An exact completely positive programming formulation for the discrete ordered median problem: an extended version

Abstract

This paper presents a first continuous, linear, conic formulation for the discrete ordered median problem (DOMP). Starting from a binary, quadratic formulation in the original space of location and allocation variables that are common in location analysis (L.A.), we prove that there exists a transformation of that formulation, using the same space of variables, that allows us to cast DOMP as a continuous, linear programming problem in the space of completely positive matrices. This is the first positive result that states equivalence between the family of continuous, convex problems and some hard combinatorial problems in L.A. The result is of theoretical interest because it allows us to share the tools from continuous optimization to shed new light into the difficult combinatorial structure of the class of ordered median problems that combines elements of the p-median, quadratic assignment and permutation polytopes.

Appendix: An explicit formulation of CP-DOMP

First of all, one can check that

$$\begin{aligned} \Phi [a_{(4)}]_1 =&\Big [{\mathop {\overbrace{\sum _{\ell =1}^n q_{111\ell },\ldots , \sum _{\ell =1}^n q_{nn1\ell }},}\limits ^{(PP)}} \; {\mathop {\overbrace{\sum _{\ell =1}^n v_{111\ell },\ldots , \sum _{\ell =1}^n v_{nn1\ell }},}\limits ^{(PX)}} \; {\mathop {\overbrace{\sum _{\ell =1}^n (P\mathcal {O})_{1\ell 1}, \ldots , \sum _{\ell =1}^n (P\mathcal {O})_{1\ell n}},}\limits ^{(P\mathcal {O})}} \\&{\mathop {\overbrace{\sum _{\ell =1}^n (PW)_{1\ell 1}, \ldots , \sum _{\ell =1}^n (PW)_{1\ell n}},}\limits ^{(PW)}}\; {\mathop {\overbrace{\sum _{\ell =1}^n (P\xi )_{1\ell 1}, \ldots , \sum _{\ell =1}^n(P\xi )_{1\ell n}},}\limits ^{(P\xi )}}\; {\mathop {\overbrace{\sum _{\ell =1}^n (P\eta )_{1\ell 11}, \ldots , \sum _{\ell =1}^n (P\eta )_{1\ell nn}},}\limits ^{(P\eta )}}\; \\&{\mathop {\overbrace{\sum _{\ell =1}^n (P\zeta )_{1\ell 11}, \ldots , \sum _{\ell =1}^n (P\zeta )_{1\ell nn}}}\limits ^{(P\zeta )}}\Big ]^T. \end{aligned}$$

Next, we present the explicit formulation of CP-DOMP, which is obtained from the original formulation replacing the original variables $\phi$ by its linear expression in terms of $\Phi$, namely $\phi =\Phi [a_{(4)}]_1$ (see theorems 3.1 and 3.2 ).

Indeed, this reformulation requires to include: 1) the linear constraints that come from MIQP1-DOMP rewritten using that $\Phi [a_{(4)}]_1=\phi$, 2) the squares of those constraints in the matrix variable $\Phi$, 3) the quadratic constraints written in the matrix variables $\Phi$ and 4) $\Phi \in {\mathcal {C}}^*$, the cone of completely positive matrices of the appropriate dimension.

In the following we check that the four conditions mentioned above give us the constraints that appear in the explicit representation of CP-DOMP included below. Indeed,

• 1. Constraints (39) are $[a_{(4)}]^T_i \Phi [a_{(4)}]_1=1$ for all $i=1,\ldots ,n$. Analogously, constraints (40) are $[a_{(5)}]^T_i \Phi [a_{(4)}]_1=1$ for all $i=1,\ldots ,n$; constraint (41) is $[a_{(7)}]^T\Phi [a_{(4)}]_1=p$; constraints (42) are $[a_{(8)}]^T_i \Phi [a_{(4)}]_1=1$ for all $i=1,\ldots ,n$; constraints (44) are $[a_{(17)}]^T_k \Phi [a_{(4)}]_1=1$ for all $k=1,\ldots ,n-1$; constraints (43) are $[a_{(12)}]^T_{j\ell } \Phi [a_{(4)}]_1=1$ for all $j,\ell =1,\ldots ,n$; and constraints (45) are $[a_{(13)}]^T_{jk} \Phi [a_{(4)}]_1=1$ for all $j,k=1,\ldots ,n$. This proves that the block $A\phi =A\Phi [a_{(4)}]_1=b$ appears in CP-DOMP-Explicit.

