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Exploiting Chordal Sparsity for Fast Global Optimality with Application to Localization
Frederike Dümbgen, Connor Holmes, Timothy D. Barfoot
TL;DR
Robotics estimation methods that seek global optimality face the computational cost of SDP solvers, while local methods can reach poor minima without good initialization. The paper uses chordal decomposition to split these SDPs into smaller problems and enable distributed solution, reporting approximately linear complexity and performance comparable to efficient local solvers while retaining certified global optimality for reasonable noise levels.
Problem
Off-the-shelf SDP solvers have approximately cubic complexity in unknowns and constraints, while local robotics methods may converge to poor local minima without well-informed initialization.
Source: §1 · pp. 1–2Method
The paper decomposes chordally sparse SDPs into interconnected smaller problems and adapts the formulation to distributed solution using consensus ADMM.
Source: §1 · p. 2, §4.4 · p. 11, §4.5 · p. 12Results
The decomposed methods have approximately linear complexity and are competitive with efficient local solvers while providing certified global optimality for reasonable noise levels.
Source: §1 · p. 2, §6 · p. 15Takeaways & Limitations
Chordal decomposition offers centralized and ADMM-based distributed routes to faster SDP computation for robotics problems with suitable sparsity.
Source: §1 · p. 2, §6 · p. 15The tested dSDP-admm setup fixed the thread count at 10 because of hardware limitations and communication overhead, and performance was expected to improve with more computational nodes.
Source: §5.2 · p. 14
Abstract
from arXiv · showhide
In recent years, many estimation problems in robotics have been shown to be solvable to global optimality using their semidefinite relaxations. However, the runtime complexity of off-the-shelf semidefinite programming (SDP) solvers is up to cubic in problem size, which inhibits real-time solutions of problems involving large state dimensions. We show that for a large class of problems, namely those with chordal sparsity, we can reduce the complexity of these solvers to linear in problem size. In particular, we show how to replace the large positive-semidefinite variable with a number of smaller interconnected ones using the well-known chordal decomposition. This formulation also allows for the straightforward application of the alternating direction method of multipliers (ADMM), which can exploit parallelism for increased scalability. We show for two example problems in simulation that the chordal solvers provide a significant speed-up over standard SDP solvers, and that global optimality is crucial in the absence of good initializations.
1 Introduction
Robotics estimation often uses fast local methods that can reach poor minima without good initialization, while globally optimal SDP methods face a cubic-complexity bottleneck. The paper applies chordal decomposition to robotics SDPs to obtain approximately linear complexity and distributed solutions, while addressing the role of redundant constraints.
1 Introduction
- Global solvers had been applied to computer-vision tasks and robotics problems including range-only localization, SLAM, pose-graph optimization, and non-smooth planning [60].The paper positions its work amid these advances toward globally optimal solvers.Source: §1 · pp. 1–2
- Off-the-shelf interior-point SDP solvers have approximately cubic complexity in unknowns and constraints, whereas local solvers can reach poor minima without well-informed initialization.Local methods such as Gauss-Newton and iSAM can exploit sparsity for linear-time computation, motivating more efficient globally optimal solvers.Source: §1 · pp. 1–2
- Chordal decomposition splits a large SDP into interconnected smaller problems, reducing solver complexity to approximately linear in the number of variables and enabling distribution with ADMM.The paper applies the decomposed SDP solver to robotics problems using factor-graph modeling.Source: §1 · p. 2, Figure 1 · p. 2
- The paper adapts decomposed and distributed SDP methods to robotics state estimation, where redundant constraints can be important for tightness and may affect the approach’s effectiveness.The methods had previously been considered in control and applied to optimal power flow.Source: §1 · pp. 2–3
2 Related Work
Prior work has accelerated certifiably optimal SDP solvers through low-rank factorization and sparsity exploitation, while distributed methods have also scaled robotics optimization. This paper focuses on chordal decomposition as a sparsity-based alternative suited to state-estimation problems with redundant constraints.
