Convex mixed-integer nonlinear programs derived from generalized disjunctive programming using cones

Abstract

We propose the formulation of convex Generalized Disjunctive Programming (GDP) problems using conic inequalities leading to conic GDP problems. We then show the reformulation of conic GDPs into Mixed-Integer Conic Programming (MICP) problems through both the big-M and hull reformulations. These reformulations have the advantage that they are representable using the same cones as the original conic GDP. In the case of the hull reformulation, they require no approximation of the perspective function. Moreover, the MICP problems derived can be solved by specialized conic solvers and offer a natural extended formulation amenable to both conic and gradient-based solvers. We present the closed form of several convex functions and their respective perspectives in conic sets, allowing users to formulate their conic GDP problems easily. We finally implement a large set of conic GDP examples and solve them via the scalar nonlinear and conic mixed-integer reformulations. These examples include applications from Process Systems Engineering, Machine learning, and randomly generated instances. Our results show that the conic structure can be exploited to solve these challenging MICP problems more efficiently. Our main contribution is providing the reformulations, examples, and computational results that support the claim that taking advantage of conic formulations of convex GDP instead of their nonlinear algebraic descriptions can lead to a more efficient solution to these problems.

Appendices

Appendix A: Background

In this manuscript, we use a similar notation to the one used by Ben-Tal and Nemirowski [11] and Alizadeh and Goldfarb [81]. We use lowercase boldface letters, e.g., $\textbf{x,c}$, to denote column vector, and uppercase boldface letters, e.g., ${\textbf{A}}, {\textbf{X}}$, to denote matrices. Sets are denoted with uppercase calligraphic letters, e.g., ${\mathcal {S}}, {\mathcal {K}}$. Subscripted vectors denote $\mathbf {x_i}$ denote the ith block of ${\textbf{x}}$. The jth component of the vectors ${\textbf{x}}$ and $\mathbf {x_i}$ are indicated as $x_j$ and $x_{ij}$. The set $\{1,\ldots ,J\}$ is represented by the symbol $\llbracket J \rrbracket$. Moreover, the subscript $\llbracket J \rrbracket$ of a vector ${\textbf{x}}$ is used to define the set ${\textbf{x}}_{\llbracket J \rrbracket }:= \{{\textbf{x}}_1,\ldots ,{\textbf{x}}_J\}$. We use ${\textbf{0}}$ and ${\textbf{1}}$ for the all zeros and all ones vector, respectively, and 0 and I for the zero and identity matrices, respectively. The vector $e_j$ will be the vector with a single 1 in position j, and its remaining elements are 0. The dimensions of the matrices and vectors will be clear from the context. We use ${\mathbb {R}}^k$ to denote the set of real numbers of dimension k, and for set ${\mathcal {S}} \subseteq {\mathbb {R}}^k$, we use $\text {cl}({\mathcal {S}})$ and $\text {conv}({\mathcal {S}})$ to denote the closure and convex hull of ${\mathcal {S}}$, respectively.

For concatenated vectors, we use the notation that “,” is row concatenation of vectors and matrices, and “;” is column concatenation. For vectors, $\textbf{x,y}$ and ${\textbf{z}}$, the following are equivalent.

$$\begin{aligned} \begin{pmatrix} {\textbf{x}} \\ {\textbf{y}} \\ {\textbf{z}} \end{pmatrix} = ({\textbf{x}}^{\top },{\textbf{y}}^{\top },{\textbf{z}}^{\top })^{\top } = ({\textbf{x}};{\textbf{y}};{\textbf{z}}). \end{aligned}$$

(A1)

The projection of a set ${\mathcal {S}} \subseteq {\mathbb {R}}^k$ onto the vector ${\textbf{x}} \in X \subseteq {\mathbb {R}}^n$, with $n \le k$ is denoted as $\text {proj}_{{\textbf{x}}}({\mathcal {S}}):= \{ {\textbf{x}} \in X: \exists {\textbf{y}}: ({\textbf{x}};{\textbf{y}}) \in {\mathcal {S}} \}$.

