The Bernstein Branch-and-Bound Unconstrained Global Optimization Algorithm for MINLP Problems

Abstract

In this work a Bernstein global optimization algorithm to solve unconstrained polynomial mixed-integer nonlinear programming (MINLP) problems is proposed. The proposed algorithm use a branch-and-bound framework and possesses several new features, such as a modified subdivision procedure, the Bernstein box consistency and the Bernstein hull consistency procedures to prune the solution search space. The performance of the proposed algorithm is numerically investigated and compared with previously reported Bernstein global optimization algorithm on a set of 10 test problems. The findings of the tests establishes the efficacy of the proposed algorithm over the previously reported Bernstein algorithm in terms of the chosen performance metrics.

Appendix

We list below the test problems studied in this work for conducting different numerical experiments. We denote the test function as $f(x_{k})$, and the initial bounds as $\mathbf {x}_{k}$, $k=1,2,\ldots ,l$.

• 1. Camel back: The six hump camel back function

  $$\begin{aligned} \min ~f(x)&= 4x_{1}^{2}-2.1x_{1}^{4}+(1/3)x_{1}^{6}+x_{1}x_{2}-4x_{2}^{2}+4x_{2}^{4} \\& x_{1} \in \mathbb {Z}, x_{2} \in \mathbb {R} \end{aligned}$$

  where $\mathbf {x}_{k}=[-3,3], k=1,2.$

• 2. Booth: The function defined by Booth

  $$\begin{aligned} \min ~f(x)&= 74-38x_{1}+5x_{1}^{2}-34x_{2}+8x_{1}x_{2}+5x_{2}^{2} \\& x_{1} \in \mathbb {Z}, x_{2} \in \mathbb {R} \end{aligned}$$

  where $\mathbf {x}_{k}=[-5,5],k=1,2.$

• 3. Reaction diffusion: A three dimensional reaction diffusion problem

  $$\begin{aligned} \min ~f(x)&= -x_{1}+2x_{2}-x_{3}-0.835634534x_{2}(1-x_{2}) \\& x_{1}, x_{2} \in \mathbb {Z}, x_{3} \in \mathbb {R} \end{aligned}$$

  where $\mathbf {x}_{k}=[-5,5], k=1,2,3.$

• 4. Caprasse’s: The system defined by Caprasse

  $$\begin{aligned} \min ~f(x) =&-x_{1}x_{3}^{3}+4x_{2}x_{3}^{2}x_{4}+4x_{1}x_{3}x_{4}^{2}+2x_{2}x_{4}^{3}+4x_{1}x_{3}\\&+ 4x_{3}^{2}-10x_{2}x_{4}-10x_{4}^{2}+2 \\& x_{1}, x_{3} \in \mathbb {Z}, x_{2}, x_{4} \in \mathbb {R} \end{aligned}$$

  where $\mathbf {x}_{k}=[-1,1], k=1,3$ and $\mathbf {x}_{k}=[-0.5,0.5], k=2,4.$

• 5. Adaptive LV: A neural network modeled by an adaptive Lotka-Volterra system

  $$\begin{aligned} \min ~f(x)&= x_{1}x_{2}^{2}+x_{1}x_{3}^{2}+x_{1}x_{4}^{2}-1.1x_{1}+1 \\& x_{1}, x_{2} \in \mathbb {Z}, x_{3}, x_{4} \in \mathbb {R} \end{aligned}$$

  where $\mathbf {x}_{1}=[0, 1], \mathbf {x}_{2}=[-20, 20]$, $\mathbf {x}_{k}=[-2, 2], k=3,4.$

• 6. AH Wright: The system defined by Wright

  $$\begin{aligned} \min ~f(x)&= x_{1}+x_{2}+x_{3}+x_{4}-x_{5}+x_{5}^{2}-10 \\&x_{1}, x_{2} \in \mathbb {R}, x_{3}, x_{4}, x_{5} \in \mathbb {Z} \end{aligned}$$

  where $\mathbf {x}_{k}=[-5,5], k=1,\ldots ,5.$

• 7. Magnetism in Physics (6): A six variable magnetism in physics problem

  $$\begin{aligned} \min ~f(x)&= 2x_{1}^{2}+2x_{2}^{2}+2x_{3}^{2}+2x_{4}^{2}+2x_{5}^{2}+x_{6}^{2}-x_{6} \\& x_{1}, x_{2}, x_{3} \in \mathbb {Z}, x_{4}, x_{5}, x_{6} \in \mathbb {R} \end{aligned}$$

  where $\mathbf {x}_{k}=[-1,1], k=1,\ldots ,6.$

• 8. Butcher: A function defined by Butcher

  $$\begin{aligned} \min ~f(x)&= x_{6}x_{2}^{2}+x_{5}x_{3}^{2}-x_{1}x_{4}^{2}+x_{4}^{3}+x_{4}^{2}-(1/3)x_{1}+(4/3)x_{4} \\& x_{1} \in \mathbb {Z}, x_{k} \in \mathbb {R}, k=2,\ldots ,6 \end{aligned}$$

  where $\mathbf {x}_{k} = [-1,1], k=1,2,3, \mathbf {x}_{4}=[-0.1,0.2], \mathbf {x}_{5} = [-0.3,1.1], \mathbf {x}_{6}=[-1.1,-0.3].$

• 9. Magnetism in physics (7): Seven variable magnetism in physics problem

  $$\begin{aligned} \min ~f(x)&= x_{1}^{2}+2x_{2}^{2}+2x_{3}^{2}+2x_{4}^{2}+2x_{5}^{2}+2x_{6}^{2}+2x_{7}^{2}-x_{1} \\& x_{1} \in \mathbb {R}, x_{k} \in \mathbb {Z}, k=2,\ldots ,7 \end{aligned}$$

  where $\mathbf {x}_{k}=[-1,1], k=1,\ldots ,7.$

• 10. Heart dipole: A heart dipole problem

  $$\begin{aligned} \min ~f(x) =&-x_{1}x_{6}^{3}+3x_{1}x_{6}x_{7}^{2}-x_{3}x_{7}^{3}+3x_{3}x_{7}x_{6}^{2}-x_{2}x_{5}^{3}\\&+3x_{2}x_{5}x_{8}^{2}-x_{4}x_{8}^{3}+\ 3x_{4}x_{8}x_{5}^{2}-0.9563453 \\&x_{k} \in \mathbb {Z}, k=1,\ldots ,5, x_{k} \in \mathbb {R}, k=5,7,8 \end{aligned}$$

  where $\mathbf {x}_{k} = [-1,1], k=1,2,3, \mathbf {x}_{4}=[-1,0],\mathbf {x}_{5}=[0,1], \mathbf {x}_{6}=[-0.1,0.2]$$\mathbf {x}_{7}=[-0.3,1.1], \mathbf {x}_{8}=[-1.1,-0.3].$