Quadratic Programming for Multi-Target Tracking

We consider the problem of tracking multiple, partially observed targets using multiple sensorsarranged in a given con guration. We model the problem as a special case of a ( nite horizon) DEC-POMDP. We present a quadratic program whose globally optimal solution yields an optimal trackingjoint policy, one that maximizes the expected targets detected over the given horizon. However, aglobally optimal solution to the QP cannot always be found since the QP is nonconvex. To remedythis, we present two linearizations of the QP to equivalent 0-1 mixed integer linear programs (MIPs)whose optimal solutions, which may be always found through the branch and bound method, forexample, yield optimal joint policies. Computational experience on di erent sensor con gurationsshows that nding an optimal joint policy by solving the proposed MIPs is much faster than usingexisting algorithms for the problem. 1 Introduction This paper addresses a special case of nite horizon DEC-POMDPs. The special case has been called anetwork distributed POMDP [4] or a factored DEC-POMDP [5]. Lately, this special case has receivedattention in these pages, especially for the problem of detecting multiple targets passing through a givencon guration of locations using multiple sensors, and specialized algorithms have been conceived for it[4], [6], [3]. Our focus too shall be on the multi-target tracking problem.In this problem, the set of agents (sensors) is partitioned into subsets. It is assumed that for eachsubset, we can de ne immediate rewards that are dependent on the actions of the agents of the subsetbut not on the actions of agents outside the subset. It is furthermore assumed that the probabilities withwhich an agent receives observations are independent of probabilities with which other agents receiveobservations. Finally, it is assumed that the probabilities of transitions between states are independentof actions of the agents.The purpose of the above partitioning scheme is to model autonomy for agents in one subset fromthose in other subsets. In the multi-target tracking problem (Figure 1), only the two sensors surroundingFigure 1: Sensor con gurations (reprised from [6]).1