If ${\mathcal {A}} \subseteq {\mathbb {R}}^k$ and ${\mathcal {B}} \subseteq {\mathbb {R}}^l$ we denote their Cartesian product as ${\mathcal {A}} \times {\mathcal {B}}:= \{({\textbf{x}};{\textbf{y}}): {\textbf{x}} \in {\mathcal {A}}, {\textbf{y}} \in {\mathcal {B}} \}$.

For ${\mathcal {A}}_1, {\mathcal {A}}_2 \subseteq {\mathbb {R}}^k$ we define the Minkowski sum of the two sets as ${\mathcal {A}}_1 + {\mathcal {A}}_2 = \{ {\textbf{u}} + {\textbf{v}}: {\textbf{u}} \in {\mathcal {A}}_1, {\textbf{v}} \in {\mathcal {A}}_2 \}$.

1.1 Cones

For a thorough discussion about convex optimization and conic programming, we refer the reader to [11]. The following definitions are required for the remainder of the manuscript.

The set ${\mathcal {K}} \subseteq {\mathbb {R}}^k$ is a cone if $\forall ({\textbf{z}},\lambda ) \in {\mathcal {K}} \times {\mathbb {R}}_+, \lambda {\textbf{z}} \in {\mathcal {K}}$. The dual cone of ${\mathcal {K}} \subseteq {\mathbb {R}}^k$ is

$$\begin{aligned} {\mathcal {K}}^* = \left\{ {\textbf{u}} \in {\mathbb {R}}^k: {\textbf{u}}^T{\textbf{z}} \ge {\textbf{0}}, \forall {\textbf{z}} \in {\mathcal {K}} \right\} , \end{aligned}$$

(A2)

and it is self-dual if ${\mathcal {K}} = {\mathcal {K}}^*$. The cone is pointed if ${\mathcal {K}} \cap (-{\mathcal {K}}) = \{ {\textbf{0}} \}$. A cone is proper if it is closed, convex, pointed, and with a non-empty interior. If ${\mathcal {K}}$ is proper, then its dual ${\mathcal {K}}^*$ is proper too. ${\mathcal {K}}$ induces a partial order on ${\mathbb {R}}^k$:

$$\begin{aligned} {\textbf{x}} \succcurlyeq _{{\mathcal {K}}} {\textbf{y}} \iff {\textbf{x}} - {\textbf{y}} \in {\mathcal {K}}, \end{aligned}$$

(A3)

which allows us to define a conic inequality as

$$\begin{aligned} {\textbf{A}}{\textbf{x}} \succcurlyeq _{{\mathcal {K}}} {\textbf{b}}, \end{aligned}$$

(A4)

where ${\textbf{A}} \in {\mathbb {R}}^{m \times k}$, ${\textbf{b}} \in {\mathbb {R}}^m$, and ${\mathcal {K}}$ a cone.

When using a cone that represents the cartesian product of others, i.e., ${\mathcal {K}} = {\mathcal {K}}_{n_1} \times \cdots \times {\mathcal {K}}_{n_r}$ with each cone ${\mathcal {K}}_{n_i} \subseteq {\mathbb {R}}^{n_i}$, its corresponding vectors and matrices are partitioned conformally, i.e.,

$$\begin{aligned} \begin{aligned} {\textbf{x}}&= (\mathbf {x_1}; \ldots ; \mathbf {x_r}) \quad \text {where}\quad \mathbf {x_i} \in {\mathbb {R}}^{n_i}, \\ {\textbf{y}}&= (\mathbf {y_1}; \ldots ; \mathbf {y_r}) \quad \text {where}\quad \mathbf {y_i} \in {\mathbb {R}}^{n_i}, \\ {\textbf{c}}&= (\mathbf {c_1}; \ldots ; \mathbf {c_r}) \quad \text {where}\quad \mathbf {c_i} \in {\mathbb {R}}^{n_i}, \\ {\textbf{A}}&= ({\textbf{A}}_1; \ldots ; {\textbf{A}}_r) \quad \text {where}\quad {\textbf{A}} \in {\mathbb {R}}^{m \times n_i}. \end{aligned} \end{aligned}$$

(A5)

Furthermore, if each cone ${\mathcal {K}}_{n_i} \subseteq {\mathbb {R}}^{n_i}$ is proper, then ${\mathcal {K}}$ is proper too.

