Graphs of Convex Sets (GCS) Optimization

• Graphs of Convex Sets (GCS) are a mixed discrete-continuous framework that integrates graph search with convex optimization for path planning and control.
• They generalize classical problems, including shortest path and TSP, by coupling discrete decisions with continuous convex program solvers.
• Advanced search algorithms and temporal-logic integrations showcase GCS's practical impact in robotics and autonomous systems.

Graphs of Convex Sets (GCS) are a mixed discrete-continuous optimization framework in which graph structure represents combinatorial choices and each selected vertex or edge carries a convex optimization problem. In the shortest-path specialization, a directed graph $G=(V,E)$ is paired with convex vertex sets $X_v$, continuous vertex variables $x_v\in X_v$, and convex edge costs $\ell_e(x_u,x_v)$; the optimization must choose both an admissible path and the continuous points along it. In the more general formulation, the decision is an admissible subgraph $H\in\mathcal H$ together with continuous variables on the selected vertices, so GCS generalizes ordinary weighted-graph problems such as shortest path, traveling salesman, spanning tree, assignment, and several classes of hybrid-control and motion-planning problems (Marcucci et al., 2021, Marcucci, 23 Oct 2025).

1. Foundational model

A general GCS optimization problem can be written as

$$\min \sum_{v \in V} f_v(x_v) + \sum_{e=[v,w]\in E} f_e(x_v,x_w)$$

subject to

$$H=(V_H,E_H)\in \mathcal H,\qquad x_v\in X_v,\qquad (x_v,x_w)\in X_e,$$

where each vertex $v$ has a convex feasible set $X_v$ and convex objective $f_v$, while each edge $X_v$0 has a convex coupling set $X_v$1 and convex objective $X_v$2 (Marcucci, 23 Oct 2025). In the canonical shortest-path problem in GCS, one chooses a path $X_v$3 from source to target and minimizes $X_v$4 over the points assigned to visited vertices (Marcucci et al., 2021).

This model interpolates between discrete graph optimization and continuous convex programming. If the continuous points are fixed, the problem reduces to an ordinary weighted-graph problem; if the graph path is fixed, the remaining problem is a convex program. The difficulty lies in their joint selection: the shortest-path problem in GCS is NP-hard, and the same hybrid structure recurs in motion planning, optimal control of hybrid systems, and skill chaining (Marcucci et al., 2021, Sundar et al., 2024, Morozov et al., 15 Jul 2025).

A defining feature of GCS is that edge costs and feasibility need not be static. An edge may encode a true distance, a control-theoretic stage cost, a continuity constraint between trajectory segments, or feasibility through an infinite cost outside a convex coupling set. This is why GCS is often used when graph vertices stand for convex regions of free space, contact modes, skills, or time-indexed safe sets rather than isolated configurations (Marcucci et al., 2021, Morozov et al., 15 Jul 2025).

2. Exact mixed-integer convex formulations and relaxations

The standard exact formulation of shortest paths in GCS is a mixed-integer convex program built from perspective or homogenization operators. For a convex set $X_v$5, the homogenized set is

$X_v$6

and for bounded $X_v$7, $X_v$8 implies $X_v$9 (Marcucci, 23 Oct 2025). In the shortest-path setting, binary edge variables $x_v\in X_v$0 indicate whether an edge is active, auxiliary variables such as $x_v\in X_v$1 and $x_v\in X_v$2 couple the discrete and continuous decisions, and perspective edge costs $x_v\in X_v$3 preserve convexity (Marcucci et al., 2021).

This perspective construction yields an exact MICP for the shortest-path problem in GCS with the same optimal value as the original mixed discrete-continuous problem (Marcucci et al., 2021). More generally, an integer linear program over an ordinary graph can be lifted automatically into an MICP over a GCS, which gives a unified route from classical shortest path, TSP, minimum spanning arborescence, facility location, and matching formulations to globally solvable GCS models (Marcucci, 23 Oct 2025). The resulting programs are solved to global optimality with off-the-shelf branch-and-bound solvers (Marcucci, 23 Oct 2025).

