A finite $\epsilon$-convergence algorithm for two-stage stochastic convex nonlinear programs with mixed-binary first and second-stage variables

Abstract

In this paper, we propose a generalized Benders decomposition-based branch and bound algorithm (GBDBAB) to solve two-stage convex mixed-binary nonlinear stochastic programs with mixed-binary variables in both first and second-stage decisions. In order to construct the convex hull of the MINLP subproblem for each scenario in closed-form, we first represent each MINLP subproblem as a generalized disjunctive program in conjunctive normal form (CNF). Second, we apply basic steps to convert the CNF of the MINLP subproblem into disjunctive normal form to obtain the convex hull of the MINLP subproblem. We prove that GBD is able to converge for the problems with pure binary variables given that the convex hull of each subproblem is constructed in closed-form. However, for problems with mixed-binary first and second-stage variables, we propose an algorithm, GBDBAB, where we may have to branch and bound on the continuous first-stage variables to obtain an optimal solution. We prove that the algorithm GBDBAB can converge to $\epsilon$-optimality in a finite number of steps. Since constructing the convex hull can be expensive, we propose a sequential convexification scheme that progressively applies basic steps to the CNF. Computational results on a problem with quadratic constraints, a constrained layout problem, and a planning problem, demonstrate the effectiveness of the algorithm.

Appendices

Appendix 1: Benders cuts for the illustrative example

The Benders cuts that are generated in each node of the branch-and-bound tree for the illustrative example in Sect. 4 are shown in Tables 9, 10 and 11.

Table 9 Benders cuts that are generated at the root node

Table 10 Benders cuts that are generated at node 1

Table 11 Benders cuts that are generated at node 2

Appendix 2: Constrained layout problem under price uncertainty

The constrained layout problem is adapted from the PhD thesis of Sawaya [32]. In this problem, we need to decide the layout of some units represented by rectangles with known lengths and widths. The units have to be manufactured before we install them at certain locations. In the manufacturing process, we need to provide the relative position of the units in order to produce the pipes that connect those units. The first-stage decisions include the designed relative position of the units. After the units are manufactured, they can be installed at several locations. In order to install some unit(s) at a given location, we need to purchase a circled area around the center of that location. The rectangle(s) installed at that location has to be constrained within the circle that is purchased. The price of each area is uncertain and the designed units are installed according to the prices.

The sets, parameters, and variables that are used in the model is defined as follows: Sets

• $i\in N=$ rectangles (units)

• $q\in Q=$ areas

• $\omega \in \varOmega =$ scenarios

Parameters

• $L_{i}=$ length of rectangle i

• $H_{i}=$ width of rectangle i

• $xbar_{q}=$ the x coordinate of area q

• $ybar_{q}=$ the y coordinate of area q

• $\tau _{\omega }=$ probability of scenario $\omega$

• $d_{q\omega }=$ unit price of area q in scenario $\omega$

first-stage decisions

• $x_{i}=$ the designed x coordinate of the center of rectangle i

• $y_{i}=$ the designed y coordinate of the center of rectangle i

• $delx_{ij}=$ the designed distance of the center of rectangle i and the center of rectangle j along the x axis

• $dely_{ij}=$ the designed distance of the center of rectangle i and the center of rectangle j along the y axis

• $Z_{ij}^{1}$, $Z_{ij}^{2}$, $Z_{ij}^{3}$, $Z_{ij}^{4}\in \{True,False\}$ decides the relative position of rectangle i and rectangle j

second-stage decisions

• $xs_{i\omega }=$ the installed x coordinate of the center of rectangle i in scenario $\omega$

• $ys_{i\omega }=$ the installed y coordinate of the center of rectangle i in scenario $\omega$

• $W_{qi\omega }=\{True, False\}$ whether to position rectangle i in area q or not