A Conic Programming (CP) problem is then defined as:

$$\begin{aligned} \begin{aligned} \min _{{\textbf{x}}}&\quad {\textbf{c}}^\top {\textbf{x}}\\ \text {s.t. }&\quad {\textbf{A}}{\textbf{x}} = {\textbf{b}},\\&\quad {\textbf{x}} \in {\mathcal {K}} \subseteq {\mathbb {R}}^k. \end{aligned} \end{aligned}$$

(CP)

Examples of proper cones are:

• The nonnegative orthant

  $$\begin{aligned} {\mathbb {R}}_+^k = \left\{ {\textbf{z}} \in {\mathbb {R}}^k: {\textbf{z}} \ge {\textbf{0}} \right\} . \end{aligned}$$

  (A6)

• The positive semi-definite cone

  $$\begin{aligned} {\mathbb {S}}_+^k = \left\{ Z \in {\mathbb {R}}^{k \times k}: Z = Z^T, \lambda _{min}(Z) \ge 0 \right\} , \end{aligned}$$

  (A7)

  where $\lambda _{min}(Z)$ denotes the smallest eigenvalue of Z.

• The second-order cone, Euclidean norm cone, or Lorentz cone

  $$\begin{aligned} {\mathcal {Q}}^k = \left\{ {\textbf{z}} \in {\mathbb {R}}^{k}: z_1 \ge \sqrt{\sum _{i=2}^k z_i^2} \right\} . \end{aligned}$$

  (A8)

• The exponential cone [82]

  $$\begin{aligned} \begin{aligned} {\mathcal {K}}_{exp}&= \text {cl} \left\{ (z_1,z_2,z_3) \in {\mathbb {R}}^3: z_1 \ge z_2 e^{z_3 / z_2}, z_1 \ge 0, z_2> 0 \right\} \\&= \left\{ (z_1,z_2,z_3) \in {\mathbb {R}}^3: z_1 \ge z_2 e^{z_3 / z_2}, z_1 \ge 0, z_2 > 0 \right\} \\&\quad \bigcup {\mathbb {R}}_+ \times \{ {\textbf{0}} \} \times (-{\mathbb {R}}_+) \\&= \left\{ (z_1,z_2,z_3) \in {\mathbb {R}}^3: z_1 \ge z_2 e^{z_3 / z_2}, z_2 \ge 0 \right\} . \end{aligned} \end{aligned}$$

  (A9)

Of these cones, the only one not being self-dual or symmetric is the exponential cone.

Other cones that are useful in practice are

• The rotated second-order cone or Euclidean norm-squared cone

  $$\begin{aligned} {\mathcal {Q}}_r^k = \left\{ {\textbf{z}} \in {\mathbb {R}}^{k}: 2z_1z_2 \ge \sqrt{\sum _{i=3}^k z_i^2}, z_1, z_2 \ge 0 \right\} , \end{aligned}$$

  (A10)

  This cone can be written as a rotation of the second-order cone, i.e., ${\textbf{z}} \in {\mathcal {Q}}^k \iff R_k{\textbf{z}} \in {\mathcal {Q}}_r^k$ with $R_k:= \begin{bmatrix} \sqrt{2}/2 &{} \sqrt{2}/2 &{} 0 \\ \sqrt{2}/2 &{} \sqrt{2}/2 &{} 0 \\ 0 &{} 0 &{} I_{k-2} \end{bmatrix}$ [15].

