S-D assignment using Lagrangian relaxation
returns a table of assignments,
solutionGap] = assignsd(
assignments, of detections to tracks by
finding a suboptimal solution to the S-D assignment problem using Lagrangian relaxation. The
cost of each potential assignment is contained in the cost matrix,
costmatrix. The algorithm terminates when the gap reaches below 0.01
(1 percent) or if the number of iterations reaches 100.
costmatrix is an n-dimensional cost matrix
costmatrix(i,j,k ...) defines the cost of the n-tuple
...) in assignment. The index '1' on all dimensions in
costmatrix represents dummy measurement or a false track and is used to
complete the assignment problem. The index 1, being a dummy, can be a part of multiple n-tuples.
The index can be assigned more than once. A typical cost value for
costmatrix(1,1,1,1, ...) is 0.
All inputs can be single or double precision, but they all must be of the same precision.
The function also returns the solution gap,
solutionGap, and the
total cost of assignments,
specifies the desired maximum gap,
solutionGap] = assignsd(
desiredGap, between the dual and the
feasible solutions as a scalar. The gap controls the quality of the solution. Values usually
range from 0 to 1. A value of 0 means the dual and feasible solutions are the same.
assignsd to perform strict assignment without index 1.
Not having dummy index means that no entity is left unassigned. Therefore, define the cost matrix to be equi-dimensional.
costMatrix = rand(6,6,6);
fullmatrix to all
fullmatix is one size larger than the cost matrix in all dimensions.
fullMatrix = inf(7,7,7);
Set the inner matrix to costMatrix to force the assignments involving index 1 to have infinite cost.
fullMatrix(2:end,2:end,2:end) = costMatrix; fullMatrix(1,1,1) = 0; [assignments,cost,gapAchieved] = assignsd(fullMatrix,0.01,100);
Restore the actual indices.
assignments = assignments - 1
assignments = 6x3 uint32 matrix 1 6 6 2 4 3 3 3 4 4 1 2 5 2 1 6 5 5
costmatrix— Cost matrix
Cost matrix, specified as an n-dimensional array where
...) defines the cost of the n-tuple
(i,j,k, ...) in an
assignment. The index '1' on all dimensions in
represents a dummy measurement or a false track and is used to complete the assignment
problem. The index 1, being a dummy, can be a part of multiple n-tuples. The index can
be assigned more than once. A typical cost value for
...) is 0.
desiredGap— Desired maximal gap
0.01(default) | nonnegative scalar
Desired maximum gap between the dual and feasible solutions, specified as a nonnegative scalar.
maxIterations— Maximum number of iterations
100(default) | positive integer
Maximum number of iterations, specified as a positive integer.
algorithm— Assignment algorithm
Assignment algorithm for solving the 2-D assignment problem, specified as
'munkres' for the Munkres algorithm,
the Jonker-Volgenant algorithm, or
'auction' for the Auction
assignments— Assignment of tracks to detections
Assignments of tracks to detections, returned as a
P-by-N list of assignments. Assignments of the
[1 1 Q 1] from a four-dimensional cost matrix can be seen as a
Q-1 entity from dimension 3 that was left unassigned. The cost
(1,1,Q,1) defines the cost of not assigning the
(Q-1)th entity from dimension 3.
cost— Total cost of assignment solution
Total cost of solutions, returned as a K-element vector where K is the number of best solutions. Each element is a scalar value summarizing the total cost of the solution to the assignment problem.
solutionGap— Solution gap
Solution gap, returned as a positive scalar. The solution gap is the duality gap achieved between the feasible and dual solution. A gap value near zero indicates the quality of solution.
The Lagrangian relaxation method computes a suboptimal solution to the S-D assignment problem. The method relaxes the S-D assignment problem to a 2-D assignment problem using a set of Lagrangian multipliers. The relaxed 2-D assignment problem is commonly known as the dual problem, which can be solved optimally using algorithms like the Munkres algorithm. Constraints are then enforced on the dual solution by solving multiple 2-D assignment problems to obtain a feasible solution to the original problem. The cost of the dual solution and the feasible solution serves as lower and upper bounds on the optimal cost, respectively. The algorithm iteratively tries to minimize the gap between the dual and feasible solutions, commonly known as the dual gap. The iteration stops when the dual gap is below a desired gap or the maximum number of iterations have reached.
When using the auction algorithm, the
assignsd function uses the
Heuristic Price Update algorithm to update the Lagrangian multipliers. When using the
Munkres and JV algorithms, the function uses the Accelerated Subgradient Update
For cost matrices with well-defined solutions, such as passive association with high-precision sensors, the solution gap converges to within 0.05 (5 percent) in approximately 100 iterations.
As the optimal solution is unknown, the solution gap can be non-zero even when the returned solution is optimal.
 Deb, S., Yeddanapudi, M., Pattipati, K., and Bar-Shalom, Y. (1997). A generalized SD assignment algorithm for multisensor-multitarget state estimation. IEEE Transactions on Aerospace and Electronic Systems, 33(2), 523-538.
 Blackman, Samuel, and Robert Popoli. Design and analysis of modern tracking systems. Norwood, MA: Artech House, 1999. (1999)