Reformulations for utilizing separability when solving convex MINLP problems

Abstract

Several deterministic methods for convex mixed integer nonlinear programming generate a polyhedral approximation of the feasible region, and utilize this approximation to obtain trial solutions. Such methods are, e.g., outer approximation, the extended cutting plane method and the extended supporting hyperplane method. In order to obtain the optimal solution and verify global optimality, these methods often require a quite accurate polyhedral approximation. In case the nonlinear functions are convex and separable to some extent, it is possible to obtain a tighter approximation by using a lifted polyhedral approximation, which can be achieved by reformulating the problem. We prove that under mild assumptions, it is possible to obtain tighter linear approximations for a type of functions referred to as almost additively separable. Here it is also shown that solvers, by a simple reformulation, can benefit from the tighter approximation, and a numerical comparison demonstrates the potential of the reformulation. The reformulation technique can also be combined with other known transformations to make it applicable to some nonseparable convex functions. By using a power transform and a logarithmic transform the reformulation technique can for example be applied to p-norms and some convex signomial functions, and the benefits of combining these transforms with the reformulation technique are illustrated with some numerical examples.

Appendices

Almost additively separable test problems

In the numerical comparison in Sect. 5, we have included three test problems currently not available in MINLPLib2. These test problems have an almost additively separable objective function which is nonlinear in all variables and can be written in the following form

$$\begin{aligned} \begin{aligned}&\text {minimize } \sum _{i=1}^{N-1} \ \left\| \ \left[ \begin{matrix} x_{i+1} \\ x_{i} \end{matrix}\right] - \left[ \begin{matrix} 5.5 \\ 5.5 \end{matrix}\right] \ \right\| _2\\&\text {subject to}\\&-10 \le x_i \le 10 \quad \forall \ i=1, \ldots , N, \\&\mathbf {x} \in {\mathbb R}^N,\\&x_i \in {\mathbb Z}\quad \forall \ i=1, 3, 5, \ldots , N-1. \end{aligned} \end{aligned}$$

(MINLP5)

The objective function is almost additively separable according to the definition in Sect. 2 and each component of the function is convex. To avoid numerical issues with the integer relaxation of problem (MINLP5), we have added a small constant of $10^{-5}$ to each square root in the objective function. Otherwise, there might be difficulties due to the non-differentiability of the Euclidean norm at zero. The number of continuous and integer variables in the problems are given in Table 4. Even though these problems might appear as simple, they are challenging for solvers based on methods such as ECP, ESH and OA.

Signomial test problems

In Sect. 6, we considered a logarithmic transform of some signomial functions. Here we present a couple of simple test problems to whom the transformation is applicable. Consider an optimization problem with a negative signomial function of the following type

$$\begin{aligned} \begin{aligned}&\text {minimize} \ \mathbf {c}^T\mathbf {x} \\&\ \text {subject to}\\&\ -a \prod _{i=1}^{N}{x_i^{p_i}} \le -1,\\&\ 0.00001 \le x_i \le 10 \quad \forall \ i=1, \ldots , N, \\&\ \mathbf {x} \in {\mathbb R}^N,\\&\ x_i \in {\mathbb Z}\quad \forall \ i=1, \ldots , J, \end{aligned} \end{aligned}$$

(N-sig)

where a is a strictly positive constant, which was chosen as 0.2 for the test problems. The signomial constraint is convex if all the exponents $p_i$ are positive and $\sum _{i=1}^N{p_i}=1$. By applying the transforms from Sect. 6, the problem can be reformulated into a separable form to which the lifted reformulation from Sect. 4 can be applied. Furthermore, we also consider signomial MINLP problems of the following type

$$\begin{aligned} \begin{aligned}&\text {minimize} \ a\prod _{i=1}^{N}{x_i^{-p_i}} + \sum _{i=1}^{N} x_i\\&\text {subject to}\\&1 \le x_i \le 10 \quad \forall i=1,2,\ldots ,N,\\&\ \mathbf {x} \in {\mathbb R}^N,\\&x_i \in {\mathbb Z}\quad \forall i=1, \ldots J, \end{aligned} \end{aligned}$$

(P-sig)

where $a \in {\mathbb R}_+$ and $p_i \in {\mathbb R}_+$. By applying the log transform, the problem can be transformed into the same form as problem (MINLP3-R). We have generated three test problems of the same type as both problems (N-sig) and (P-sig). The test problems vary in size and both the exponential terms $p_i$ and the vectors $\mathbf {c}$ have been chosen randomly. These test problems have been submitted to MINLPLib2, where they should be commonly available. The number of continuous and integer variables for these test problems are presented in Table 4. The exponential terms $p_i$ and the components in the vector $\mathbf {c}$ is presented for all test problems in Table 5.

Table 4 Number of variables for the test problems

Table 5 Parameters for the test problems in Sect. 6

Test problems for the power transform

As shown in Sect. 6, it is possible to transform some nonseparable constraints into a separable form by a simple power transform. One such type of MINLP problems can be written as

$$\begin{aligned} \begin{aligned}&\text {minimize}\ -\mathbf {c}^T\mathbf {x} \\&\text {subject to}\\&\Vert \mathbf {x} \Vert _2 \le 10,\\&0 \le x_i \le 5 \quad \forall i=1,2, \ldots , N,\\&\ \mathbf {x} \in {\mathbb R}^N,\\&x_i \in {\mathbb Z}\quad \forall i=1, \ldots J. \end{aligned} \end{aligned}$$

(Norm-con)

By applying the power transform from Sect. 6, the problem can transformed into a separable form. Another type of test problems considered are given by

$$\begin{aligned} \begin{aligned}&\text {minimize}\ -\mathbf {c}^T\mathbf {x} \\&\text {subject to}\\&\left( \sum _{i=1}^{N-1}{a^{x_i+x_{i+1}}}\right) ^p \le b,\\&0 \le x_i \le 5 \quad \forall i=1,2, \ldots , N.\\&\ \mathbf {x} \in {\mathbb R}^N,\\&x_i \in {\mathbb Z}\quad \forall i=1, \ldots J, \end{aligned} \end{aligned}$$

(P-con)

where $a,\,p > 1$ and constants. This is also a convex nonseparable problem that can be made separable by a power transform. We have generated three test problems of both the same type as problem (Norm-con) and problem (P-con). The vectors $\mathbf {c}$ have been chosen randomly as positive numbers between 0 and 1. These test problems have also been submitted to MINLPLib2. The size of these test problems are given in Table 4 and all the constants are given in Table 5.