2 Related Work
- SDP acceleration methods exploit either low-rank structure through Burer-Monteiro factorization or matrix sparsity, and distributed variants of low-rank methods have been applied to pose-graph optimization and semantic SLAM [56] [55].The cited Burer-Monteiro methods had not been successfully applied to problems requiring redundant constraints because they break common constraint qualifications.Source: §2 · p. 3
- Redundant constraints are important for tight semidefinite relaxations in many robotics instances, motivating sparsity-exploiting alternatives when they do not significantly affect chordal sparsity [60] [19].The paper distinguishes decomposition methods, which reduce an SDP to smaller instances, from sparsity-based acceleration of operations within a solver.Source: §2 · p. 3
- Decomposition methods can solve smaller SDP instances in one shot or alternation, and the paper identifies state-estimation problems as candidates when redundant constraints preserve chordal sparsity.These are distinct from methods that exploit sparse constraint matrices to accelerate solver operations.Source: §2 · p. 3
- Distributed robotics methods include scalable estimation and collaborative multi-robot approaches, but methods that distribute local solvers may converge to poor local minima; the paper presents ADMM as a step toward a distributed certifiably optimal alternative [22].The related work also describes distributed ADMM-based quadratic-program solvers in control and a certifiably optimal trajectory-optimization solver [33].Source: §2 · pp. 3–4
3 Background
The background establishes notation for matrices and graphs, then defines the sparsity graph and chordal structure used to decompose semidefinite constraints. For a chordal extension, positive semidefiniteness of the sparse matrix’s completion is characterized by positive semidefinite submatrices on maximal cliques.
3.1 Notation and definitions
- The notation distinguishes scalars, vectors, matrices, and sets, defines positive semidefiniteness, and introduces graph terms including induced subgraphs, cliques, cycles, and chords.The vectorization operator vech returns the upper-diagonal elements of a symmetric matrix, with off-diagonal elements multiplied by a factor specified in the text.Source: §3.1 · p. 4
3.2 Chordal sparsity in SDPs
- The aggregate sparsity pattern collects index pairs where at least one problem matrix has a nonzero entry, and its sparsity graph has those pairs as edges.The pattern is defined over the cost and constraint matrices of the SDP.Source: §3.2 · p. 4
- A graph is chordal when every cycle of length at least four contains a chord; a graph can be made chordal by adding edges, but finding a minimum chordal extension is NP-hard.The paper uses deterministic chordal extensions for its typical robotics sparsity patterns, while efficient heuristics exist.Source: §3.2 · p. 5
- A matrix is semidefinite-completable in a sparsity pattern when changing entries outside that pattern can make it positive semidefinite.This notion connects sparse matrix structure to the clique-based decomposition property.Source: §3.2 · p. 5
- For the chordal extension of an SDP’s aggregate sparsity graph, semidefinite-completability is equivalent to positive semidefiniteness of every maximal-clique submatrix.This property enables replacing one large SDP variable with smaller clique submatrices [61].Source: §3.2 · p. 5
4 Method
The method reformulates localization estimation from nonlinear least squares (NLS) through a quadratically constrained quadratic program (QCQP), preparing it for semidefinite optimization. It focuses on two localization examples and factor graphs whose small cliques support sparsity-aware methods.
4.1 Nonlinear least squares (NLS) formulation
- Nonlinear least squares estimates states from absolute and relative measurements or priors, but nonlinear errors or non-convex constraints can make the problem non-convex and local solvers initialization-sensitive.The paper considers factor graphs with small maximum cliques, including chain-structured examples.Source: §4.1 · p. 6, Factor-graph representation · p. 6
Factor-graph representation.
- The factor graph Gf represents states and measurements or priors as nodes, with edges encoding their bipartite connectivity.The paper focuses on chordally sparse graphs with small maximum cliques.Source: Factor-graph representation · p. 6
Example CT-RO.
- CT-RO estimates a moving device’s position over time from distances to known landmarks, with a Gaussian-process motion prior encouraging smoothness and supporting recovery in underdetermined cases.The state includes position and velocity; local solvers may reach local minima when landmarks are degenerate.Source: Example CT-RO · p. 6
Example MW.
