A computationally useful algebraic representation of nonlinear disjunctive convex sets using the perspective function

Abstract

Nonlinear disjunctive convex sets arise naturally in the formulation or solution methods of many discrete–continuous optimization problems. Often, a tight algebraic representation of the disjunctive convex set is sought, with the tightest such representation involving the characterization of the convex hull of the disjunctive convex set. In the most general case, this can be explicitly expressed through the use of the perspective function in higher dimensional space—the so-called extended formulation of the convex hull of a disjunctive convex set. However, there are a number of challenges in using this characterization in computation which prevents its wide-spread use, including issues that arise because of the functional form of the perspective function. In this paper, we propose an explicit algebraic representation of a fairly large class of nonlinear disjunctive convex sets using the perspective function that addresses this latter computational challenge. This explicit representation can be used to generate (tighter) algebraic reformulations for a variety of different problems containing disjunctive convex sets, and we report illustrative computational results using this representation for several nonlinear disjunctive problems.

Appendix: description of nonlinear GDP problem classes

1.1 Synthesis of process networks

These problems consist in determining those process units $Y_{k}$ to be included in the design of a process network such that the structure and operating conditions of this network will meet certain design specifications, while minimizing the sum of fixed costs $c_{k}$ and variable costs $a^{T} x$ of the overall network (the variable $x$ represents material flow). The following example represents a GDP model of a 5-process network, and can be found in Sawaya’s thesis [44]. This example is a slightly modified version of the original form proposed by Duran and Grossmann [13] (MINLP form), Turkay and Grossmann [48] and Lee and Grossmann [37] (GDP forms).

$$\hbox{min} Z = \sum\limits_{k = 1}^{5} {c_{k} } + a^{T} x$$

$$\begin{aligned} & s.t. \\ & x_{1} - x_{2} - x_{3} = 0 \\ \end{aligned}$$

(25)

$$x_{6} - x_{4} - x_{5} = 0$$

(26)

$$x_{6} - x_{7} - x_{8} - x_{11} = 0$$

(27)

$$x_{8} - x_{9} - x_{10} - x_{11} = 0$$

(28)

$$\left[ {\begin{array}{*{20}c} {Y_{1} } \\ {\exp (x_{4} ) - 1 - x_{2} \le 0} \\ {c_{1} = 5} \\ \end{array} } \right] \vee \left[ {\begin{array}{*{20}c} {\neg Y_{1} } \\ {x_{4} = x_{2} = 0} \\ {c_{1} = 0} \\ \end{array} } \right]$$

(29)

$$\left[ {\begin{array}{*{20}c} {Y_{2} } \\ {\exp (x_{5} /1.2) - 1 - x_{3} \le 0} \\ {c_{2} = 8} \\ \end{array} } \right] \vee \left[ {\begin{array}{*{20}c} {\neg Y_{2} } \\ {x_{3} = x_{5} = 0} \\ {c_{2} = 0} \\ \end{array} } \right]$$

(30)

$$\left[ {\begin{array}{*{20}c} {Y_{3} } \\ {x_{13} - 0.75x_{9} = 0} \\ {c_{3} = 6} \\ \end{array} } \right] \vee \left[ {\begin{array}{*{20}c} {\neg Y_{3} } \\ {x_{9} = x_{13} = 0} \\ {c_{3} = 0} \\ \end{array} } \right]$$

(31)

$$\left[ {\begin{array}{*{20}c} {Y_{4} } \\ {\exp (x_{14} /1.5) - 1 - x_{10} \le 0} \\ {c_{4} = 10} \\ \end{array} } \right] \vee \left[ {\begin{array}{*{20}c} {\neg Y_{4} } \\ {x_{10} = x_{14} = 0} \\ {c_{4} = 0} \\ \end{array} } \right]$$

(32)

$$\left[ {\begin{array}{*{20}c} {Y_{5} } \\ {x_{15} - x_{11} = 0} \\ {x_{15} - 0.5x_{12} = 0} \\ {c_{5} = 6} \\ \end{array} } \right] \vee \left[ {\begin{array}{*{20}c} {\neg Y_{5} } \\ {x_{11} = x_{12} = x_{15} = 0} \\ {c_{5} = 0} \\ \end{array} } \right]$$