• 2. Constraints (46)–(53) are obtained squaring (4),(5), (7),(8), (17), (12), (18) and (13), respectively. This proves that $diag(A^T\Phi A)=b\circ b$ also appears in CP-DOMP-Explicit.

• 3. Constraints (54), (55) and (56) are the quadratic constraints (14), (15) and (16) replacing the quadratic terms by the corresponding matrix variables in $\Phi$ (recall that $\Phi$ was introduced in (22)). Hence, CP-DOMP-Explicit includes the quadratic constraints in MIQP1-DOMP written in terms of the matrix variables $\Phi$.

• 4. $\Phi \in {\mathcal {C}}^*_{(4n^2+3n)\times (4n^2+3n)}$, the appropriate dimension of the space of variables.

We note in passing that we do not need to add the constraint $[a_{(4)}]^T_1 \Phi [a_{(4)}]_1=1$ since it is already included. Indeed, it is the first constraint in the block (39).

Based on the above discussion, the explicit reformulation of CP-DOMP as a completely positive convex problem in the essential matrix variables $\Phi$ is:

$$\begin{aligned} \min \quad&\langle F, V \rangle + 1/2 \langle D,Q \rangle +1/2\langle H,U \rangle \end{aligned}$$

(CP-DOMP-Explicit)

$$\begin{aligned}&\text{ s.t. } \sum _{j=1}^n\sum _{\ell =1}^n q_{ij1\ell }=1, \quad \forall i=1,\ldots ,n, \end{aligned}$$

(39)

$$\begin{aligned}&\quad \sum _{k=1}^n\sum _{\ell =1}^nq_{ki1\ell }=1, \quad \forall i=1,\ldots ,n, \end{aligned}$$

(40)

$$\begin{aligned}&\quad \sum _{k=1}^n\sum _{\ell =1}^n (P\mathcal {O})_{1k\ell }=p, \quad \end{aligned}$$

(41)

$$\begin{aligned}&\quad \sum _{j=1}^n\sum _{\ell = 1}^n v_{ij1\ell }=1, \quad \forall i=1,\ldots ,n, \end{aligned}$$

(42)

$$\begin{aligned}&\quad \sum _{k=1}^n v_{1kj\ell }-\sum _{k=1}^n (P\mathcal {O})_{1k\ell }+\sum _{k=1}^n (P\zeta )_{1kj\ell }=0, \quad \forall j,\ell =1,\ldots ,n, \end{aligned}$$

(43)

$$\begin{aligned}&\quad \sum _{\ell =1}^n \rho _{1\ell k}-\sum _{\ell =1}^n \rho _{1\ell k+1}+\sum _{\ell = 1}^n (P\xi )_{1\ell k}=0, \quad \forall k=1,\ldots , n-1, \end{aligned}$$

(44)

$$\begin{aligned}&\quad \sum _{\ell =1}^n \rho _{1\ell k} -\sum _{\ell =1}^n\left( \sum _{r=1}^n v_{1rj\ell }\right) c_{j\ell }+\sum _{\ell =1}^n c_{j\ell } \left( 1-\sum _{r=1}^n q_{jk1r}\right) \nonumber \\&\qquad +\sum _{r=1}^n \nu _{1rjk}=0, \quad \forall j,k=1,\ldots ,n, \end{aligned}$$