A central empirical fact in the literature is that the convex relaxation obtained by dropping integrality is often very tight, but it is not exact in general. The original shortest-path paper shows that carefully constructed symmetric instances can make the relaxation arbitrarily loose (Marcucci et al., 2021). Later application papers routinely exploit the relaxed problem as a lower bound or as the basis for rounding, restricted reoptimization, or heuristic search rather than as a universal certificate of exactness (Sundar et al., 2024, Chen et al., 22 May 2026).

3. Search-based and multi-query algorithms

Several later algorithms replace monolithic batch optimization on the full graph by restricted relaxations, implicit search, or offline-to-online decomposition.

Method	Main mechanism	Guarantee or reported behavior
$x_v\in X_v$4-GCS (Sundar et al., 2024)	Uses $x_v\in X_v$5 on representative points, then solves a convex relaxation only on the explored cut-set	Provable lower bounds; on a 25,615-vertex, 56,550-edge 2D maze it terminated in 1 iteration for all heuristic weights and reduced runtime by more than 50% with $x_v\in X_v$6
GCS* (Chia et al., 2024)	Forward heuristic search on implicit GCS with ReachesCheaper and ReachesNew dominance tests	Complete and cost-optimal with exact or conservative ReachesCheaper; sampling-based version is probabilistically complete and asymptotically cost-optimal
IxG / IxG* (Natarajan et al., 2024)	Interleaves graph search with convex trajectory optimization on partial paths	IxG* is optimal for $x_v\in X_v$7 and bounded-suboptimal for $x_v\in X_v$8; in a 2D maze, GCS took 6.728 s and IxG* 1.2613 s
Multi-query SPP in GCS (Morozov et al., 2024)	Offline SDP for convex-quadratic lower bounds, online short-horizon convex programs	About 6 s offline and 2–11 ms online for a 7-DoF KUKA iiwa; up to two orders of magnitude faster than existing motion planners
Shortest-Walk Problem in GCS (Morozov et al., 15 Jul 2025)	Allows revisits, synthesizes piecewise-quadratic lower bounds, then runs incremental search	Extends GCS from shortest paths to shortest walks, unifying motion planning, skill chaining, and hybrid optimal control

These algorithms respond to a recurring structural issue: a full GCS batch problem may be far larger than the part of the graph actually used by the optimal solution. GCS* makes this point explicit by handling implicit graphs with about $x_v\in X_v$9 vertices and up to $\ell_e(x_u,x_v)$0 edges in planar pushing domains, while IxG and IxG* target cases where batch GCS formulations become memory-limited or query-independent in size (Chia et al., 2024, Natarajan et al., 2024).

A related line of work studies reusable lower bounds rather than per-query search alone. The multi-query shortest-path framework computes convex-quadratic approximations of the cost-to-go by semidefinite programming and then uses them in a greedy lookahead policy, while the shortest-walk formulation derives a Bellman equation for revisiting graphs and uses piecewise-quadratic lower bounds to guide incremental search (Morozov et al., 2024, Morozov et al., 15 Jul 2025).

4. Augmented graphs, tours, and other combinatorial generalizations

Although shortest paths are the canonical entry point, GCS has been extended to tour-type and higher-order combinatorial problems by augmenting the graph itself. For the traveling salesman problem in graphs of convex sets, an augmented graph of convex sets (AGCS-TSPS) creates one copy of the original graph for each visited-target subset and arranges these copies in layers indexed by subset cardinality. A shortest path in this augmented graph exactly solves the TSP-GCS, and the construction is closely related to the Bellman–Held–Karp dynamic-programming lattice. The drawback is the same exponential subset growth: the number of subgraphs is $\ell_e(x_u,x_v)$1, and the exact augmented construction becomes impractical beyond small instances (Luna et al., 7 Apr 2026).

That exponential structure has motivated hierarchical and bound-driven solvers. The AGCS-TSP paper develops lower bounds based on direct TSP-GCS relaxation, AGCS relaxation, minimum 1-trees, and bounded edge costs, then combines weighted 1-trees, branch-and-bound, greedy upper bounds, and 2-Opt refinement into a scalable heuristic that is often about two orders of magnitude faster than exact AGCS-TSPS while remaining near-optimal (Luna et al., 7 Apr 2026).

