A space exploration algorithm for multiparametric programming via Delaunay triangulation

Abstract

We present a novel parameter space exploration algorithm for three classes of multiparametric problems, namely linear (mpLP), quadratic (mpQP), and mixed-integer linear (mpMILP). We construct subsets of the parameter space in the form of simplices through Delaunay triangulation to facilitate identification of the optimal partitions that describe the solution space. The presented exploration strategy prioritizes identifying volumetrically larger critical regions compared to existing methods. We demonstrate the exploration algorithm on an illustrative example, and compare the volumetrically identified parameter space against existing solvers on randomly generated problems in all three classes.

Appendix

1.1 Details of the motivating example

The multiparametric quadratic programming problem used as the motivating example is defined as follows.

$$\begin{aligned} \begin{array}{lll} z^*(\theta )=&{}\underset{x}{\min }&{}(Qx+H\theta +c)^T x\\ &{} s.t. &{}Ax \le b+F\theta \\ &{} &{} \underline{x} \le x \le \bar{x}\\ &{} &{} \underline{\theta } \le \theta \le \bar{\theta } \end{array} \end{aligned}$$

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$$\begin{aligned} Q= & {} \left[ \begin{array}{cccccccccc} 24.97 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 4 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 4.72 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1.1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0.11 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1.38 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 3.24 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1.3 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 14.37 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 759.27\\ \end{array}\right] \\ H= & {} \left[ \begin{array}{cc} 0 &{}\quad 0 \\ 1 &{}\quad 1 \\ 0 &{}\quad 1 \\ 0 &{}\quad -\,1 \\ 0 &{}\quad 0 \\ -\,1 &{}\quad -\,1 \\ 0 &{}\quad 0 \\ -\,1 &{}\quad 0 \\ 0 &{}\quad 0 \\ 0 &{}\quad 0 \\ \end{array}\right] \quad c = \left[ \begin{array}{c} 5\\ -\,3\\ 1\\ -\,2\\ 2\\ 3\\ 4\\ 3\\ 5\\ 1\\ \end{array}\right] \end{aligned}$$

