Multi-period liability clearing via convex optimal control

Abstract

We consider the problem of determining a sequence of payments among a set of entities that clear (if possible) the liabilities among them. We formulate this as an optimal control problem, which is convex when the objective function is, and therefore readily solved. For this optimal control problem, we give a number of useful and interesting convex costs and constraints that can be combined in any way for different applications. We describe a number of extensions, for example to handle unknown changes in cash and liabilities, to allow bailouts, to find the minimum time to clear the liabilities, or to minimize the number of non-cleared liabilities, when fully clearing the liabilities is impossible.

A Sparsity preserving formulation

In this section we describe a sparsity-preserving formulation of problem (8). We make use of the fact that $L_t$ and $P_t$ are at least as sparse as $L^\mathrm {init}$ (see Sect. 3).

First, let $m=\mathbf {nnz}(L^\mathrm {init})$ and $I_k \in \{1,\ldots ,n\} \times \{1,\ldots ,n\}$, $k=1,\ldots ,m$, be the sparsity pattern of $L^\mathrm {init}$, meaning $(L^\mathrm {init})_{ij}=0$ for all $(i,j)\not \in I_k$, $k=1,\ldots ,m$. Instead of working with the matrix variables $L_t$ and $P_t$, we work with the vector variables $l_t\in {\text{ R }}_+^m$ and $p_t\in {\text{ R }}_+^m$, which represent the nonzero entries of $L_t$ and $P_t$ (in the same order). That is,

$$\begin{aligned} (l_t)_k = (L_t)_{ij}, \quad (p_t)_k = (P_t)_{ij}, \quad (i,j) = I_k, \quad k=1,\ldots ,m. \end{aligned}$$

The initial liability is given by $l^\mathrm {init}\in {\text{ R }}_+^m$, which contains the nonzero entries of $L^\mathrm {init}$. The sparsity preserving formulation of the optimal control problem (8) has the form

$$\begin{aligned} \begin{array}{ll} \text{ minimize } &{} \sum _{t=1}^{T-1} g_t(c_t,l_t,p_t) + g_T(c_T,l_T)\\ \text{ subject } \text{ to } &{} l_{t+1} = l_t - p_t, \quad t=1,\ldots ,T-1,\\ &{} c_{t+1} = c_t - S^\mathrm {row}p_t + S^\mathrm {col} p_t, \quad t=1,\ldots ,T-1,\\ &{} S^\mathrm {row} p_t \le c_t, \quad t=1,\ldots ,T-1,\\ &{} p_t \ge 0, \quad t=1,\ldots ,T-1,\\ &{} l_t \ge 0, \quad t=1,\ldots ,T,\\ &{} c_1 = c^\mathrm {init}, \quad l_1 = l^\mathrm {init}, \quad c_T \ge 0, \end{array} \end{aligned}$$

(19)

where $S^\mathrm {row}\in {\text{ R }}^{n \times m}$ sums the rows of $P_t$, i.e., $S^\mathrm {row}p_t=P_t\mathbf{1}$, and $S^\mathrm {col}\in {\text{ R }}^{n \times m}$ sums the columns of $P_t$, i.e., $S^\mathrm {col}p_t=P_t^T\mathbf{1}$. The cost functions are applied only to the nonzero entries of $L_t$ and $P_t$, so they take the form $g_t:{\text{ R }}_+^n \times {\text{ R }}_+^m \times {\text{ R }}_+^m\rightarrow {\text{ R }}\cup \{+\infty \}$ and $g_T:{\text{ R }}_+^n \times {\text{ R }}_+^m\rightarrow {\text{ R }}\cup \{+\infty \}$. Problem (19) has just $2T(n+m)$ variables, which can be much fewer than the original $2T(n+n^2)$ variables when $m \ll n^2$.