Location of a conservative hyperplane for cutting plane methods in disjoint bilinear programming

Abstract

Although several classes of cutting plane methods for deterministically solving disjoint bilinear programming (DBLP) have been proposed, the frequently encountered computational issue regarding the generation of a suitable cut from a degenerate vertex in a pseudo-global minimizer (PGM) still remains. Among the approaches to dealing with degeneracy, the most recent one is to generate a conservative cut. Nevertheless, the computational performance of the corresponding distance-following algorithm for its location seems far from satisfactory. This paper proposes several approaches that can be utilized to efficiently locate a conservative hyperplane from a degenerate vertex in a PGM. Extensive experiments are conducted to evaluate their performance from the dimensionality as well as the degree of degeneracy. From the computational viewpoint, these new approaches can outperform the earlier developed distance-following algorithm, and thereby can be incorporated into cutting plane methods for solving DBLP.

Appendix: A more general illustration

As an illustration, we provide a numerical example based on a cutting plane method for solving DBLP, where Augmented Mountain Climbing (AMC) Method is the local optimization algorithm in a Cutting Plane Method, and $\alpha$ is the current best objective value.

Example 3

$$\begin{aligned}&\begin{array}{cl} \min &{} f(\varvec{x},\varvec{y}) = \left( \begin{matrix} x_1 \\ x_2 \\ x_3 \end{matrix}\right) ^t\left( \begin{matrix} 2 &{} -3 &{} 1 \\ -1 &{} 2 &{} 4 \\ 3 &{} 5 &{} -3 \end{matrix}\right) \left( \begin{matrix} y_1 \\ y_2 \\ y_3 \end{matrix}\right) +\left( \begin{matrix} 1 \\ 2 \\ 3 \end{matrix}\right) ^t\left( \begin{matrix} x_1 \\ x_2 \\ x_3 \end{matrix}\right) + \left( \begin{matrix} 3 \\ 5 \\ 6 \end{matrix}\right) ^t\left( \begin{matrix} y_1 \\ y_2 \\ y_3 \end{matrix}\right) , \\ \\ \text{ s.t. } &{}\left( \begin{array}{rrr} -\frac{2}{3}&{} 0 &{}\frac{1}{3} \\ &{}&{}\\ 0 &{}-\frac{2}{11}&{} \frac{1}{11}\\ &{}&{}\\ \frac{3}{22}&{} \frac{1}{22} &{}\frac{1}{11}\\ &{}&{}\\ -\frac{1}{6}&{} -\frac{1}{2} &{}\frac{1}{3} \\ &{}&{}\\ -\frac{2}{9} &{}0 &{}\frac{1}{9}\\ &{}&{}\\ 0 &{}\frac{2}{5} &{}-\frac{1}{5} \\ &{}&{}\\ \frac{1}{2}&{} -\frac{1}{10}&{} -\frac{1}{5} \\ &{}&{}\\ \frac{1}{18} &{}-\frac{5}{18}&{} -\frac{1}{9}\\ \end{array}\right) \left( \begin{matrix} x_1\\ x_2\\ x_3 \end{matrix}\right) \le \left( \begin{array}{r} -1\\ 1\\ 1\\ -1\\ -1\\ 1\\ 1\\ -1 \end{array} \right) , \quad \left( \begin{array}{rrr} -\frac{2}{7}&{} 0 &{}\frac{1}{7} \\ &{}&{}\\ 0 &{}-\frac{2}{23}&{} \frac{1}{23}\\ &{}&{}\\ \frac{3}{46}&{} \frac{1}{46} &{}\frac{1}{23}\\ &{}&{}\\ -\frac{1}{14}&{} -\frac{3}{14} &{}\frac{1}{7} \\ &{}&{}\\ -\frac{2}{21} &{}0 &{}\frac{1}{21}\\ &{}&{}\\ 0 &{}\frac{2}{9} &{}-\frac{1}{9} \\ &{}&{}\\ \frac{5}{18}&{} -\frac{1}{18}&{} -\frac{1}{9} \\ &{}&{}\\ \frac{1}{42} &{}-\frac{5}{42}&{} -\frac{1}{21}\\ \end{array}\right) \left( \begin{matrix} y_1\\ y_2\\ y_3 \end{matrix}\right) \le \left( \begin{array}{r} -1\\ 1\\ 1\\ -1\\ -1\\ 1\\ 1\\ -1 \end{array} \right) , \end{array}\\&\varvec{x},\varvec{y}\ge \varvec{0}. \end{aligned}$$

In Example3, all vertices are degenerate in $\varvec{X}_0$; see the left sub-figure in Fig. 5. It should be noted that for Example3, the degeneracy removal procedure for a cutting plane method proposed in Alarie et al. [1] completely fails at the very beginning since none of the vertices is non-degenerate. As a result, the entire algorithm developed therein reduces to a pure branch and bound method, which is unable to utilize the advantage from a cutting plane method in feasible region reduction. On the contrary, the concept of a conservative cut can circumvent such a problem.