$$\begin{aligned} A= & {} \left[ \begin{array}{cccccccccc} 0.14 &{}\quad 0.21 &{}\quad -\,0.33 &{}\quad 0.2 &{}\quad 0 &{}\quad -\,0.52 &{}\quad -\,0.24 &{}\quad 0 &{}\quad -\,0.5 &{}\quad 0.26 \\ -\,0.64 &{}\quad -\,0.05 &{}\quad -\,0.41 &{}\quad -\,0.24 &{}\quad 0 &{}\quad -\,0.3 &{}\quad 0.3 &{}\quad 0.14 &{}\quad 0 &{}\quad 0.21 \\ -\,0.57 &{}\quad -\,0.12 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0.31 &{}\quad 0 &{}\quad 0 &{}\quad -\,0.54 \\ 0 &{}\quad 0 &{}\quad 0.07 &{}\quad -\,0.69 &{}\quad 0 &{}\quad 0.36 &{}\quad 0.44 &{}\quad 0.17 &{}\quad 0 &{}\quad 0.25 \\ 0.35 &{}\quad 0 &{}\quad -\,0.21 &{}\quad 0.55 &{}\quad 0 &{}\quad -\,0.18 &{}\quad 0 &{}\quad -\,0.56 &{}\quad 0.19 &{}\quad 0.03 \\ 0 &{}\quad -\,0.3 -\,0.59 &{}\quad -\,0.38 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\,0.64 0 &{}\quad 0 &{}\quad &{}\quad \\ 0.08 &{}\quad -\,0.63 &{}\quad -\,0.45 &{}\quad 0 &{}\quad -\,0.33 &{}\quad -\,0.25 &{}\quad 0 &{}\quad -\,0.23 &{}\quad 0.15 &{}\quad 0 \\ 0.24 &{}\quad -\,0.43 &{}\quad -\,0.17 &{}\quad 0.28 &{}\quad -\,0.51 &{}\quad -\,0.23 &{}\quad -\,0.36 &{}\quad 0.34 &{}\quad 0 &{}\quad 0.28\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -\,0.17 &{}\quad -\,0.03 &{}\quad 0.44 &{}\quad 0 &{}\quad 0.22 &{}\quad -\,0.72 &{}\quad 0.19 \\ 0 &{}\quad 0 &{}\quad -\,0.61 &{}\quad -\,0.28 &{}\quad 0 &{}\quad 0.3 &{}\quad 0.41 &{}\quad -\,0.39 &{}\quad -\,0.37 &{}\quad 0 \\ -\,0.14 &{}\quad 0 &{}\quad -\,0.53 &{}\quad 0 &{}\quad 0.14 &{}\quad 0.11 &{}\quad 0 &{}\quad -\,0.76 &{}\quad -\,0.29 &{}\quad 0.026 \\ 0 &{}\quad -\,0.44 &{}\quad -\,0.02 &{}\quad 0 &{}\quad -\,0.39 &{}\quad 0 &{}\quad -\,0.17 &{}\quad 0.4 &{}\quad -\,0.51 &{}\quad 0 \\ 0 &{}\quad 0.21 &{}\quad -\,0.11 &{}\quad -\,0.66 &{}\quad -\,0.14 &{}\quad 0.43 &{}\quad 0.15 &{}\quad 0.47 &{}\quad -\,0.05 &{}\quad -\,0.25 \\ 0 &{}\quad -\,0.01 &{}\quad -\,0.37 &{}\quad -\,0.35 &{}\quad -\,0.29 &{}\quad -\,0.01 &{}\quad 0.36 &{}\quad 0.02 &{}\quad 0 &{}\quad -\,0.17 \\ 0 &{}\quad 0.32 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad -\,0.55 &{}\quad 0.34 &{}\quad 0.44 &{}\quad -\,0.24 &{}\quad 0.36 \\ \end{array}\right] \\ b= & {} \left[ \begin{array}{c} 7.57 \\ 10.72 \\ 5.47 \\ 9.26 \\ 12.32 \\ 8.47 \\ 5.22 \\ 6.89 \\ 3.70 \\ 6.22 \\ 8.43 \\ 4.74 \\ 3.74 \\ 3.05 \\ 5.68 \\ \end{array}\right] \quad F = \left[ \begin{array}{cc} -\,0.38 &{}\quad 0 \\ 0 &{}\quad -\,0.34 \\ 0.46 &{}\quad -\,0.25 \\ -\,0.33 &{}\quad 0 \\ 0.38 &{}\quad -\,0.01 \\ 0 &{}\quad 0 \\ 0 &{}\quad 0.39 \\ -\,0.11 &{}\quad 0 \\ 0.41 &{}\quad 0 \\ 0 &{}\quad -\,0.04 \\ 0 &{}\quad 0.09 \\ 0 &{}\quad -\,0.45 \\ 0 &{}\quad 0.06 \\ 0.59 &{}\quad -\,0.38 \\ -\,0.27 &{}\quad -\,0.15 \\ \end{array}\right] \\ \underline{x}= & {} \left[ \begin{array}{cc} -\,1E7 \\ -\,1E7 \\ -\,1E7 \\ -\,1E7 \\ -\,1E7 \\ -\,1E7 \\ -\,1E7 \\ -\,1E7 \\ -\,1E7 \\ -\,1E7 \\ \end{array}\right] \quad \bar{x}= \left[ \begin{array}{cc} 1E7 \\ 1E7 \\ 1E7 \\ 1E7 \\ 1E7 \\ 1E7 \\ 1E7 \\ 1E7 \\ 1E7 \\ 1E7 \end{array}\right] \quad \underline{\theta } = \left[ \begin{array}{c} -\,10 \\ -\,10\end{array}\right] \quad \bar{\theta }= \left[ \begin{array}{c} 10 \\ 10 \end{array}\right] \end{aligned}$$