(33)

$$x_{1} \le 10,\quad x_{12} \le 7$$

(34)

$$Y_{1} \mathop \vee \limits_{ - } Y_{2}$$

(35)

$$\begin{aligned} & a^{T} = [0,0,0,0,0,0,5,0,0,0,0, - 2,200,250,300] \\ & x_{j} ,c_{k} \ge 0,\quad Y_{k} \in \{ True,False\} \;\;{\text{for}}\;k{ = 1,2,} \ldots 5 ;\;\; \, j{ = 1,2,} \ldots 1 5\\ \end{aligned}$$

Equations (25)–(28) represent linear mass balances around nodes N1–N4. Disjunctions (29)–(33) embody the discrete dichotomy of process selection, where a unit, along with its in-and-out flows (connected to one another via an exponential relationship) and fixed cost, is selected for inclusion in the final network only if its corresponding Boolean variable $Y_{k} = True$; otherwise, the unit is not selected, and its flows and fixed cost are set to 0. Equation (34) represents upper bounds on certain flows, and finally, logic Eq. (35) imposes the condition that either process 1 or 2 must be selected (but not both).

1.2 Retrofit–Synthesis of process networks

These problems consist in simultaneously redesigning part of an existing plant and synthesizing (from scratch) part of a new one. Specifically, one is interested in determining whether certain units should be included in the design of the new plant, and whether certain modifications such as improvements in yield, capacity and energy reduction should be performed on the existing plant. In addition it is required that economic potential be maximized given a certain time horizon and limited capital investments. The nonlinearities in this set of problems stem from the synthesis portion of the model, and correspond to logarithmic functions. Below, we only show the retrofit portion of the model (since the synthesis portion was presented above), which is a modification of work by Jackson and Grossmann [33] and which appeared in [44]:

$$Min\quad \sum\limits_{\forall t \in T} {\sum\limits_{{\forall s \in S_{prod} }} {PR_{s}^{t} } } mf_{s}^{t} - \sum\limits_{\forall t \in T} {\sum\limits_{{\forall s \in S_{raw} }} {PR_{s}^{t} } } mf_{s}^{t} - \sum\limits_{\forall t \in T} {PRSTqst^{t} } - \sum\limits_{\forall t \in T} {PRWTqwt^{t} } - \sum\limits_{\forall t \in T} {\sum\limits_{\forall p \in P} {fc_{p}^{t} } } - \sum\limits_{\forall t \in T} {ec^{t} }$$

(36)

$$s.t.\quad mf_{s}^{t} = f_{s}^{t} MW_{s} \quad \forall s \in S,\quad \forall t \in T$$

(37)

$$mf_{s}^{t} \ge DEM_{s}^{t} \quad \forall s \in S_{prod} ,\quad \forall t \in T$$

(38)

$$mf_{s}^{t} \le SUP_{s}^{t} \quad \forall s \in S_{raw} ,\quad \forall t \in T$$

(39)

$$\sum\limits_{{\forall s \in S_{{n_{in} }} }} {mf_{s}^{t} } = \sum\limits_{{\forall s \in S_{{n_{out} }} }} {mf_{s}^{t} } \quad \forall n \in N,\quad \forall t \in T$$

(40)

$$\sum\limits_{{\forall s \in S_{{p_{in} }} }} {mf_{s}^{t} } = \sum\limits_{{\forall s \in S_{{p_{out} }} }} {mf_{s}^{t} + unrct_{p}^{t} } \quad \forall p \in P,\quad \forall t \in T$$

(41)

$$\mathop \vee \limits_{\forall m \in M} \left[ {\begin{array}{*{20}c} {Y_{pm}^{t} } \\ {f_{s}^{t} = f_{{p_{lmt} }}^{t} (\frac{{GMA_{s}^{t} }}{{GMA_{{p_{lmt} }} }})ETA_{pm}^{t} } \\ {\sum\limits_{{\forall s \in S_{{p_{in} }} }} {mf_{s}^{t} } \le CAP_{pm}^{t} } \\ \end{array} } \right]\quad \forall s \in S_{{p_{out} }} ,\quad \forall p \in P,\quad \forall t \in T$$

