Theoretical filtering of RLT bound-factor constraints for solving polynomial programming problems to global optimality

Abstract

In this paper, we propose two sets of theoretically filtered bound-factor constraints for constructing reformulation-linearization technique (RLT)-based linear programming (LP) relaxations for solving polynomial programming problems. We establish related theoretical results for convergence to a global optimum for these reduced sized relaxations, and provide insights into their relative sizes and tightness. Extensive computational results are provided to demonstrate the relative effectiveness of the proposed theoretical filtering strategies in comparison to the standard RLT and a prior heuristic filtering technique using problems from the literature as well as randomly generated test cases.

Appendix

PP1 (Problem P1 in [17]):

$$\begin{aligned}&{\text{ Minimize}} -x_1 -x_2\\&{\text{ subject} \text{ to}} \nonumber \\&\qquad \qquad x_2 \le 2 + 2x_1^4 -8x_1^3 + 8x_1^2 \\&\qquad \qquad x_2 \le 4x_1^4 - 32 x_1^3 + 88 x_1^2 - 96 x_1 + 36 \\&\qquad \qquad 0 \le x_1 \le 3, \ \ \ 0 \le x_2 \le 4. \end{aligned}$$

PP2 (Problem 19 in [8]):

$$\begin{aligned}&{\text{ Minimize} } (x_1-10)^3 + (x_2-20)^3\\&{\text{ subject} \text{ to}} \nonumber \\&\qquad \qquad (x_1-5)^2 + (x_2 -5)^2 \ge 100 \\&\qquad \qquad -(x_1-6)^2 -(x_2 -5)^2 \ge -82.81 \\&\qquad \qquad 13 \le x_1 \le 100, \ \ \ 0 \le x_2 \le 100. \end{aligned}$$

PP3 (Problem P7 in [17]):

$$\begin{aligned}&{\text{ Minimize} } x_4 \\&{\text{ subject} \text{ to}} \nonumber \\&\qquad \qquad x_1^4x_2^4 -x_1^4 -x_2^4x_3 = 0 \\&\qquad \qquad 1.4 - 0.25 x_4 - x_1 \le 0 \\&\qquad \qquad -1.4 - 0.25 x_4 + x_1 \le 0 \\&\qquad \qquad 1.5 - 0.2 x_4 - x_2 \le 0 \\&\qquad \qquad -1.5 - 0.2 x_4 + x_2 \le 0 \\&\qquad \qquad 0.8 -0.2 x_4 - x_3 \le 0 \\&\qquad \qquad -0.8 -0.2 x_4 + x_3 \le 0 \\&\qquad \qquad 0 \le x_j \le 5, \ \quad \forall j=1, \ldots , 4. \end{aligned}$$

PP4 (Problem P8 in [17]):

$$\begin{aligned}&{\text{ Minimize} } 54.528 x_2 x_4 + 27.264 x_1 x_3 -54.528 x_3 x_4\\&{\text{ subject} \text{ to}} \nonumber \\&\qquad \qquad 61.01627586 - I \le 0 \\&\qquad \qquad 8 x_1 - I \le 0 \\&\qquad \qquad x_1 x_2 x_4 - x_2 x_4^2 + x_1^2x_3 + x_3 x_4^2 - 2 x_1 x_3 x_4 - 3.5 x_3 I \le 0 \\&\qquad \qquad x_1 - 3 x_2 \le 0\\&\qquad \qquad 2 x_2 - x_1 \le 0 \\&\qquad \qquad x_3 -1.5 x_4 \le 0 \\&\qquad \qquad 0.5 x_4 - x_3 \le 0 \\&\qquad \qquad 3 \le x_1 \le 20, \ 2 \le x_2 \le 15,\ 0.125 \le x_3 \le 0.75,\ 0.25 \le x_4 \le 1.25, \end{aligned}$$

where $I = 6 x_1^2x_2 x_3 - 12 x_1 x_2 x_3^2 + 8 x_2 x_3^3 + x_1^3x_4 - 6x_1^2x_3x_4 +12 x_1 x_3^2x_4 - 8 x_3^3x_4.$

PP5 (Problem 100 in [8] - with imposed/modified variable bounds):

$$\begin{aligned}&{\text{ Minimize} } (x_1-10)^2 + 5 (x_2 - 12)^2 + (x_3-1)^4 + 3 (x_4 -11)^2 + 10 (x_5-1)^6 + \\&\qquad \qquad 7 x_6^2 + x_7^4 -4 x_6 x_7 - 10x_6 - 8 x_7 \\&{\text{ subject} \text{ to}} \nonumber \\&\qquad \qquad 127 - 2x_1^2-3 x_2^4 - (x_3-1) - 4x_4^2 - 5(x_5-1) \ge 0 \\&\qquad \qquad 282 -7x_1 -3 x_2 -10 (x_3-1)^2 -x_4 + (x_5-1) \ge 0 \\&\qquad \qquad 196 -23 x_1 - x_2^2 -6 x_6^2 + 8 x_7 \ge 0 \\&\qquad \qquad -4x_1^2 - x_2^2 + 3x_1 x_2 - 2x_3^2 - 5x_6 + 11x_7 \ge 0 \\&\qquad \qquad 0 \le x_j \le 3, \quad \forall j =1, 2, 3, 5, 6,{\text{ and} }7, \ \ 0 \le x_4 \le 5. \end{aligned}$$

PP6 (Problem 117 in [8]—with imposed variable bounds, where the parameter data is given in Table 6):

$$\begin{aligned}&{\text{ Minimize}} -\sum _{j=1}^{10} b_j x_j + \sum _{j=1}^{5} \sum _{k=1}^{5} c_{kj} x_{10+k} x_{10+j} + 2 \sum _{j=1}^{5} d_j x_{10+j}^3 \\&{\text{ subject} \text{ to}} \nonumber \\&\qquad \qquad 2 \sum _{k=1}^{5} c_{kj}x_{10+k} + 3 d_j x_{10 + j}^2 +e_j - \sum _{k=1}^{10} a_{kj} x_j \ge 0, \ j=1, \ldots , 5 \\&\qquad \qquad 0 \le x_j \le 15, \quad \forall j=1, \ldots , 15. \end{aligned}$$

Table 6 Parameter data for PP6 (Problem 117 in [8])