Dantzig–Wolfe reformulations for binary quadratic problems

Abstract

The purpose of this paper is to provide strong reformulations for binary quadratic problems. We propose a first methodological analysis on a family of reformulations combining Dantzig–Wolfe and Quadratic Convex optimization principles. We show that a few reformulations of our family yield continuous relaxations that are strong in terms of dual bounds and computationally efficient to optimize. As a representative case study, we apply them to a cardinality constrained quadratic knapsack problem, providing extensive experimental insights. We report and analyze in depth a particular reformulation providing continuous relaxations whose solutions turn out to be integer optima in all our tests.

Appendices

A Details on the QCR method

1.1 A.1 Semidefinite model providing $\delta ^*$ and $\rho ^*$

Let us consider ($\hbox {BQ}_{\delta ,\rho }$) (the convexified formulation of a Binary Quadratic Problem, introduced in Sect. 4):

$$\begin{aligned} (\text {BQ}_{\delta ,\rho })~~\min ~ f_{\delta ,\rho }(x)=&\, x^\top Q x + L^\top x + \sum _{j \in J} \delta _j (x_j^2 - x_j) \\&+ \rho (A_{=} x - b_{=})^2 \nonumber \\ \text {s.t.}~~~&G x \le g \\&H x \le h \\&x \in \{0,1\}^n\;. \end{aligned}$$

The SDP used to obtain the optimal parameters $\delta ^*,\rho ^*$ (see [8, 36]) is the following:

$$\begin{aligned} \text {(SDP}_{\text {BQ}_{\delta ,\rho }}\text {)}~~\text{ max }~~&\sum _{i=1}^{n}\sum _{j=1}^{n} Q_{ij}X_{ij}&\\ \text{ s.t. }~~&X_{ii} = x_i ~~~ i=1,\dots ,n&[\delta _i] \\&\langle A_= A_=^\top , X \rangle -2b_=^\top A_= x = -b_=^2&[\rho ]\\&G x \le g&\\&H x \le h&\\&\left( \begin{array}{cc} 1&{}x^t\\ x&{}X \end{array} \right) \succeq 0&\\&x \in \mathbb R^n, ~ X \in \mathbb R^{n \times n} \;.&\end{aligned}$$

Namely, the optimal values $\delta ^*$ and $\rho ^*$ of problem ($\hbox {BQ}_{\delta ,\rho }$) are simply given by the optimal values of the dual variables of ($\text {SDP}_{\text {BQ}_{\delta ,\rho }}$).

1.2 A.2 Semidefinite model providing $\delta ^*$, $\rho ^*$ and $\varGamma ^*$

Let us consider ($\hbox {BQ}_{\delta ,\rho ,\varGamma }$):

$$\begin{aligned} (\text {BQ}_{\delta ,\rho ,\varGamma })~~\min ~ f_{\delta , \rho , \varGamma }(x,z)=&x^\top Q x + L^\top x + \sum _{j \in J} \delta _j (x_j^2 - x_j) + \rho (A_{=} x - b_{=})^2 + \\&+ \sum _{i \in J} \sum _{j \in J} \varGamma _{ij} (z_{ij} - x_i x_j) \\ \text {s.t.}~~~&G x \le g \\&H x \le h \\&z_{ij} \le x_i,\ z_{ij} \le x_j ~~~~~~~~~~~ i, j = 1, \ldots , n\\&z_{ij} \ge 0, \ z_{ij} \ge x_i + x_j - 1 ~~ i, j = 1, \ldots , n\\&x \in \{0,1\}^n\;. \end{aligned}$$

The SDP used to obtain the optimal parameters $\delta ^*,\rho ^*,\varGamma ^*$ (see [7]) is the following:

$$\begin{aligned} \text {(SDP}_{\text {BQ}_{\delta ,\rho ,\varGamma }}\text {)}~~\text{ max }~~&\sum _{i=1}^{n}\sum _{j=1}^{n} Q_{ij}X_{ij}&\\ \text{ s.t. }~~&X_{ii} = x_i ~~~ i=1,\dots ,n&[\delta _i] \\&\langle A_= A_=^\top , X \rangle -2b_=^\top A_= x = -b_=^2&[\rho ]\\&G x \le g&\\&H x \le h&\\&X_{ij} \le x_i,\ X_{ij} \le x_j ~~~~~~~~~~~~ i,j=1, \ldots ,n&[\varGamma ^+_{ij}]\\&X_{ij} \ge x_i + x_j - 1,\ X_{ij} \ge 0 ~~~ i,j=1, \ldots ,n&[\varGamma ^-_{ij}]\\&\left( \begin{array}{cc} 1&{}x^t\\ x&{}X \end{array} \right) \succeq 0&\\&x \in \mathbb R^n, ~ X \in \mathbb R^{n \times n} \;.&\end{aligned}$$