- Matrix-weighted localization estimates a moving device’s pose from landmark observations while allowing anisotropic noise, with relative measurements modeled using isotropic noise.The state comprises rotation and translation; absolute landmark measurements use a typically anisotropic inverse covariance matrix.Source: Example MW · pp. 6–7
4.2 NLS to Quadratically Constrained Quadratic Program
- The QCQP reformulation lifts the state into a vector containing state variables and sufficient higher-order functions to make costs and constraints quadratic.A homogenization variable h = 1 makes linear and constant cost terms quadratic; primary constraints depend on one variable group.Source: §4.2 · p. 7
- The QCQP factor graph extends the cost graph with constraint factors, and its connectivity reflects aggregate sparsity in the cost and constraint matrices.The variant is defined over lifted states and includes both cost and constraint factors.Source: §4.2 · p. 8, Figure 2 · p. 8
Examples.
The section derives a semidefinite relaxation and uses sparsity patterns to decompose it. It also explains how redundant constraints can affect both tightness and the sparsity available for decomposition.
4.3 QCQP to Semidefinite Program
- The cost and constraint matrices define aggregate sparsity, which can be decomposed into clique subproblems; the authors favor manual groupings that avoid fill-in between otherwise unrelated variable groups.The figure shows the cost and constraint sparsity patterns alongside used and automatically generated decompositions.Source: Figure 3 · p. 9
- The SDP relaxation drops the rank-one constraint; tightness is needed to recover a globally optimal QCQP solution, while redundant constraints can help enforce low rank but may enlarge cliques.For the two examples, the relaxations are tight with essentially unchanged sparsity patterns.Source: §4.3 · p. 9
- CT-RO’s relaxation is tight without redundant constraints.This behavior was previously observed for CT-RO.Source: §4.3 · p. 9
- For MW, redundant handedness constraints may improve tightness as noise increases and measurements decrease, while leaving the measurement-graph structure unchanged.The constraints depend on one pose at a time; five of six listed constraints are used because one is linearly dependent.Source: §4.3 · pp. 9–10
4.4 Decomposed solution: dSDP
dSDP exploits chordal sparsity by replacing a large semidefinite constraint with smaller clique-wise constraints linked by overlaps. dSDP-admm adds consensus variables so clique subproblems can be updated in a distributed fashion.
4.4 Decomposed solution: dSDP
- Replacing y ⪰0 with clique-wise ct ⪰0 constraints splits the large SDP variable into T = |Vc| smaller ones; runtime gains depend on clique sizes, counts, and connectivity.The decomposed dSDP problem is still centralized because the overlap constraints link cliques.Source: §4.4 · pp. 10–11
- Manual cliques ci = (h, yi, yi+1) avoid fill-in dependencies between variable groups, while a chordal decomposition at the block level also works beyond simple chains.Automatic maximal-clique heuristics can introduce fill-in; the authors expect additional sparsity within each clique to yield only minor improvement.Source: §4.4 · p. 10
- The dSDP formulation uses a clique tree, splits shared cost terms equally across overlapping cliques, and enforces agreement on their shared variables.The factor-graph decomposition keeps the original variable groups intact.Source: §4.4 · p. 11, Figure 4 · p. 11
4.5 Distributed solution: dSDP-admm
- dSDP-admm adapts consensus ADMM by pairing local, overlapping clique variables with a consensus variable and retaining semidefinite constraints on the smaller clique variables.Each clique's feasible set includes its local linear constraints and positive-semidefinite constraint.Source: §4.5 · p. 12
- ADMM separates the clique updates into T SDP subproblems, while each consensus update averages the cliques sharing a variable and requires communication only between overlapping cliques.The consensus update has a closed-form solution, allowing the algorithm to run in a distributed fashion.Source: §4.5 · p. 12
5 Localization Results
The simulations compare the decomposed solvers with local and standard SDP solvers on two localization problems. The decomposed methods are faster than full SDP while retaining global-optimality certificates, but the ADMM variant trades solution quality for runtime under a fixed iteration limit.
