Data-driven construction of Convex Region Surrogate models

Abstract

With the increasing trend of solving more complex and integrated optimization problems, there is a need for develo** process models that are sufficiently accurate as well as computationally efficient. In this work, we develop an algorithm for the data-driven construction of a type of surrogate model that can be formulated as a set of mixed-integer linear constraints, yet still provide good approximations of nonlinearities and nonconvexities. In such a surrogate model, which we refer to as Convex Region Surrogate (CRS), the feasible region is given by the union of convex regions in the form of polytopes, and for each region, the corresponding cost function can be approximated by a linear function. The general problem is as follows: given a set of data points in the parameter space and a scalar cost value associated with each data point, find a CRS model that approximates the feasible region and cost function indicated by the given data points. We present a two-phase algorithm to solve this problem and demonstrate its effectiveness with an extensive computational study as well as a real-world case study.

Appendices

Appendix 1: Alternative formulation for CRS model

Instead of expressing the feasible region of a CRS model as the union of feasible convex regions, we can also formulate it as the difference of the convex hull and the union of infeasible convex regions. In our example, as shown in Fig. 28, the infeasible convex regions are the two empty polytopes.

Fig. 28

The CRS can also be expressed as the difference of the convex hull around all data points and the union of infeasible (empty) convex regions

A polytope can be seen as an intersection of a set of half-spaces of which each is bounded by the hyperplane containing the corresponding facet of the polytope. A point in the polytope, x, is then a solution of the set of constraints $a_h^T x \le b_h \quad \forall h \in H$ where H is the set of half-spaces. This is commonly referred to as an H-representation of a polyhedron. By applying the H-representation to the infeasible regions, we can express a feasible point in the CRS model, x, as a solution to the following set of constraints:

$$x = \sum\limits_j \lambda_j \, v_j \quad \forall \, j \in V $$

(20a)

$$\sum\limits_{j \in V} \lambda_j = 1$$

(20b)

$$ 0 \le \lambda_j \le 1 \quad \forall \, j \in V$$

(20c)

$$a_{rh}^{{\rm T}} \, x \ge b_{rh} + {\bar{\epsilon}} - M(1-z_{rh}) \quad \forall \; r \in {\bar{R}}, \; h \in H_r$$

(20d)

$$\sum\limits_{h \in H_r} z_{rh} \ge 1 \quad \forall \; r \in {\bar{R}}$$

(20e)

$$z_{rh} \in \{0,1\} \quad \forall \; r \in {\bar{R}}, \, h \in H_r$$

(20f)

where V is the set of vertices of the convex hull, ${\bar{R}}$ is the set of infeasible convex regions, $H_r$ is the set of half-spaces associated with region r, and ${\bar{\epsilon}}$ is a small margin parameter. Equations (20a)–(20c) state that x is a point inside the convex hull, while Eqs. (20d)–(20f) enforce that x is not a point in the interior of any of the infeasible regions. The binary variable $z_{rh}$ is 1 if the constraint $a_{rh}^{\text{T}} \, x \le b_{rh}$ is violated. Note that this approach could also be used to consider “holes” in the feasible region.

Formulation (20) is not entirely equivalent to Eqs. (3b)–(3g) since points on facets shared by feasible and infeasible regions will not be feasible in (20). Moreover, the big-M parameter M in (20d) typically leads to weak LP relaxations which will likely make the alternative formulation less efficient if the numbers of feasible and infeasible convex regions are similar.

Appendix 2: Data for illustrative example

For the illustrative example, we have constructed the set of 100 data points shown in Table 7. Each data point consists of a 2-dimensional parameter vector a and a cost value g. The cost values are calculated from the parameter values using two different linear correlations, for which the corresponding constants and coefficients are listed in Table 8. The cost values for the first 52 points are generated from the first linear correlation, while the cost values for the remaining points are calculated by applying the second linear correlation.

Table 7 Complete set of data for the illustrative example

Table 8 Cost constants and coefficients used in the illustrative example