• The power cone, with $l < k, \sum _{i \in \llbracket l \rrbracket } \alpha _i = 1, \alpha _{i \in \llbracket l \rrbracket } > 0$,

  $$\begin{aligned} {\mathcal {P}}_k^{\alpha _1, \ldots , \alpha _l} = \left\{ {\textbf{z}} \in {\mathbb {R}}^k: \prod _{i = 1}^l z_i^{\alpha _i} \ge \sqrt{\sum _{i = l+1}^k z_i^2}, \quad z_i \ge 0 \quad i \in \llbracket l \rrbracket \right\} . \end{aligned}$$

  (A11)

  This cone can be decomposed using a second-order cone and $l-1$ three-dimensional power cones

  $$\begin{aligned} {\mathcal {P}}_3^{\alpha } = \left\{ (z_1,z_2,z_3) \in {\mathbb {R}}^3: z_1^\alpha z_2^{1-\alpha } \ge |z_3 |, \quad z_1, z_2 \ge 0 \right\} , \end{aligned}$$

  (A12)

  through $l - 1$ additional variables $(u,v_{1},\ldots ,v_{l-2})$,

  $$\begin{aligned} {\textbf{z}} \in {\mathcal {P}}_k^{\alpha _1, \ldots , \alpha _l} \iff {\left\{ \begin{array}{ll} (u,z_{l+1},\ldots ,z_{k}) \in {\mathcal {Q}}^{k-l+1},\\ (z_1,v_1,u) \in {\mathcal {P}}_3^{\alpha _1}, \\ (z_i,v_i,v_{i-1}) \in {\mathcal {P}}_3^{{\bar{\alpha }}_i}, \quad i=2,\ldots ,l-1, \\ (z_{l-1},z_l,v_{l-2}) \in {\mathcal {P}}_3^{{\bar{\alpha }}_{l-1}}, \end{array}\right. } \end{aligned}$$

  (A13)

  where ${\bar{\alpha }}_i=\alpha _i/(\alpha _i+\cdots +\alpha _l)$ for $i=2,\ldots ,l-1$ [15]. ${\mathcal {P}}_3^{\alpha }$ can be represented using linear and exponential cone constraints, i.e., $\lim _{\alpha \rightarrow 0} (z_1,z_2,z_2+\alpha z_3) \in {\mathcal {P}}_3^{\alpha } = (z_1,z_2,z_3) \in {\mathcal {K}}_{exp}$

Most, if not all, applications-related convex optimization problems can be represented by conic extended formulations using these standard cones [15], i.e., in problem CP, the cone ${\mathcal {K}}$ is a product ${\mathcal {K}}_1 \times \cdots \times {\mathcal {K}}_r$, where each ${\mathcal {K}}_i$ is one of the recognized cones mentioned above. Equivalent conic formulations for more exotic convex sets using unique cones can be formulated with potential advantages for improved solution performance [17, 83].

As mentioned in the introduction, an alternative to a convex optimization problem’s algebraic description as in problem MINLP is the following Mixed-Integer Conic Programming (MICP) problem:

$$\begin{aligned} \begin{aligned} \min _{\textbf{z,y}}&\quad {\textbf{c}}^T{\textbf{z}}\\ \text {s.t. }&\quad {\textbf{A}}{\textbf{z}} +{\textbf{B}}{\textbf{y}} = {\textbf{b}}, \\&\quad {\textbf{y}}^l \le {\textbf{y}} \le {\textbf{y}}^u,\\&\quad {\textbf{z}} \in {\mathcal {K}} \subseteq {\mathbb {R}}^k,\ {\textbf{y}} \in {\mathbb {Z}}^{n_y}, \end{aligned} \end{aligned}$$

(MICP)

where ${\mathcal {K}}$ is a closed convex cone.