5.1 Simulation Setup
- Simulations compare the two methods with local and standard SDP solvers on two example localization problems, evaluating efficiency, optimality, and accuracy; noise studies increase measurement noise for range-only and matrix-weighted localization.Timing studies solve one example per problem size; noise studies average ten random problem instances, with Fig. 6 using N = 100 and Nm = 8.Source: §5 · p. 12, §5.1 · p. 13, Figure 6 · p. 14
5.2 Results
- The decomposed methods are significantly faster than full SDP and competitive with local Gauss–Newton in runtime complexity while providing certified global optimality.Figure 5 reports comparable complexity and little variance for the sparsity-exploiting algorithms; the local solver is faster on average but has initialization-dependent variance.Source: Figure 5 · p. 13, §5.2 · p. 14
- For CT-RO, relaxation tightness decreases as noise rises, and fixed-iteration dSDP-admm quality deteriorates; SDP and dSDP retain consistently high EVR.The authors attribute the ADMM difficulty to weaker node connections through a motion prior rather than pose-to-pose measurements, making consensus harder at higher noise.Source: §5.2 · pp. 14–15, Figure 6 · p. 14
- For MW, redundant constraints make EVR nearly constant across the considered noise levels, while dSDP-admm is less tight but still suggests a rank-one solution.The authors report that the added computational cost for higher tightness is minimal.Source: §5.2 · p. 15
- Randomly initialized local optimization has high accuracy variance, whereas local-gt, SDP, and dSDP achieve almost identical accuracy; dSDP-admm is less accurate, especially for CT-RO.The authors link dSDP-admm’s accuracy loss to the imposed three-iteration limit and describe an accuracy–runtime trade-off.Source: §5.2 · p. 15
6 Conclusion and Future Work
The conclusion presents chordal decomposition as a route to linear-time SDP solvers for a large class of robotics optimization problems. It also identifies improvements to ADMM and behavior beyond chordal sparsity as future work.
6 Conclusion and Future Work
- Chordal decomposition yields linear-time SDP solvers for a large class of robotics optimization problems.The decomposition can speed centralized computation and also supports a tested ADMM-based distributed implementation.Source: §6 · p. 15
- The methods have computational complexity on par with efficient local solvers while remaining certifiably optimal for reasonable noise levels.Source: §6 · p. 15
- Future directions include improving vanilla ADMM and studying problems that are not chordally sparse; the current Python implementation also limits effective multithreading.The paper mentions diagonal rescaling, approximate interior-point methods, factor-width decomposition, loopy belief propagation, and chordal reduction as possible approaches.Source: §6 · p. 15
A.1 General formulation
The general formulation assumes cost terms can be represented using lifted vectors and that their terms and feasible sets can be enforced with quadratic constraints. Under these assumptions, it obtains an equivalent QCQP formulation.
A.1 General formulation
- The functions ℓ_k are assumed to contain enough higher-order terms to express the error terms as B_k x_k(ξ_k) and B_ij x_ij(ξ_i, ξ_j).Source: §A.1 · p. 16
- The lifted cost terms are rewritten in the required form by introducing matrices such as Q_k = B_k^T W_k B_k.Source: §A.1 · p. 16
- Each lifted-vector term and feasible set is assumed enforceable by quadratic constraints, yielding an equivalent QCQP formulation.The passage notes that polynomial or rational substitutions of any order permit quadratic constraints for lifted-vector terms.Source: §A.1 · p. 16
A.2 Example reformulations
The appendix reformulates the CT-RO and MW examples for the general QCQP framework. CT-RO uses a squared-position substitution, whereas MW needs no cost substitution and enforces its rotation constraints quadratically.
A.2 Example reformulations
- For CT-RO, the substitution ℓ_k(ξ_k) := ∥t_k∥² lifts the squared position norm into a polynomial form that can be written quadratically.Source: §A.2 · p. 16
- The CT-RO reformulation expresses each absolute error term using the lifted vector and a row of B_k.Source: §A.2 · p. 16
- For MW, the cost is already quadratic, so no substitutions are needed; quadratic constraints enforce rotation orthogonality and handedness.The handedness constraint replaces the cubic determinant constraint in the definition of SO(3).Source: §A.2 · p. 16