Lagrangian heuristic for simultaneous subsidization and penalization: implementations on rooted travelling salesman games

Abstract

This work examines the problem of stabilizing the grand coalition of an unbalanced cooperative game under the concept of simultaneous subsidization and penalization (S&P). We design a generic framework for developing heuristic algorithms to evaluate the trade-off between subsidy and penalty in the S&P instrument. By incorporating some Lagrangian relaxation techniques, we develop an approach for computing feasible subsidy–penalty pairs under which the grand coalition is stabilized in unbalanced cooperative games. This approach is particularly applicable when the characteristic functions of a cooperative game involve intractable integer programmes. To illustrate the performance of the Lagrangian relaxation based approach, we investigate the rooted travelling salesman game, and the computational results obtained show that our new approach is both efficient and effective.

Appendix

Proof of Theorem 1

The proof is straightforward. First, we can change the second constraint in LP(2) as an inequality, which is expressed as

$$\begin{aligned} \begin{aligned}&\min _{\beta ,z}~ z\\ s.t.~~\beta (S) \le c(S&)+z,~\forall S \in {\mathbb {S}} \setminus \big \{V\big \},\\ \beta (V) \ge&c(V)-\omega . \end{aligned} \end{aligned}$$

(19)

The conversion is equivalent since for the optimal $z^*$ of LP(2), as long as $\beta (S) = c(S) + z^*$ for some S, and $\beta (V) > c(V) - \omega$, one can find a new cost allocation $\beta ^{'} = \beta - \delta (\delta >0)$ such that $\beta ^{'}(S) < c(S) + z^*$ and $\beta ^{'}(V) = c(V) - \omega$, which conflicts to the optimum of $z^*$.

By substituting c(S) with $c_l(S)$ and c(V) with $c_u(V)$, then we can get the restricted one of LP(2) because of $\beta (S) \le c_l(S) + z \le c(S) + z$ and $\beta (V) \ge c_u(V) - \omega \ge c(V) - \omega$.

It is clear that LP(4) is an upper bound of LP(2). Thus, $z_r(\omega )$ is an upper bound of $z(\omega )$. $\square$

Proof of Remark 1

From the strong duality of LP (4), we have

$$\begin{aligned} \begin{aligned} z_r(\omega ) = \max _{\rho }~ \rho _V \big [ c_u(V)-\omega \big ]&- \sum _{S \in {\mathbb {S}} \setminus \{V\}} \rho _S c_l(S)\\ s.t.~~\sum _{S \in {\mathbb {S}} \setminus \{V\}} \rho _S =&1,\\ \sum _{S \in {\mathbb {S}} \setminus \{V\}: k \in S} \rho _S - \rho _V = 0,&~\forall k \in V,\\ \rho _S \ge 0,~\forall S \in&{\mathbb {S}}. \end{aligned} \end{aligned}$$

(20)

In the objective function of LP(1), each function

$$\begin{aligned} \rho _V \big [ c_u(V)-\omega \big ] - \sum _{S \in {\mathbb {S}} \setminus \{V\}} \rho _S c_l(S) \end{aligned}$$

can be viewed as a straight line in $\omega$ whose slope is simply $-\rho _V$. Hence, given some specific $\omega _0$, the value of $z_r(\omega _0)$ is the maximum of $\big \{\rho _V \big [ c_u(V)-\omega _0 \big ] - \sum _{S \in {\mathbb {S}} \setminus \{V\}} \rho _S c_l(S):~\rho \in \text {feasible region of LP}(1) \big \}$. This makes $z_r(\omega )$ be a point-wise maximum of straight lines $\rho _V \big [c_u(V)-\omega \big ] - \sum _{S \in {\mathbb {S}} \setminus \{V\}} \rho _S c_l(S)$. Thus, $z_r(\omega )$ is a piece-wise linear and convex function $\omega$. Moreover, since the slope $-\rho _V$ of each straight line constructing $z_r(\omega )$ is negative ($\rho _V$ is clearly positive), the R-SPF $z_r(\omega )$ must be strictly decreasing in $\omega$.

