RLT-POS: Reformulation-Linearization Technique-based optimization software for solving polynomial programming problems

Abstract

In this paper, we introduce a Reformulation-Linearization Technique-based open-source optimization software for solving polynomial programming problems (RLT-POS). We present algorithms and mechanisms that form the backbone of RLT-POS, including constraint filtering techniques, reduced RLT representations, and semidefinite cuts. When implemented individually, each model enhancement has been shown in previous papers to significantly improve the performance of the standard RLT procedure. However, the coordination between different model enhancement techniques becomes critical for an improved overall performance since special structures in the original formulation that work in favor of a particular technique might be lost after implementing some other model enhancement. More specifically, we discuss the coordination between (1) constraint elimination via filtering techniques and reduced RLT representations, and (2) semidefinite cuts for sparse problems. We present computational results using instances from the literature as well as randomly generated problems to demonstrate the improvement over a standard RLT implementation and to compare the performances of the software packages BARON, COUENNE, and SparsePOP with RLT-POS.

Appendix

PP1 (Problem 119 in [10], where the parameter data is specified in (22) and in Table 13):

$$\begin{aligned} \text {Minimize}&\quad \sum _{i=1}^{16} \sum _{j=1}^{16} a_{ij}\left( x_i^2 + x_i + 1\right) \left( x_j^2 + x_j + 1\right) \\ \text {subject to}&\quad \nonumber \\&\quad \sum _{j = 1}^{16} b_{ij} x_j - c_i = 0, \quad \forall i = 1,\ldots , 8 \\&\quad 0 \le x_i \le 5, \quad \forall j=1, \ldots , 16. \end{aligned}$$

$$\begin{aligned} a_{ij} \!=\! 1, \forall (\!i, j\!)\!\in & {} \! \{ (\!1, 1), (\!1, 4), (\!1, 7\!), (\!1, 8), (\!1, 16), (\!2, 2), (\!2, 3), (\!2, 7), (\!2, 10), (\!3, 3), \nonumber \\&(\!3, 7), (\!3, 9), (\!3, 10), (3, 14), (4, 4), (4, 7),(4, 11), (4, 15), (5, 5),\nonumber \\&(5, 6), (5, 10), (5, 12), (5, 16), (6, 6), (6, 8), (6, 15), \, (7, 7), \, (7, 11),\nonumber \\&(7, 13), \, (8, 8),\, (8, 10), \, (8, 15), \, (9, 9), \, (9, 12), \, (9, 16), (10, 10), \nonumber \\&(\!10, 14\!), (\!11, 11\!), (\!11, 13\!), (\!12, 12\!), (\!12, 14\!), (\!13, 13\!), (\!13, 14\!), (\!14, 14\!), \nonumber \\&(\!15, 15\!), (16, 16) \} \end{aligned}$$

(22)

Table 13 Parameter data for Problems PP1, PP2, and PP3 (Problem 119 in [10])

PP2 (Problem 119 in [10]—as given by Problem PP1, but with a modified objective function):

$$\begin{aligned} \text {Minimize}&\quad \sum _{i=1}^{16} \sum _{j=1}^{16} a_{ij}\left( x_i^3 + x_i + 1\right) \left( x_j^2 + x_j + 1\right) \\ \text {subject to}&\nonumber \\&\quad \sum _{j = 1}^{16} b_{ij} x_j - c_i = 0, \quad \forall i = 1, \ldots , 8 \\&\quad 0 \le x_i \le 5,\quad \forall j=1, \ldots , 16. \end{aligned}$$

PP3 (Problem 119 in [10]—as given by Problem PP1, but with a modified objective function):

$$\begin{aligned} \text {Minimize}&\quad \sum _{i=1}^{16} \sum _{j=1}^{16} a_{ij}\left( x_i^3 + x_i^2 + 1\right) \left( x_j^3 + x_j^2 + 1\right) \\ \text {subject to}&\nonumber \\&\quad \sum _{j = 1}^{16} b_{ij} x_j - c_i = 0, \quad \forall i = 1, \ldots , 8 \\&\quad 0 \le x_i \le 5,\quad \forall j=1, \ldots , 16. \end{aligned}$$

PP4 (Problem 49 in [10]—, along with imposed variable bounds):

$$\begin{aligned} \text {Minimize}&\quad (x_1 - x_2)^2 + (x_3 -1)^2 + (x_4 -1)^4 + (x_5-1)^6 \\ \text {subject to}&\nonumber \\&\quad x_1 + x_2 + x_3 + 4x_4 = 7 \\&\quad x_3 + 5x_5 = 6 \\&\quad 0 \le x_j \le 5, \quad \forall j=1, \ldots , 5. \end{aligned}$$

PP5 (Problem 50 in [10]—, along with imposed variable bounds):

$$\begin{aligned} \text {Minimize}&\quad (x_1 - x_2)^2 + (x_2 - x_3)^2 + (x_3 -x_4)^4 + (x_4 -x_5)^2 \\ \text {subject to}&\nonumber \\&\quad x_1 + 2x_2 + 3x_3 = 6 \\&\quad x_2 + 2x_3 + 3x_4 = 6 \\&\quad x_3 + 2x_4 + 3x_5 = 6 \\&\quad 0 \le x_j \le 5,\quad \forall j=1, \ldots , 5. \end{aligned}$$