• $S_{q\omega }=$ the size of area q that is purchased in scenario $\omega$

Constraint (62) relates the designed relative distance of rectangle i and rectangle j in both x and y direction to the designed x and y coordinate of rectangle i and rectangle j. There are variable costs associated with the relative designed distances, which corresponds to the pipes that connect those units. Constraint (63) prevents rectangle i and rectangle j from overlapping with each other. After the design decisions are made, the actual position to install each unit $i\in N$ are decided. Constraint (64) enforces that the actual installed positions of rectangle i and rectangle j in each scenario $\omega \in \varOmega$ have to coincide with the designed relative position. Moreover, in each scenario, each rectangle i has to be constrained in exactly one circle $q\in Q$ that is purchased, which is enforced by constraint (65). The objective also includes the expected cost of the purchase of circles. Note that the problem is expressed as a GDP and can be reformulated with big-M or hull reformulation (see Table 7 in [16]) as a convex MINLP, which is a two-stage stochastic program with mixed integer variables in both first and second stage.

$$\begin{aligned} \min&\displaystyle \sum _{i}\sum _{j,j>i}c_{ij}(delx_{ij}+dely_{ij})+\sum _{\omega \in \varOmega }\tau _{\omega }\sum _{q\in Q}(d_{q\omega }S_{q\omega }) \end{aligned}$$

(61)

$$\begin{aligned}&\displaystyle s.t. \nonumber \\&\displaystyle delx_{ij}\ge x_{i}-x_{j}\quad \forall i,j\in N, i<j \nonumber \\&\displaystyle delx_{ij}\ge x_{j}-x_{i}\quad \forall i,j\in N, i<j \nonumber \\&\displaystyle dely_{ij}\ge y_{i}-y_{j}\quad \forall i,j\in N, i<j \nonumber \\&\displaystyle dely_{ij}\ge y_{j}-y_{i}\quad \forall i,j\in N, i<j \nonumber \\ \end{aligned}$$

(62)

$$\begin{aligned}&\displaystyle \begin{bmatrix} Z_{ij}^1\\ x_{i}+L_{i}/2\le x_{j}-L_{j}/2 \end{bmatrix} \vee \begin{bmatrix} Z_{ij}^2\\ x_{j}+L_{j}/2\le x_{i}-L_{i}/2 \end{bmatrix} \nonumber \\&\displaystyle \quad \vee \begin{bmatrix} Z_{ij}^3\\ y_{i}+H_{i}/2\le y_{j}-H_{j}/2 \end{bmatrix}\vee \begin{bmatrix} Z_{ij}^4\\ y_{j}+H_{j}/2\le y_{i}-H_{i}/2 \end{bmatrix}\nonumber \\&\displaystyle \quad \forall i,j\in N, i<j\end{aligned}$$

(63)

$$\begin{aligned}&\displaystyle xs_{i\omega }-xs_{j\omega }=x_{i}-x_{j}\quad \forall i,j\in N, i<j,\omega \in \varOmega \nonumber \\&\displaystyle ys_{i\omega }-ys_{j\omega }=y_{i}-y_{j}\quad \forall i,j\in N, i<j,\omega \in \varOmega \end{aligned}$$

(64)

$$\begin{aligned}&\displaystyle \vee _{q\in Q} \begin{bmatrix} W_{qi\omega }\\ (xs_{i\omega }-L_{i}/2-xbar_{q})^2+(ys_{i\omega }+H_{i}/2-ybar_{q})^2\le S_{q\omega }\\ (xs_{i\omega }-L_{i}/2-xbar_{q})^2+(ys_{i\omega }-H_{i}/2-ybar_{q})^2\le S_{q\omega }\\ (xs_{i\omega }+L_{i}/2-xbar_{q})^2+(ys_{i\omega }+H_{i}/2-ybar_{q})^2\le S_{q\omega }\\ (xs_{i\omega }+L_{i}/2-xbar_{q})^2+(ys_{i\omega }-H_{i}/2-ybar_{q})^2\le S_{q\omega } \end{bmatrix} \quad \forall i\in N,\omega \in \varOmega \nonumber \\ \end{aligned}$$

(65)

$$\begin{aligned}&\displaystyle delx_{ij},dely_{ij},S_{q\omega }\in \mathrm{I\!R}^{1}_{+}, Z_{ij}^{1},Z_{ij}^{2},Z_{ij}^{3},Z_{ij}^{4}, W_{qi\omega }\nonumber \\&\displaystyle \quad \in \{True, False\}\quad \forall i,j\in N, i<j,q\in Q,\omega \in \varOmega \end{aligned}$$

(66)