(42)

$$\mathop \vee \limits_{\forall m \in M} \left[ {\begin{array}{*{20}c} {W_{pm}^{t} } \\ {fc_{p}^{t} = FC_{pm}^{t} } \\ \end{array} } \right]\quad \forall p \in P,\quad \forall t \in T$$

(43)

$$q_{sk}^{t} = mf_{s}^{t} CP_{s} (T_{{s_{{out_{k} }} }}^{t} - T_{{s_{{in_{k} }} }}^{t} )\quad \forall s \in S_{cold} ,\quad \forall k \in K,\quad \forall t \in T$$

(44)

$$q_{sk}^{t} = mf_{s}^{t} CP_{s} (T_{{s_{{in_{k} }} }}^{t} - T_{{s_{{out_{k} }} }}^{t} )\quad \forall s \in S_{hot} ,\quad \forall k \in K,\quad \forall t \in T$$

(45)

$$\left[ {\begin{array}{*{20}c} {X_{1}^{t} } \\ {qst^{t} = \sum\limits_{\forall k \in K} {\sum\limits_{{\forall s \in S_{cold} }} {q_{sk}^{t} } } } \\ {q_{sk}^{t} = \sum\limits_{\forall k \in K} {\sum\limits_{{\forall s \in S_{hot} }} {q_{sk}^{t} } } } \\ \end{array} } \right] \vee \left[ {\begin{array}{*{20}c} {X_{2}^{t} } \\ {r_{k}^{t} - r_{k - 1}^{t} - qst_{k}^{t} + qwt_{k}^{t} = \sum\limits_{{\forall s \in S_{hot} }} {q_{sk}^{t} } - \sum\limits_{{\forall s \in S_{cold} }} {q_{sk}^{t} } } \\ {qst^{t} = \sum\limits_{\forall k \in K} {qst_{k}^{t} } } \\ {qwt^{t} = r_{\left| K \right|}^{t} + \sum\limits_{\forall k \in K} {qwt_{k}^{t} } } \\ \end{array} } \right]\quad \forall k \in K,\quad \forall t \in T$$

(46)

$$\left[ {\begin{array}{*{20}c} {V_{1}^{t} } \\ {ec^{t} = EFC_{1}^{t} } \\ \end{array} } \right] \vee \left[ {\begin{array}{*{20}c} {V_{2}^{t} } \\ {ec^{t} = EFC_{2}^{t} } \\ \end{array} } \right]\quad \forall t \in T$$

(47)

$$\sum\limits_{\forall p \in P} {fc_{p}^{t} } + ec^{t} + \sum\limits_{{\forall s \in S_{raw} }} {PR_{s}^{t} mf_{s}^{t} } + PRSTqst^{t} + PRWTqwt^{t} \le INV^{t} \quad \forall t \in T$$

(48)

$$Y_{pm}^{t} \to \mathop \wedge \limits_{{\tau \text{ > }t}} Y_{pm}^{\tau } \quad \forall p \in P,\quad \forall t \in T,\quad \forall m \in M\backslash m_{1}$$

(49)

$$W_{pm}^{t} \to \mathop \wedge \limits_{\tau \ne t} W_{p1}^{t} \quad \forall p \in P,\quad \forall t \in T,\quad \forall m \in M\backslash m_{1}$$

(50)

$$Y_{p1}^{t} \to W_{p1}^{t} \quad \forall p \in P,\quad \forall t \in T$$

(51)

$$Y_{pm}^{t} \mathop \wedge \limits_{{\tau \text{ < }t}} \, \neg Y_{pm}^{\tau } \to W_{pm}^{t} \quad \forall p \in P,\quad \forall t \in T,\quad \forall m \in M\backslash m_{1}$$

(52)

$$X_{j}^{t} \to \mathop \wedge \limits_{{\tau \text{ > }t}} X_{j}^{\tau } \quad \forall t \in T,\quad \forall j \in J\backslash j_{1}$$

(53)

$$V_{j}^{t} \to \mathop \wedge \limits_{\tau \ne t} V_{1}^{t} \quad \forall t \in T,\quad \forall j \in J\backslash j_{1}$$