Solution:

We initialize $\alpha =10^8$, $i=0$, $(\varvec{x}_0, \varvec{y}_0)=(\varvec{0},\varvec{0})$, and $\varvec{X}_0 = \varvec{X}_0^0$. Let the cutting plane method start from vertex $A=(3, 3, 3)^t$ in $\varvec{X}_0$. In the following figures, PGM is used to indicate the $\varvec{x}$ part of a true PGM $(\overline{\varvec{x}}^i,\overline{\varvec{y}})$ identified by AMC.

• Iteration 1. Figure 5 illustrates the first iteration. We set $i=1$ as $\varvec{X}_0\ne \emptyset$. By AMC, we locate the PGM as $\overline{\varvec{x}}^1=F=(3.5, 3.5, 2)^t$, $\overline{\varvec{y}}=(7.5, 7.5, 6)^t$, with $\alpha _1=301.5$; see the middle sub-figure. Since $\alpha _1<\alpha$, update with $\alpha =301.5$ and set $\left( \hat{\varvec{x}}^1,\hat{\varvec{y}}^1\right) =(\overline{\varvec{x}}^1,\overline{\varvec{y}})$.

Fig. 5

Iteration 1

• (a) The information provided by AMC helps us readily identify the four adjacent vertices of the degenerate vertex F. They are $A=(3,3,3)^t$, $B=(4,3,3.5)^t$, $C=(4,4,3)^t$, and $D=(3,4,3)^t$.

• (b) By Algorithm 5, we can generate a conservative plane with A, C, and D, as shown in the middle sub-figure. The corresponding plane is $H_1: -x_3/3+1=0$. Update the constraint set with the new cut, and set $\varvec{X}_0=\varvec{X}_0^1$ shown in the right sub-figure.

• Iteration 2. Figure 6 demonstrates the second iteration. We can set $i=2$ as $\varvec{X}_0 \ne \emptyset$. By AMC, the PGM is $\overline{\varvec{x}}^2=A=(3, 3, 3)^t$, $\overline{\varvec{y}}=(7, 7, 7)^t$, with $\alpha _2=326$; see the left sub-figure. Since $\alpha _2>\alpha$, keep $\alpha =301.5$, and set $\left( \hat{\varvec{x}}^2,\hat{\varvec{y}}^2\right) =\left( \hat{\varvec{x}}^1,\hat{\varvec{y}}^1\right)$.

  • (a) With the information provided by AMC, we can readily locate the four adjacent vertices of the degenerate vertex A. They are $B=(4,3,3.5)^t$, $D=(3, 4,3)^t$, $E=(3.5,3.5,4)^t$, and $G=(3.83, 3.13,3)^t$. Moreover, these four vertices lie on the same plane.

  • (b) By Algorithm 5, we can generate a conservative plane with E, D, and G, on which lies B as well; see the left sub-figure. The corresponding plane is $H_2: -0.14x_1-0.15x_2+0.01x_3+1=0$. Update the constraint set with the new cut, and set $\varvec{X}_0=\varvec{X}_0^2$ shown in the right sub-figure.

Fig. 6

Iteration 2

• Iteration 3. Figure 7 illustrates the third iteration. We set $i=3$ as $\varvec{X}_0 \ne \emptyset$. By AMC, the PGM is $\overline{\varvec{x}}^3=G=(3.81, 3.13, 3)^t$, $\overline{\varvec{y}}=(7, 7, 7)^t$, with $\alpha _3=332.7$; see the left sub-figure. Since $\alpha _3>\alpha$, keep $\alpha =301.5$ and set $\left( \hat{\varvec{x}}^3,\hat{\varvec{y}}^3\right) =\left( \hat{\varvec{x}}^1,\hat{\varvec{y}}^1\right)$.

  • (a) The information provided by AMC helps us readily locate the three adjacent vertices of the non-degenerate vertex G. They are $B=(4,3,3.5)^t$, $C=(4, 4,3)^t$, and $D=(3,4,3)^t$.

  • (b) Although G is non-degenerate, it is not an original vertex of $\varvec{X}_0$, and we thus only need to generate a simple cut to cut off G, as demonstrated by the left sub-figure. The corresponding plane is $H_3: 0.01x_1-0.10x_2-0.21x_3+1=0$. Update the constraint set with the new cut, and set $\varvec{X}_0=\varvec{X}_0^3$ shown in the right sub-figure.

• Iteration 4. Finally, we set $i=4$ as $\varvec{X}_0 \ne \emptyset$. AMC identifies the PGM $\overline{\varvec{x}}^4=B=(4, 3, 3.5)^t$, $\overline{\varvec{y}}=(7, 7, 7)^t$, with $\alpha _4=345$; see the right sub-figure in Fig. 7. Since $\alpha _4>\alpha$, keep $\alpha =301.5$ and set $\left( \hat{\varvec{x}}^4,\hat{\varvec{y}}^4\right) =\left( \hat{\varvec{x}}^1,\hat{\varvec{y}}^1\right)$. Because B is a non-degenerate original vertex, a regular polar cut terminates the entire algorithm with the global optimum as $\alpha =301.5$ and its minimizer as $\hat{\varvec{x}}^4=(3.5, 3.5, 2)^t$, $\hat{\varvec{y}}^4=(7.5,7.5, 6)^t$.

Fig. 7

Iteration 3