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Global optimality conditions and optimization methods for polynomial programming problems

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This paper is concerned with the general polynomial programming problem with box constraints, including global optimality conditions and optimization methods. First, a necessary global optimality condition for a general polynomial programming problem with box constraints is given. Then we design a local optimization method by using the necessary global optimality condition to obtain some strongly or \(\varepsilon \)-strongly local minimizers which substantially improve some KKT points. Finally, a global optimization method, by combining the new local optimization method and an auxiliary function, is designed. Numerical examples show that our methods are efficient and stable.

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Correspondence to Z. Y. Wu.

Additional information

This research is partially supported by cstc2013jjB00001, cstc2011jjA00010 and SRF for ROCS, SEM, NSFC 11471062.

Appendix

Appendix

1.1 Problem 1: Beale function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}\quad f(x):=(1.5-x_{1}+x_{1}x_{2})^2+(2.25-x_{1}+x_{1}x^2_{2})^2+(2.625-x_{1}+x_{1}x^3_{2})^2\\ &{}s.t. &{}\quad -4.5 \le x_i \le 4.5,\quad i=1,2. \end{array} \end{aligned}$$

1.2 Problem 2: Booth function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}\quad f(x):=(x_{1}+2x_{2}-7)^2+(2x_{1}+x_{2}-5)^2\\ &{}s.t.&{}\quad -10 \le x_i \le 10,\quad i=1,2. \end{array} \end{aligned}$$

1.3 Problem 3: Matyas function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}\quad f(x):=0.26(x^2_{1}+x^2_{2})-0.48x_{1}x_{2}\\ &{}s.t.&{}\quad -10 \le x_i \le 10,\quad i=1,2. \end{array} \end{aligned}$$

1.4 Problem 4: Goldstein and Price function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}\quad f(x):=\left[ 1+(x_{1}+x_{2}+1)^2(19-14x_{1}+3x^2_{1}-14x_{2}+6x_{1}x_{2}+3x^2_{2})\right] \\ &{}&{}\qquad \times \,\left[ 30+(2x_{1}-3x_{2})^2(18-32x_{1}+12x^2_{1}+48x_{2}-36x_{1}x_{2}+27x^2_{2})\right] \\ &{}s.t.&{}\quad -2 \le x_i \le 2,\quad i=1,2. \end{array} \end{aligned}$$

1.5 Problem 5: Six-hump Camelback function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}\quad f(x):=(4-2.1x^2_{1}+x^4_{1}/3)x^2_{1}+x_{1}x_{2}+(-4+4x^2_{2})x^2_{2}\\ &{}s.t.&{}\quad -3\le x_1\le 3, -2\le x_{2}\le 3. \end{array} \end{aligned}$$

1.6 Problem 6: Perm(3, 0.5) function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=\sum \limits _{i=1}^n\left[ \sum \limits _{j=1}^n(j^i+0.5)((x_{j}/j)^i-1)\right] ^2 \\ &{}s.t.&{}x_i\in [-n,n],\quad i=1,2,\ldots ,n, \\ &{} \text{ where }&{} n=3. \end{array} \end{aligned}$$

1.7 Problem 7: Perm0(3, 10) function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=\sum \limits _{ k=1}^n\left[ \sum \limits _{i=1}^n(i+10)(x_{i}^k-(1/i)^{k})\right] ^2 \\ &{}s.t.&{}x_i\in [-n,n],\quad i=1,2,\ldots ,n,\\ &{} \text{ where }&{} n=3. \end{array} \end{aligned}$$

1.8 Problem 8: Perm(4, 0.5) function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=\sum \limits _{i=1}^n\left[ \sum \limits _{j=1}^n(j^i+0.5)((x_{j}/j)^i-1)\right] ^2 \\ &{}s.t.&{}x_i\in [-n,n],\quad i=1,2,\ldots ,n,\\ &{} \text{ where }&{} n=4. \end{array} \end{aligned}$$

1.9 Problem 9: Perm0(4, 10) function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=\sum \limits _{ k=1}^n\left[ \sum \limits _{i=1}^n(i+10)(x_{i}^k-(1/i)^{k})\right] ^2 \\ &{}s.t.&{}x_i\in [-n,n],\quad i=1,2,\ldots ,n,\\ &{} \text{ where }&{} n=4. \end{array} \end{aligned}$$

1.10 Problem 10: Colville function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}\quad f(x):=100(x^2_{1}-x^2_{2})^2+(x_{1}-1)^2+(x_{3}-1)^2+90(x^2_{3}-x_{4})^2\\ &{}&{}\quad +\,10.1((x_{2}-1)^2+(x_{4}-1)^2)+19.8(x_{2}-1)(x_{4}-1)\\ &{}s.t.&{}\quad -10 \le x_i \le 10,\quad i=1,2,3,4. \end{array} \end{aligned}$$

1.11 Problem 11: Powersum function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=\sum \limits _{i=1}^4\left[ \left( \sum \limits _{i=1}^4 x_{j}^i\right) -b_{i}\right] ^2 \\ &{}s.t.&{}0 \le x_i \le 4,\quad i=1,\ldots , 4,\\ &{} \text{ where }&{} b=(8,18,44,114). \end{array} \end{aligned}$$