From the constraints in LP (20), we have

$$\begin{aligned} 1 = \sum _{S \in {\mathbb {S}} \setminus \{V\}} \rho _S \le \sum _{k \in V}\sum _{S \in {\mathbb {S}} \setminus \{V\}: k \in S} \rho _S \le (v-1)\sum _{S \in {\mathbb {S}} \setminus \{V\}} \rho _S = v-1. \end{aligned}$$

Note that $\sum _{k \in V}\sum _{S \in {\mathbb {S}} \setminus \{V\}: k \in S} \rho _S = v\rho _V$; therefore, we have $1 \le v\rho _V \le v-1$, that is $-(v-1)/v\le -\rho _V \le -1/v$. $\square$

Proof of Lemma 1

Denote an optimal solution of (6) as $\big [ \beta _{\lambda }(\ \cdot \ ,\omega ), z_{\lambda }(\omega ) \big ]$. Now, the vector $\big [ {\bar{\beta }}_{\lambda }(\ \cdot \ ,\omega ), z_{\lambda }(\omega ) \big ]$ with $\big \{{\bar{\beta }}_{\lambda }(k,\omega ) = \beta _{\lambda }(k,\omega ) - c_{\lambda 1}(k):\forall k \in V\big \}$ is clearly a feasible solution for (7) since first, for all $S \in {\mathbb {S}} \setminus V$,

$$\begin{aligned} {\bar{\beta }}_{\lambda }(S,\omega ) - z_{\lambda }(\omega ) = \beta _{\lambda }(S,\omega ) - z_{\lambda }(\omega ) - c_{\lambda 1}(S) \le c_{\lambda }(S) - c_{\lambda 1}(S) = c_{\lambda 2}(S), \end{aligned}$$

and second,

$$\begin{aligned} -{\bar{\beta }}_{\lambda }(V,\omega ) = -\beta _{\lambda }(V,\omega ) + c_{\lambda 1}(V) \le -c_{u}(V) + \omega + c_{\lambda 1}(V), \end{aligned}$$

where $\sum _{k \in S}{\bar{\beta }}_{\lambda }(k,\omega ) = {\bar{\beta }}_{\lambda }(S,\omega )$ because of the modularity of IM sub-game 1 $(V,c_{\lambda 1})$.

Similarly, given the optimal solution $\big [ \beta _{\lambda 2}(\ \cdot \ ,\omega ), z_{\lambda }(\omega ) \big ]$ of (7), we can show that vector $\big [ {\bar{\beta }}_{\lambda 2}(\ \cdot \ ,\omega ), z_{\lambda }(\omega ) \big ]$ with $\big \{{\bar{\beta }}_{\lambda 2}(k,\omega ) = \beta _{\lambda 2}(k,\omega ) + c_{\lambda 1}(k):\forall k \in V\big \}$ is feasible for (6). Hence, the optimal objective values $z_{\lambda }(\omega )$ of LPs (7) and (6) are the same. $\square$

Proof of Lemma 2

Note that from Theorem 1, $z_{\lambda }(\omega )$ is an upper bound of $z(\omega )$ for any given $\omega$. In addition, from Remark 1, the LR-SPF $z_{\lambda }(\omega )$ is convex in $\omega$, implying that $U_{\lambda }(\omega )$ is an upper bound of LR-SPF $z_{\lambda }(\omega )$ and SPF $z(\omega )$. $\square$

Proof of Lemma 3

The proof is straightforward. Similar to the proof of Theorem 1, we can show that

$$\begin{aligned} z_{l}(\omega ) = \min _{\beta ,z} \big \{ z \in {\mathbb {R}}:\beta (V) \ge c_\lambda (V)-\omega \text{ and } \beta (S) \le c_u(S)+z, \forall S \in {\mathbb {S}} \setminus \{V\} \big \}, \end{aligned}$$

is a lower bound of $z(\omega )$ with the relaxation of $\beta (V) \ge c(V)-\omega \ge c_\lambda (V)-\omega$ and $\beta (S) \le c(S)+z \le c_u(S)+z$.

In addition, we argue that $z_k(\omega ) \le z_{l}(\omega )$, since the latter feasible region is a subset of the former one. Therefore, we have $z_k(\omega )\le z_{l}(\omega ) \le z(\omega )$. $\square$