Without loss of generality, integer variables need not be restricted to cones, given that corresponding continuous variables can be introduced via equality constraints. Notice that for an arbitrary convex function $f: {\mathbb {R}}^k \rightarrow {\mathbb {R}} \cup \{ \infty \}$, one can define a closed convex cone using its recession,

$$\begin{aligned} {\mathcal {K}}_f = \text {cl} \{({\textbf{z}},\lambda ,t): \lambda f({\textbf{z}} / \lambda ) = \tilde{f}({\textbf{z}},\lambda ) \le t, \lambda > 0 \}, \end{aligned}$$

(A14)

where the function $\tilde{f}({\textbf{z}},\lambda )$ is the perspective function of function $f({\textbf{z}})$, and whose algebraic representation is a central piece of this work. Closed convex cones can also be defined as the recession of convex sets. On the other hand, a conic constraint can be equivalent to a convex inequality,

$$\begin{aligned} {\textbf{A}}{\textbf{x}} \succcurlyeq _{{\mathcal {K}}} {\textbf{b}} \iff {\textbf{g}}({\textbf{x}}) \le {\textbf{0}}. \end{aligned}$$

(A15)

Although certain cones can be non-smooth, e.g., SOC cones, these can be reformulated using appropriately chosen smooth convex functions ${\textbf{g}}({\textbf{x}})$ [35, 84].

We can therefore reformulate problem MINLP in the following parsimonious manner [22]:

$$\begin{aligned} \begin{aligned} \min _{\begin{array}{c} \textbf{x,y},s_{\llbracket J \rrbracket } \\ {\textbf{x}}_f,{\textbf{y}}_f,t_f, \\ {\textbf{x}}_{\llbracket J \rrbracket },{\textbf{y}}_{\llbracket J \rrbracket } \end{array}}&\quad t_f\\ \text {s.t. }&\quad {\textbf{x}} = \mathbf {x_f},\quad {\textbf{y}} = \mathbf {y_f},\\&\quad ((\mathbf {x_f};\mathbf {y_f}),1,t_f) \in {\mathcal {K}}_f,\\&\quad {\textbf{x}} = \mathbf {x_j}, {\textbf{y}} = \mathbf {y_j}, \\&\quad \left( (\mathbf {x_j};\mathbf {y_j}),1,s_j \right) \in {\mathcal {K}}_{g_j}, \quad s_j \in {\mathbb {R}}_+,\quad j \in \llbracket J \rrbracket , \\&\quad {\textbf{y}}^l \le {\textbf{y}} \le {\textbf{y}}^u,\\&\quad {\textbf{x}}\in {\mathbb {R}}_+^{n_x},\quad {\textbf{y}} \in {\mathbb {Z}}^{n_y},\\ \end{aligned} \end{aligned}$$

(MINLP-Cone)

where copies of the original variables ${\textbf{x}}$ and ${\textbf{y}}$ are introduced for the objective function and each constraint, $\mathbf {x_f},\mathbf {y_f},\mathbf {x_j},\mathbf {y_j}, j \in \llbracket J \rrbracket$, such that each belongs to the recession cone of each constraint defined as in (A14). Each conic set requires the introduction of an epigraph variable t and a recession variable $\lambda$. The epigraph variable from the objective function, $t_f$, is used in the new objective, and the ones corresponding to the constraints are set as nonnegative slack variables $s_j$. The recession variables $\lambda$ in (A14) are fixed to one in all cases.

Notice that problem MINLP-Cone is in MICP form with ${\mathcal {K}} = {\mathbb {R}}_+^{n_x + J} \times {\mathcal {K}}_f \times {\mathcal {K}}_{g_1} \times \cdots \times {\mathcal {K}}_{g_J}$. As mentioned above, the case when ${\mathcal {K}} = {\mathcal {K}}_1 \times \cdots \times {\mathcal {K}}_r$ where each ${\mathcal {K}}_i$ is a recognized cone is more useful from practical purposes. Lubin et al. [22] showed that all the convex MINLP instances at the benchmark library MINLPLib [65] could be represented with nonnegative, second-order, and exponential cones.