1.12 Problem 12: Dixon and Price function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=(x_{1}-1)^2+\sum \limits _{i=2}^n i(2x_{i}^2-x_{i-1})^2\\ &{}s.t.&{}x_i\in [-10,10], \quad i=1,2,\ldots ,n,\\ &{} \text{ where }&{} n=5. \end{array} \end{aligned}$$

1.13 Problem 13: Dixon and Price function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=(x_{1}-1)^2+\sum \limits _{i=2}^n i(2x_{i}^2-x_{i-1})^2\\ &{}s.t.&{}x_i\in [-10,10], \quad i=1,2,\ldots ,n,\\ &{} \text{ where }&{} n=10. \end{array} \end{aligned}$$

1.14 Problem 14: Trid function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=\sum \limits _{i=1}^n(x_{i}-1)^2-\sum \limits _{i=2}^n x_{i}x_{i-1}\\ &{}s.t.&{}-n^2 \le x_i \le n^2,\quad i=1,2,\ldots ,n,\\ &{} \text{ where }&{} n=10. \end{array} \end{aligned}$$

1.15 Problem 15: Rosenbrock function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=\sum \limits _{i=1}^{n-1}\left[ 100(x_{i+1}-x^2_{i})^2+(x_{i}-1)^2\right] \\ &{}s.t.&{}-5 \le x_i \le 10,\quad i=1,2,\ldots ,n,\\ &{} \text{ where }&{} n=20. \end{array} \end{aligned}$$

1.16 Problem 16: Sum squares function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=\sum \limits _{i=1}^{n}ix^2_{i}\\ &{}s.t.&{}-10 \le x_i \le 10,\quad i=1,2,\ldots ,n,\\ &{} \text{ where }&{} n=20. \end{array} \end{aligned}$$

1.17 Problem 17: Zakharov function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=\sum \limits _{i=1}^{n}x^2_{i}+\left( \sum \limits _{i=1}^{n}0.5ix_{i}\right) ^2+(\sum \limits _{i=1}^{n}0.5ix_{i})^4\\ &{}s.t.&{}-5 \le x_i \le 10,\quad i=1,2,\ldots ,n,\\ &{} \text{ where }&{} n=20. \end{array} \end{aligned}$$

1.18 Problem 18: Powell function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=\sum \limits _{i=1}^{n/4}\left[ (x_{4i-3}+10x_{4i-2})^2+5(x_{4i-1}-x_{4i})^2\right. \\ &{}&{}\left. +\,(x_{4i-2}-2x_{4i-1})^4+10(x_{4i-3}-x_{4i})^4\right] \\ &{}s.t.&{}-4 \le x_i \le 5,\quad i=1,2,\ldots ,n,\\ &{} \text{ where }&{} n=24. \end{array} \end{aligned}$$

1.19 Problem 19: Sphere function

$$\begin{aligned} \begin{array}{lll} &{}\min &{}f(x):=\sum \limits _{i=1}^{n}x^2_{i}\\ &{}s.t.&{}-5.12 \le x_i \le 5.12,\quad i=1,2,\ldots ,n,\\ &{} \text{ where }&{} n=30. \end{array} \end{aligned}$$

1.20 Problem 20: Example 4.1 in [23]

$$\begin{aligned} \begin{array}{lll} &{}\min &{} f(x):=\sum _{k=1}^{3}\left( \sum _{i=1}^{n}x_{i}^{k}-1\right) ^{2}+\sum _{i=1}^{n}(x_{i-1}^{2}+x_{i}^{2}+x_{i+1}^{2}-x_{i}^{3}-1)^{2}\\ &{}s.t.&{} x_i\in [-500,500] ,\quad i=1,\ldots , n,\\ &{}\text{ where }&{} x_{0}=x_{n+1}=0,\ n=16. \end{array} \end{aligned}$$

1.21 Problem 21: Example 5.2 in [23]

$$\begin{aligned} \begin{array}{lll} &{}\min &{} f(x):=\sum _{i=1}^{m}f^2_{i}(x)\\ &{}s.t.&{} x_i\in [-500,500] ,\quad i=1,\ldots , n, \end{array} \end{aligned}$$

where \(n=30\) and the polynomials \(f_{i}\) are defined as follows:

$$\begin{aligned} f_{i}(x):=\sum _{j=2}^{n}(j-1)x_{j}t_{i}^{j-2}-\left( \sum _{j=1}^{n}x_{j}t_{i}^{j-1}\right) ^{2}-1,\quad t_{i}=\frac{i}{29},\quad 1\le i\le 29, \end{aligned}$$

and \(f_{30}=x_{1}\), \(f_{31}=x_{2}-x_{1}^2-1\).

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Wu, Z.Y., Tian, J. & Ugon, J. Global optimality conditions and optimization methods for polynomial programming problems. J Glob Optim 62, 617–641 (2015). https://doi.org/10.1007/s10898-015-0292-5

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