Namely, the optimal values $\delta ^*$, $\rho ^*$ and $\varGamma ^*$ of problem ($\hbox {BQ}_{\delta ,\rho ,\varGamma }$) are simply given by the optimal values of the dual variables of ($\text {SDP}_{\text {BQ}_{\delta ,\rho ,\varGamma }}$), with ${\varGamma }^* = {\varGamma ^{+}}^* + {\varGamma ^{-}}^*$.

B Dual of a nonlinear model reformulated with DWR

Let us consider the following formulation

$$\begin{aligned} \text {({F-NLM})}~~\min ~~&f(x) \\ \text {s.t.}~~&g_i(x) \le 0&i \in I&~[\mu _i]~~ \\ ~~&x- \sum _{p \in P} x_p y^p = 0&~[\pi ]\\ ~~&\sum _{p \in P} y^p - 1 = 0&~[\pi _0]\\ ~~&y^p \ge 0&\forall p \in \mathcal {P}&~[\nu _p] \end{aligned}$$

(we recall that the constraints $x \in [0,1]^n$ are included in the constraints $g_i(x) \le 0$).

The formulation of its Wolfe dual is

$$\begin{aligned} \max ~~&L(x, y, \mu , \pi , \pi _0, \nu )\\ s.t.~~&\nabla _x L(x, y, \mu , \pi , \pi _0, \nu ) =0\\&\nabla _yL(x, y, \mu , \pi , \pi _0, \nu ) =0\\&y^p \ge 0&\forall p \in \mathcal {P}\\&\mu _i \ge 0&i \in I\\&\nu _p \le 0&\forall p \in \mathcal {P} \end{aligned}$$

with

$$\begin{aligned}&L(x, y, \mu , \pi , \pi _0, \nu ) = f(x) + \sum _{i \in I} \mu _i g_i(x) + \pi ^\top \left( x- \sum _{p \in \mathcal {P}} x_p y_p\right) \\&+ \pi _0 \left( \sum _{p \in \mathcal {P}} y_p - 1\right) + \sum _{p \in \mathcal {P}} \nu _p y_p \end{aligned}$$

which leads to

$$\begin{aligned} \max ~~&L(x, y, \mu , \pi , \pi _0, \nu )&\\ \text {s.t.}~~&\nabla _x f(x) + \sum _{i \in I} \mu _i \nabla _x g_i(x) + \pi = 0&\\&- \pi ^\top x_p + \pi _0 +\nu _p =0&\forall p \in \mathcal {P}\\&y^p \ge 0&\forall p \in \mathcal {P}\\&\mu _i \ge 0&i \in I\\&\nu _p \le 0&\forall p \in \mathcal {P} \end{aligned}$$

If we substituting the definition of $\nu _p= \pi ^\top x_p - \pi _0$ in $L(x, y, \mu , \pi , \pi _0, \nu )$ we have:

$$\begin{aligned}&L(x, y, \mu , \pi , \pi _0, \nu ) = f(x) + \sum _{i \in I} \mu _i g_i(x) + \pi ^\top \left( x- \sum _{p \in \mathcal {P}} x_p y_p\right) \\&\qquad + \pi _0 \left( \sum _{p \in \mathcal {P}} y_p - 1\right) + \sum _{p \in \mathcal {P}} \nu _p y_p\\&\quad = f(x) + \sum _{i \in I} \mu _i g_i(x) + \pi ^\top \left( x- \sum _{p \in \mathcal {P}} x_p y_p\right) + \pi _0 \left( \sum _{p \in \mathcal {P}} y_p - 1\right) \\&\qquad + \sum _{p \in \mathcal {P}} (\pi ^\top x_p - \pi _0) y_p\\&\quad = f(x) + \sum _{i \in I} \mu _i g_i(x) + \pi ^\top \left( x- \sum _{p \in \mathcal {P}} x_p y_p\right) - \pi _0 + \sum _{p \in \mathcal {P}} \left( \pi ^\top x_p \right) y_p\\&\quad = f(x) + \sum _{i \in I} \mu _i g_i(x) + \pi ^\top x - \pi _0 \end{aligned}$$

leading to the following dual:

$$\begin{aligned} \max ~~&f(x) + \sum _{i \in I} \mu _i g_i(x) + \pi ^\top x - \pi _0 \\ \text {s.t.}~~&\nabla _x f(x) + \sum _{i \in I} \mu _i \nabla _x g_i(x) + \pi = 0\\&- \pi ^\top x_p + \pi _0 \ge 0&\forall p \in \mathcal {P}\\&y^p \ge 0&\forall p \in \mathcal {P}\\&\mu _i \ge 0&i \in I\;. \end{aligned}$$