1.2 Perspective function

For a convex function $h({\textbf{x}}): {\mathbb {R}}^n \rightarrow {\mathbb {R}} \cup \{ \infty \}$ its perspective function $\tilde{h}({\textbf{x}},\lambda ): {\mathbb {R}}^{n+1} \rightarrow {\mathbb {R}} \cup \{ \infty \}$ is defined as

$$\begin{aligned} \tilde{h}({\textbf{x}},\lambda ) = {\left\{ \begin{array}{ll} \lambda h({\textbf{x}}/\lambda ) &{} \text {if}\quad \lambda > 0 \\ \infty &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

(A16)

The perspective of a convex function is convex but not closed. Hence, consider the closure of the perspective function $(\text {cl}~\tilde{h})({\textbf{x}},\lambda )$ defined as

$$\begin{aligned} \left( \text {cl }\tilde{h} \right) ({\textbf{x}},\lambda ) = {\left\{ \begin{array}{ll} \lambda h({\textbf{x}}/\lambda ) &{} \text {if}\quad \lambda > 0 \\ h'_\infty ({\textbf{x}}) &{} \text {if}\quad \lambda = 0 \\ \infty &{} \text {otherwise} \end{array}\right. }, \end{aligned}$$

(A17)

where $h'_\infty ({\textbf{x}})$ is the recession function of function $h({\textbf{x}})$ [85, Section B Proposition 2.2.2], and which in general does not have a closed-form.

The closure of the perspective function of a convex function is relevant for convex MINLP on two ends. On the one hand, it appears when describing the closure of the convex hull of disjunctive sets. On the other hand, as seen above, it can be used to define closed convex cones ${\mathcal {K}}$ that determine the feasible region of conic programs. Relying on the amenable properties of convex cones, conic programs can be addressed with specialized algorithms, allowing for more efficient solution methods.

The closure of the perspective function presents a challenge when implementing it for nonlinear optimization models, given that it is not defined at $\lambda =0$. As seen below, modeling this function becomes necessary when writing the convex hull of the union of convex sets. Several authors have addressed this difficulty in the literature through $\varepsilon$-approximations. The first proposal was made by Lee and Grossmann [45], where

$$\begin{aligned} \left( \text {cl } \tilde{h} \right) ({\textbf{x}},\lambda ) \approx (\lambda + \varepsilon ) h \left( \frac{{\textbf{x}}}{\lambda + \varepsilon } \right) . \end{aligned}$$

(A18)

This approximation is exact when $\varepsilon \rightarrow 0$. However, it requires values for $\varepsilon$, which are small enough to become numerically challenging when implemented in a solution algorithm.

Furman et al. [43] propose another approximation for the perspective function such that

$$\begin{aligned} \left( \text {cl } \tilde{h} \right) ({\textbf{x}},\lambda ) \approx ((1-\varepsilon )\lambda + \varepsilon )h \left( \frac{{\textbf{x}}}{(1-\varepsilon )\lambda + \varepsilon } \right) - \varepsilon h(0) (1-\lambda ), \end{aligned}$$

(A19)

which is exact for values of $\lambda = 0$ and $\lambda = 1$, is convex for $h({\textbf{x}})$ convex, and is exact when $\varepsilon \rightarrow 0$ as long as $h({\textbf{0}})$ is defined. Using this approximation in the set describing the system of equations of the closed convex hull of a disjunctive set also has properties beneficial for mathematical programming.

This approximation is used in software implementations when reformulating a disjunctive set using its hull relaxation [40, 78]. Notice that even with its desirable properties, the approximation introduces some error for values $\varepsilon > 0$; hence, it is desirable to circumvent its usage. As shown in [42] and the Sect. 4, using a conic constraint to model the perspective function allows for a more efficient solution of convex MINLP problems.

1.3 Disjunctive programming

Optimization over disjunctive sets is denoted as Disjunctive Programming [34, 60]. The system of inequalities gives a disjunctive set joined by logical operators of conjunction ($\wedge$, “and”) and disjunction ($\vee$, “or”). These sets are non-convex and usually represent the union of convex sets. The main reference on Disjunctive Programming is the book by Balas [34].