(45)

$$\begin{aligned}&\quad \sum _{i=1}^n q_{ikik}=1, \quad \forall k=1,\ldots ,n, \end{aligned}$$

(46)

$$\begin{aligned}&\quad \sum _{i=k}^n q_{ikik}=1, \quad \forall i=1,\ldots ,n, \end{aligned}$$

(47)

$$\begin{aligned}&\quad \sum _{i=1}^n\sum _{j=1}^n\sigma _{ij}=p^2, \end{aligned}$$

(48)

$$\begin{aligned}&\quad \sum _{k=1}^n\sum _{\ell =1}^n u_{jkj\ell }=1, \quad \forall j=1,\ldots ,n, \end{aligned}$$

(49)

$$\begin{aligned}&\quad \omega _{kk}-2\omega _{k,k+1}+2\delta _{kk}+\omega _{k+1,k+1}-2\delta _{k,k+1}+\psi _{kk}=0, \quad \forall k=1,\dots ,n-1, \end{aligned}$$

(50)

$$\begin{aligned}&\quad u_{j\ell j\ell }-2 \chi _{j\ell \ell }+2 \tau _{j \ell j \ell }-2\beta _{\ell j \ell } +\sigma _{\ell \ell }+z_{j\ell j \ell }=0, \quad \forall j,\ell =1,\dots ,n, \end{aligned}$$

(51)

$$\begin{aligned}&\quad \sum _{r=1}^n\sum _{s=1}^n c_{rs} u_{rsrs}+\sum _{\begin{array}{c} i,j,r,s=1\\ (i,j)\ne (r,s)\end{array}}^n c_{ij} c_{rs} u_{ijrs}-2\sum _{i,j=1} n c_{ij} \left( \sum _{\ell =1}^n \gamma _{ij\ell }\right) +\sum _{r,s=1} ^n\omega _{rs}=0, \end{aligned}$$

(52)

$$\begin{aligned}&\quad \left( \sum _{\ell =1} ^n c_{j\ell }\right) ^2 q_{jkjk}+ \sum _{\ell =1} ^n c_{j\ell }^2 u_{j\ell j\ell }-2 \left( \sum _{\ell =1} ^n c_{j\ell }\right) \sum _{\ell =1}^nc_{j\ell } v_{jkj\ell }\nonumber \\&\qquad +2 \sum _{\ell =1}^nc_{j\ell } \rho _{jkk}+2\left( \sum _{\ell =1}^nc_{j\ell }\right) \nu _{jkjk}+2\sum _{r<s}^n c_{jr} c_{js} u_{jrjs}-2\sum _{\ell =1}^nc_{j\ell }\gamma _{j\ell k} \nonumber \\&\qquad -2\sum _{\ell =1}^n c_{j\ell }\kappa _{j\ell jk}+2\epsilon _{kjk}+\omega _{kk}+{\pi }_{jkjk}= \left( \sum _{\ell =1} ^n c_{j\ell }\right) ^2, \quad \forall j,k=1,\ldots ,n, \end{aligned}$$

(53)

$$\begin{aligned}&\quad \sum _{\ell =1}^n \rho _{1\ell k} -\sum _{j=1}^n\sum _{\ell =1}^n c_{j\ell } v_{jkj\ell }=0, \quad \forall k=1,\ldots ,n, \end{aligned}$$

(54)

$$\begin{aligned}&\quad \sum _{j,k=1}^n \left( \sum _{\ell =1}^n q_{1\ell jk}-q_{jkjk}\right) =0, \end{aligned}$$

(55)

$$\begin{aligned}&\quad \sum _{j,\ell =1}^n \left( \sum _{r=1}^n v_{j\ell 1r} - u_{j\ell j\ell }\right) =0, \end{aligned}$$

(56)

$$\begin{aligned}&\quad \Phi \in {\mathcal {C}}^*_{(4n^2+3n)\times (4n^2+3n)}. \end{aligned}$$

(57)