C Reinterpretation of the new reformulations for (BQP) with Lagrangian duality

Let us consider the following reformulation of (BQP):

$$\begin{aligned} (\text {Q}')~~\min ~~&w^\top S w + L^\top w&\\ \text {s.t.}~~~&x_i - w_i = 0&i= 1, \ldots , n&~~~[\phi _i]\\ ~~~&G w \le g&~~~[\mu ]\\&H x \le h&~~~[\psi ]\\&x^2_i - x_i = 0&i = 1, \ldots , n&~~~[\delta _i]\\&(A_= x -\delta b)^2 = 0&~~~[\rho ]\\&z_{ij} \le x_i,\ z_{ij} \le x_j&i, j = 1, \ldots , n&~~~[\varXi _{ij}]\\&z_{ij} \ge 0, \ z_{ij} \ge x_i + x_j - 1&i, j = 1, \ldots , n&~~~[\varUpsilon _{ij}]\\&z_{ij} - x_i x_j = 0&i, j = 1, \ldots , n&~~~[\varGamma _{ij}] \end{aligned}$$

where for each constraint we indicate in squared brakets the corresponding Lagrangian multiplier.

Sections 3 and 4 show that the continuous relaxations of the reformulations showed in Table 1 can be viewed as a specific Lagrangian Dual of ($\hbox {Q}'$) where some of the constraints are relaxed in the Lagrangian Function, some are kept in the Lagrangian subproblem and other are dropped. For simplicitly, each constraint is represented by its Lagrangian multiplier. For example, ($\hbox {Q}_{\delta ,\rho }$-QM) is obtained by relaxing constraints $[\phi ]$,$[\mu ]$,$[\delta ]$,$[\rho ]$, keeping constraints $[\psi ]$,$[\delta ]$ and dropping constraints $[\varXi ]$,$[\varUpsilon ]$,$[\varGamma ]$:

$$\begin{aligned} \text {(Q}_{\delta ,\rho }\text {-QM)}~~~\max _{\phi ,\mu ,\delta ,\rho } \theta (\phi ,\mu ,\delta ,\rho ) = \max _{\phi ,\mu ,\delta ,\rho } \underset{\begin{array}{c} x,w \in \mathbb {R}^n:~ H x \le h \\ x^2_i - x_i = 0,~ i = 1, \ldots , n \end{array}}{\min } L(x,w,\phi ,\mu ,\delta ,\rho ) \end{aligned}$$

with $L(x,w,\phi ,\mu ,\delta ,\rho ) = w^\top Q w + L^\top w + \phi ^\top (x- w) + \mu ^\top (G w - g) + \delta ^\top (x^2 - x) + \rho (A_= x -b_=)^2$.

D Comparing (K$_{\delta ^*,\rho ^*}$-QP-kn) with CPLEX

In Table 5 we compare the time needed for computing the root node of (K$_{\delta ^*,\rho ^*}$-QP-kn) (that in each test corresponds to the time needed to solve the problem to global integer optimality) with the overall time needed by CPLEX to solve the original formulation with general purpose techniques (BK) and ($\hbox {BK}_{\delta ^*,\rho ^*}$). For (BK) the number of instances yielding timeout is also reported: the average computing time does not include them.

Table 5 DD instances, DWR versus CPLEX

The results of Table 5 provide the following observation:

Experimental Observation 13

With (K$_{\delta ^*,\rho ^*}$-QP-kn) we are able to close the whole integrality gap faster than the time needed by CPLEX to solve to optimality the same instance. The time needed by (K$_{\delta ^*,\rho ^*}$-QP-kn) is similar to that of ($\hbox {BK}_{\delta ^*,\rho ^*}$), that is CPLEX after applying an ad-hoc convexification of the objective function.

We remark that none of these techniques uses problem specific algorithms. In this regard, (K$_{\delta ^*,\rho ^*}$-QP-kn) has some additional potential, as the column generation scheme can strongly benefit from the use of ad-hoc pricing algorithms.