Consider the following disjunctive set

$$\begin{aligned} {\mathcal {C}} = \left\{ {\textbf{x}} \in {\mathbb {R}}^n: {\textbf{x}} \in \bigvee _{i \in I} {\mathcal {C}}_i \right\} = \bigcup _{i \in I} \left\{ {\textbf{x}} \in {\mathbb {R}}^n: {\textbf{x}} \in {\mathcal {C}}_i\right\} , \end{aligned}$$

(A20)

where $|I |$ is finite. Each set defined as ${\mathcal {C}}_i:= \{ {\textbf{x}} \in {\mathbb {R}}^n \mid {\textbf{h}}_i({\textbf{x}}) \le {\textbf{0}} \}$ is a convex, bounded, and nonempty set defined by a vector-valued function ${\textbf{h}}_i: {\mathbb {R}}^n \rightarrow \left( {\mathbb {R}} \cup \{ \infty \} \right) ^{J_i}$. Notice that is it sufficient for ${\mathcal {C}}_i$ to be convex that each component of ${\textbf{h}}_i$, $h_{i \llbracket J_i \rrbracket }$, is a proper closed convex function, although it is not a necessary condition. A proper closed convex function is one whose epigraph is a nonempty closed convex set [86].

Ceria and Soares [55] characterize the closure of the convex hull of ${\mathcal {C}}$, $\text {cl conv}({\mathcal {C}})$, with the following result.

Theorem 2

[55] Let ${\mathcal {C}}_i = \{ {\textbf{x}} \in {\mathbb {R}}^n \mid {\textbf{h}}_i({\textbf{x}}) \le {\textbf{0}} \} \ne \emptyset$, assume that each component of ${\textbf{h}}_i$, $h_{i\llbracket J_i \rrbracket }$, is a proper closed convex function, and let

$$\begin{aligned} {\mathcal {H}} = \left\{ \begin{aligned}&{\textbf{x}} = \sum _{i \in I}{\textbf{v}}_{i}, \\&\sum _{i \in I}\lambda _{i} = 1, \\&\left( \text {cl } \tilde{{\textbf{h}}}_i \right) ({\textbf{v}}_{i},\lambda _{i})\le 0,&i \in I, \\&{\textbf{v}}_{i} \in {\mathbb {R}}^n,&i \in I, \\&\lambda _{i} \in {\mathbb {R}}_+,&i \in I \end{aligned} \right\} . \end{aligned}$$

(A21)

Then $\text {cl conv}(\bigcup _{i \in I} {\mathcal {C}}_i) = \text {proj}_{{\textbf{x}}}({\mathcal {H}})$.

Proof

See [55, Theorem 1] and [38, Theorem 1]. $\square$

Theorem 2 provides a description of $\text {cl conv}({\mathcal {C}})$ in a higher dimensional space, an extended formulation. This Theorem generalizes the result by [34, 60, 87, 88] where all the convex sets ${\mathcal {C}}_i$ are polyhedral. Even though the extended formulations induce growth in the size of the optimization problem, some of them have shown to be amenable for MINLP solution algorithms [22, 59, 89, 90].

A similar formulation was derived by Stubbs and Mehrotra [46] in the context of a Branch-and-cut method for Mixed-binary convex programs. These authors notice that the extended formulation might not be computationally practical; hence, they derive linear inequalities or cuts from this formulation to be later integrated into the solution procedure. Similar ideas have been explored in the literature [47]. In particular cases, the dimension of the extended formulation can be reduced to the original size of the problem, e.g., when there are only two terms in the disjunction, i.e., $|I |=2$, and one of the convex sets ${\mathcal {C}}_i$ is a point [42]. A description in the original space of variables has also been given for the case when one set ${\mathcal {C}}_1$ is a box and the constraints defining the other ${\mathcal {C}}_2$ is determined by the same bounds as the box and nonlinear constraints being isotone [44]. This has been extended even further by Bonami et al. [38] with complementary disjunctions. In other words, the activation of one disjunction implies that the other one is deactivated, in the case that the functions that define each set ${\textbf{h}}_{\{1,2\}}$ are isotone and share the same indices on which they are non-decreasing. The last two cases present the formulation in the original space of variables by paying a prize of exponentially many constraints required to represent $\text {cl conv}({\mathcal {C}})$.