(54)

$$X_{1}^{t} \to V_{1}^{t} \quad \forall t \in T$$

(55)

$$X_{j}^{t} \mathop \wedge \limits_{{\tau \text{ < }t}} \, \neg X_{j}^{\tau } \to V_{j}^{t} \quad \forall t \in T,\quad \forall j \in J\backslash j_{1}$$

(56)

$$\begin{aligned} & mf_{s}^{t} ,f_{s}^{t} \in {\mathbf{\mathbb{R}}}^{1}_{ + } \quad \forall s \in S,\quad \forall t \in T \\ & f_{lmt,p}^{t} ,unrct_{p}^{t} ,fc_{p}^{t} \in {\mathbf{\mathbb{R}}}^{1}_{ + } \quad \forall p \in P,\quad \forall t \in T \\ & q_{sk}^{t} \in {\mathbf{\mathbb{R}}}^{1}_{ + } \quad \forall s \in S,\quad \forall k \in K,\quad \forall t \in T \\ & qst^{t} ,qwt^{t} ,ec^{t} \in {\mathbf{\mathbb{R}}}^{1}_{ + } \quad \forall t \in T \\ & qst_{k}^{t} ,qwt_{k}^{t} ,r_{k}^{t} \in {\mathbf{\mathbb{R}}}^{1}_{ + } \quad \forall k \in K,\quad \forall t \in T \\ & Y_{{^{pm} }}^{t} ,W_{{^{pm} }}^{t} \in {\{ }True,False{\} }\quad \forall p \in P,\quad \forall t \in T,\quad \forall m \in M \\ & X_{{^{j} }}^{t} ,V_{{^{j} }}^{t} \in {\{ }True,False{\} }\quad \forall j \in J,\quad \forall t \in T \\ \end{aligned}$$

The objective function (36) includes revenues from sales, costs of raw material, utility costs, as well as capital costs $fc_{p}^{t}$ and energy costs $ec^{t}$ over time periods $t \in T$. Equation (37) represents an equivalence relation between mass and molar flow rates, Eqs. (38) and (39) ensure that mass flow rates for products and raw materials are respectively bounded by demand and supply parameters, and Eqs. (40) and (41) serve as mass balances around nodes $n \in N$ and processes $p \in P$, respectively. The first set of disjunctions (42) selects one of the operating modes for the retrofit project $m \in M$, for every process $p \in P$, in every time period $t \in T$, where projects m include modifying either nothing at all ($m_{1} \in M$), process conversion ($m_{2} \in M$), capacity ($m_{3} \in M$) or both ($m_{4} \in M$). The second set of disjunctions (43) enforces the cost of the aforementioned modifications, where capital costs are set to zero ($fc_{p}^{t}$ = 0) if nothing is modified. Equations (44) and (45) serve as equivalence relations between energy and mass flow rate variables, while disjunction(s) (46) select the appropriate operating mode $X_{{^{j} }}^{t}$$\forall j \in J$ so that $X_{1}^{t}$ corresponds to no energy integration and $X_{2}^{t}$ enforces the transshipment equations. Through Boolean variables $V_{{^{j} }}^{t}$, the set of disjunctions (47) enforce the cost associated with energy reduction, where these costs are set to zero ($ec^{t}$ = 0) if nothing is modified ($V_{{^{1} }}^{t}$ = True). Equation (48) limits the expenses for the retrofit project. Equations (51) and (52), (55) and (56) are logical conditions that connect, respectively, disjunctions (42)–(43) and disjunctions (44)–(45) with each other, and Eqs. (49) and (50), (53) and (54) impose logical conditions between disjuncts in every set of corresponding disjunctions. Essentially, these logical equations constrain the problem such that costs associated with conversion and/or capacity are enforced exactly once for every process $p \in P$ in every time period $t \in T$, and such that costs associated with energy reduction are enforced exactly once per time period $t \in T$.

