x_{1}^{6} - 2x_{2}^{6} + 2x_{3}^{6} - 2x_{4}^{6} + 2x_{5}^{6} $$

  x₁ = x₂ = x₃ = x₄ = x₅ = [− 1.0,1.0]

• 5. Quad-10: A 10-variable function with power 2, l = 10 (Vrahatis et al. 1997)

  $$ p(x) = x_{1}^{2} + x_{2}^{2} + \cdots + x_{10}^{2} - r,\quad {\text{for}}\quad r = - \,2 $$

  x₁ = x₂ =···= x₁₀ = [− 1.0, 1.0]

• 6. Reim-6: A 6-variable Reimer’s system, l = 6 (Verschelde 2001)

  $$ p(x) = - \,1 + 2x_{1}^{7} - 2x_{2}^{7} + 2x_{3}^{7} - 2x_{4}^{7} + 2x_{5}^{7} - 2x_{6}^{7} $$

  x₁ = x₂ =···= x₆ = [− 5.0, 5.0]

• 7. Holzmann-8: A Holzmann function with 8-variables, l = 8 (Himmelblau and Yetes 1972)

  $$ p(x) = \int\limits_{i = 1}^{l} {i*x_{i}^{4} } $$

  x₁ = x₂ =···= x₈ = [− 10.0, 10.0]

• 8. PowerSum-6: A power sum function with 6-variables l = 6 (Sahil and Queen 2004)

  $$ p(x) = \int\limits_{i = 1}^{L} {x_{i}^{g} } ,\quad {\text{where}},\,\,g = 8 $$

  x₁ = x₂ =···= x₆ = [− 5.0,5.0]