In the case that ${\mathcal {C}}_i$ is compact, its recession cone is the origin, i.e., ${\mathcal {C}}_{i\infty }= \{ {\textbf{x}} \in {\mathbb {R}}^n \mid {\textbf{h}}'_{i\infty }({\textbf{x}}) \le {\textbf{0}} \} = \{ {\textbf{0}} \}$ [85, Section A, Proposition 2.2.3]. This fact, together with (A17) and Theorem 2, forces that for a compact ${\mathcal {C}}_i$, a value of $\lambda _i=0$ implies ${\textbf{v}}_i=0$. This fact has been used to propose mixed-integer programming formulations for expressing the disjunctive choice between convex sets by setting the interpolation variables to be binary $\lambda _i \in \{0,1\}, i \in I$ [45, 91], i.e.,

$$\begin{aligned} {\mathcal {H}}_{\{0,1\}} = \left\{ \begin{aligned}&{\textbf{x}} = \sum _{i \in I}{\textbf{v}}_{i}, \\&\sum _{i \in I}\lambda _{i} = 1, \\&\left( \text {cl } \tilde{{\textbf{h}}}_i \right) ({\textbf{v}}_{i},\lambda _{i})\le 0, \quad I \in I, \\&{\textbf{v}}_{i} \in {\mathbb {R}}^n, \quad i \in I, \\&\lambda _{i} \in \{ 0, 1 \}, \quad i \in I \end{aligned} \right\} . \end{aligned}$$

(A22)

An interesting observation is that using the approximation of the closure of the perspective function from Furman et al. [43], for any value of $\varepsilon \in (0,1)$, $\text {proj}_{\textbf{x}}({\mathcal {H}}_{\{0,1\}})={\mathcal {C}}$ when ${\textbf{h}}_i({\textbf{0}})$ is defined $\forall i \in I$ and

$$\begin{aligned} \left\{ {\textbf{x}} \in {\mathbb {R}}^n: {\textbf{h}}_i({\textbf{x}}) - {\textbf{h}}_i({\textbf{0}}) \le {\textbf{0}} \right\} = \{ {\textbf{0}} \},\quad \forall i \in I \end{aligned}$$

(A23)

see [43, Proposition 1].

The condition on (A23) is required to ensure that if $\lambda _i=1$, then ${\textbf{v}}_{i'}=0, \forall i' \in I {\setminus } \{ i \}$. This condition is not valid in general for a disjunctive set ${\mathcal {C}}$, but it is sufficient to have a bounded range on ${\textbf{x}} \in {\mathcal {C}}_i, i \in I$. Moreover, when these conditions are satisfied, ${\mathcal {C}} \subseteq \text {proj}_{\textbf{x}}({\mathcal {H}})$ using the approximation in (A19) for $\varepsilon \in (0,1)$, with $\text {cl conv} ({\mathcal {C}}) = \text {proj}_{\textbf{x}}({\mathcal {H}})$ in the limit when $\varepsilon \rightarrow 0$ [43, Proposition 3].

The problem formulation HR is derived by replacing each disjunction with set ${\mathcal {H}}_{\{0,1\}}$ (A22). Notice that to guarantee the validity of the formulation, the condition on (A23) is enforced implicitly by having the bounds over ${\textbf{x}}$ included in each disjunct, leading to constraint ${\textbf{x}}^{l}y_{ik} \le {\textbf{v}}_{ik} \le {\textbf{x}}^{u}y_{ik}$.

Appendix B: Detailed computational results

Table 2 Results for quadratic GDPs using different mixed-integer reformulations and solvers

Table 3 Results for exponential GDPs using the Big-M reformulation and different solvers

Table 4 Results for Exponential GDPs using the Hull reformulation and different solvers