1.3 Constrained Layout

In these problems, which first appeared in [45] (see also [44]), non-overlapping process units represented by rectangles must be placed within the confines of certain designated areas formulated as circular nonlinear constraints, such that the cost of connecting these units is minimized. The nonlinearities in this set of problems are all quadratic and correspond to Euclidean-distance constraints, and the integrality gap for all instances presented is equal to 100%. Note that these problem are intentionally poorly modeled in order to have a large integrality gap and no feasible solution near the optimal solution of the continuous relaxation.

$$Min\quad Q = \sum\limits_{i} {\sum\limits_{j} {c_{ij} (delx_{ij} + \, dely_{ij} )} }$$

(57)

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

(58)

$$delx_{ij} \ge x_{j} - x_{i} \quad \forall i,j \in N,\;\;i\text{ < }j$$

(59)

$$dely_{ij} \ge y_{i} - y_{j} \quad \forall i,j \in N,\;\;i\text{ < }j \,$$

(60)

$$dely_{ij} \ge y_{j} - y_{i} \quad \forall i,j \in N,\;\;i\text{ < }j$$

(61)

$$\left[ {\begin{array}{*{20}c} {Z^{1}_{ij} } \\ {x_{i} + L_{i} /2 \le x_{j} - L_{j} /2} \\ \end{array} } \right] \vee \left[ {\begin{array}{*{20}c} {Z^{2}_{ij} } \\ {x_{j} + L_{j} /2 \le x_{i} - L_{i} /2} \\ \end{array} } \right] \vee \left[ {\begin{array}{*{20}c} {Z^{3}_{ij} } \\ {y_{i} + H_{i} /2 \le y_{j} - H_{j} /2} \\ \end{array} } \right] \vee \left[ {\begin{array}{*{20}c} {Z^{4}_{ij} } \\ {y_{j} + H_{j} /2 \le y_{i} - H_{i} /2} \\ \end{array} } \right]\quad \forall i,j \in N,\;\;i\text{ < }j$$

(62)

$$\mathop \vee \limits_{{area \in J_{i} }} \left[ {\begin{array}{*{20}c} {W_{area,i} } \\ {(x_{i} - L_{i} /2 - xbar_{area} )^{2} + (y_{i} + H_{i} /2 - ybar_{area} )^{2} \le r_{area}^{2} } \\ {(x_{i} - L_{i} /2 - xbar_{area} )^{2} + (y_{i} - H_{i} /2 - ybar_{area} )^{2} \le r_{area}^{2} } \\ {(x_{i} + L_{i} /2 - xbar_{area} )^{2} + (y_{i} + H_{i} /2 - ybar_{area} )^{2} \le r_{area}^{2} } \\ {(x_{i} + L_{i} /2 - xbar_{area} )^{2} + (y_{i} - H_{i} /2 - ybar_{area} )^{2} \le r_{area}^{2} } \\ \end{array} } \right]\quad \forall i \in N$$

(63)

$$x_{i} \le UB^{1}_{i} \quad \forall i \in N$$

(64)

$$x_{i} \ge LB^{1}_{i} \quad \forall i \in N$$

(65)

$$y_{i} \le UB^{2}_{i} \quad \forall i \in N$$

(66)

$$y_{i} \ge LB^{2}_{i} \quad \forall i \in N$$

(67)

$$delx_{ij} ,dely_{ij} \in {\mathbf{\mathbb{R}}}^{1}_{ + } , {\text{ Z}}^{1}_{ij} ,Z^{2}_{ij} ,Z^{3}_{ij} ,Z^{4}_{ij} ,W_{area,i} \in {\{ }True,False{\} }\quad \forall i,j \in N,\;\;i\text{ < }j$$

Every process unit is represented by a rectangle $i \in N$ that has length L_i, height H_i and coordinates (x_i, y_i), where the point of reference corresponds to the center of every rectangle. By constraining every pair of rectangles (i, j) where $(i,j \in N,\;i < j)$ such that no overlap occurs, we obtain a series of disjunctions with four terms each—Eqs. (62)—where each term represents the position of rectangle i in relation to rectangle j. Furthermore, per Eqs. (63), every rectangle i must be placed within some circular constrained area centered at $(xbar,ybar)_{area}$ with radius $r_{area}$, and must also satisfy the upper and lower bounds represented by inequalities (64)–(67). Finally, there is a cost $c_{ij}$ that needs to be paid between every pair of rectangles (i, j). The objective of the problem, then, is to minimize the overall cost of laying out the rectangles [represented by the objective function (57) and the inequalities (58)–(61)] such that no two rectangles overlap and every rectangle is placed within some constrained circular area.