Discretization and global optimization for mixed integer bilinear programming

Abstract

We consider global optimization of mixed-integer bilinear programs (MIBLP) using discretization-based mixed-integer linear programming (MILP) relaxations. We start from the widely used radix-based discretization formulation (called R-formulation in this paper), where the base R may be any natural number, but we do not require the discretization level to be a power of R. We prove the conditions under which R-formulation is locally sharp, and then propose an $R^+$-formulation that is always locally sharp. We also propose an H-formulation that allows multiple bases and prove that it is also always locally sharp. We develop a global optimization algorithm with adaptive discretization (GOAD) where the discretization level of each variable is determined according to the solution of previously solved MILP relaxations. The computational study shows the computational advantage of GOAD over general-purpose global solvers BARON and SCIP.

Appendices

Appendix A: Parameters for the multiperiod pooling problems

See Table 7.

Table 7 Supply and demand parameters

Appendix B: The SB formulation for the multiperiod pooling problems

See Table 8 for the list of symbols in the SB formulation

Table 8 List of symbols for the SB formulation

The SB formulation

Objective:

$$\begin{aligned} \max .\;\;\sum _{t\in T}\left[ \sum _{i:(i,d)\in A}\sum _{d\in D}\beta _d f_{i,d,t}-\sum _{s\in S}\sum _{i:(s,i)\in A}\beta _s f_{s,i,t}-\sum _{(i,i')\in A}(\alpha _{i,i'}y_{l,i,i',t}+\beta _{i,i'}f_{i,i',t}) \right] .\end{aligned}$$

s.t. Bilinear terms:

$$\begin{aligned} f_{l,b,i,t}&=I_{l,b,t-1}x_{b,i,t},&\quad \forall (b,l)\in L, (b,i)\in A, t>1. \end{aligned}$$

(SB1)

Mass Balance:

$$\begin{aligned}&I_{l,s,t}=I_{l,s}^0/I_{l,s,t-1}+F^{IN}_{s,t}-\sum _{i:(s,i)\in A}f_{l,s,i,t},&\quad \forall (l,s)\in L, t\in T,x \end{aligned}$$

(SB1)

$$\begin{aligned}&I_{l,b,t}=I^0_{l,b}/I_{l,b,t-1}-\sum _{i:(b,i)\in A}f_{l,b,i,t} \end{aligned}$$

$$\begin{aligned}&\qquad \qquad +\sum _{i:(i,b)\in A}f_{l,i,b,t},&\quad \forall b\in B, (l,b)\in L, t\in T, \end{aligned}$$

(SB3)

$$\begin{aligned}&I_{l,d,t}=I^0_{l,d}/I_{l,d,t-1}-f^{OUT}_{l,d,t}+\sum _{i:(i,d)\in A}f_{l,i,d,t},&\quad \forall d\in D, t\in T. \end{aligned}$$

(SB4)

Quality Bounds:

$$\begin{aligned}&\sum _{l}I_{l,b,t-1}C^0_{q,l}\ge C^L_{q,d}\sum _{l}I_{l,b,t-1}-M(1-y_{b,d,t}),&\quad \forall (b,d)\in A, q\in Q, t>1, \end{aligned}$$

(SB5)

$$\begin{aligned}&\sum _{l}I_{l,b,t-1}C^0_{q,l}\le C^U_{q,d}\sum _{l}I_{l,b,t-1}+M(1-y_{b,d,t}),&\quad \forall (b,d)\in A, q\in Q, t>1,\end{aligned}$$

(SB5)

$$\begin{aligned}&C^0_{q,s}y_{s,d,t}\ge C^L_{q,d}y_{s,d,t},&\quad \forall (s,d)\in A, t\in T,\end{aligned}$$

(SB7)

$$\begin{aligned}&C^0_{q,s}y_{s,d,t}\le C^U_{q,d}y_{s,d,t},&\quad \forall (s,d)\in A, t\in T,\end{aligned}$$

(SB8)

$$\begin{aligned}&C^0_{q,b}y_{b,d,t}\ge C^L_{q,d}y_{b,d,t},&\quad \forall (b,d)\in A, t=1.\end{aligned}$$

(SB9)

$$\begin{aligned}&C^0_{q,b}y_{b,d,t}\le C^U_{q,d}y_{b,d,t},&\quad \forall (b,d)\in A, t=1. \end{aligned}$$

(SB10)

Tank Capacity:

$$\begin{aligned} I^L_i\le I_{i,t}&\le I^U_i,&\quad \forall i\in N, t\in T. \end{aligned}$$

(SB11)

Pipe Capacity:

$$\begin{aligned} F^L_{i,i',t}y_{i,i',t}&\le f_{i,i',t}\le F^U_{i,i',t}y_{i,i',t},&\quad \forall (i,i')\in A, t\in T. \end{aligned}$$

(SB12)

Individual Flow Constraints:

$$\begin{aligned} \sum _{l}I_{l,b,t}&=I_{b,t},&\quad \forall (b,l)\in L, t\in T, \end{aligned}$$

(SB13)

$$\begin{aligned} \sum _{l}f^{out}_{l,d,t}&=F^{OUT}_{d,t},&\quad \forall d\in D, t\in T. \end{aligned}$$

(SB14)

Operation Mode:

$$\begin{aligned}&y_{i,b,t}+y_{b,i',t}\le 1,&\quad \forall (i,b),(b,i')\in A, t\in T. \end{aligned}$$

(SB15)

Variable Bounds:

$$\begin{aligned}&f_{i,i',t}\ge 0,&\quad \forall (i,i')\in A, t\in T, \end{aligned}$$

(SB16)

$$\begin{aligned}&0\le x_{b,i,t}\le 1,&\quad \forall (b,i)\in A, t\in T,\end{aligned}$$

(SB17)

$$\begin{aligned}&y_{i,i',t}\in \{0,1\},&\quad \forall (i,i')\in A, t\in T. \end{aligned}$$

(SB18)