GHOST formulates the same GCS-TSP problem as a hierarchical search over tours on the complete graph induced by the GCS, combined with low-level unfolding into feasible GCS paths. Its key technical device is a triplet-based lower-bound graph over $\ell_e(x_u,x_v)$2 passages rather than static pairwise edges, which produces admissible bounds for both high-level tour search and low-level path realization. GHOST guarantees optimality, includes a bounded-suboptimal variant, and is reported to be orders-of-magnitude faster than unified mixed-integer convex programming baselines on simple cases while also handling incomplete GCSs and high-order continuity constraints (Tang et al., 9 Nov 2025).

5. Temporal logic, timed automata, and precedence constraints

A prominent use of GCS is the integration of symbolic task specifications with continuous motion planning. For LTL motion planning, a transition system built from labeled convex regions is combined with a DFA or DBA for the temporal-logic specification, yielding a product graph

$\ell_e(x_u,x_v)$3

Each product-graph vertex carries a Bézier trajectory segment whose control points lie in one convex region, and edge constraints enforce continuity and, if desired, derivative matching. Because a Bézier curve lies in the convex hull of its control points, the whole segment remains in the associated convex region, which avoids the clipping and passthrough failures of time-discretized MICP methods. The method guarantees soundness for co-safe and full LTL, probabilistic completeness under the stated relaxation-and-rounding assumptions, polynomial complexity in the number of control points and the configuration dimension for the GCS solve itself, and it scales to problems including a 30-DoF humanoid; one reported benchmark solves a “5 keys, 5 doors” problem in 5.82 seconds, and the 30-DoF humanoid example in about 7.72 seconds (Kurtz et al., 2023).

For STL, the analogous construction uses timed automata rather than finite automata. An STL formula is represented by a timed automaton $\ell_e(x_u,x_v)$4, combined with a convex decomposition $\ell_e(x_u,x_v)$5 of configuration space to form a joint transition system with states $\ell_e(x_u,x_v)$6, where $\ell_e(x_u,x_v)$7 is an automaton state and $\ell_e(x_u,x_v)$8 a convex region. Each GCS vertex then includes Bézier control points for geometry, Bézier control points for time, and clock values. Edge constraints encode $\ell_e(x_u,x_v)$9 continuity, clock evolution, timed-automaton guards and resets, and convex velocity constraints. Once the timed automaton and decomposition are fixed, the convex relaxation scales polynomially in $H\in\mathcal H$0, $H\in\mathcal H$1, $H\in\mathcal H$2, and $H\in\mathcal H$3, and the reported experiments include six 2D benchmarks, a 3D quadrotor solved in about 15.68 s versus 489 s for a PWL method, a 30-DoF humanoid solved in 10.2 s, and a UR-3 hardware experiment solved in 9.65 s (Chen et al., 22 May 2026).

Temporal precedence constraints can also be encoded directly by graph augmentation rather than by generic automaton synthesis. In the key-door setting, a layered augmented GCS is indexed by collected-key subsets; doors are traversable only in layers where their unlocking keys have already been collected, and zero-cost inter-layer edges at key vertices encode the event of key collection. The resulting shortest path simultaneously chooses the key-collection order and the continuous trajectory. The method is described as exact up to a finite Bézier curve parameterization, and the restricted STL fragment used for precedence admits a singly exponential progress monitor in the number of logical atoms rather than the doubly exponential worst-case cost of generic temporal-logic compilation (You et al., 30 May 2026).

6. Space-time planning, autonomous driving, and dynamic multi-agent systems

A major extension of GCS is the explicit inclusion of time as an optimization dimension. In Space-Time Graphs of Convex Sets (ST-GCS), each vertex is a convex set in $H\in\mathcal H$4, with piecewise-linear or Bézier-parameterized motion constrained by forward-time and velocity inequalities. Exact Convex Decomposition is used to reserve previously planned trajectories as spatiotemporal obstacles for subsequent robots. Within prioritized multi-robot planning, ST-GCS yields higher success rates and better solution quality than sampling-based baselines and is often orders-of-magnitude faster in the reported benchmarks (Tang et al., 1 Mar 2025). A complementary formulation systematizes the derivation of GCS-compatible constraints, shows that ST-GCS produces equivalent trajectories to standard GCS in static environments, and demonstrates minimum-distance collision-free trajectories in dynamic environments without requiring an initial guess (Osburn et al., 13 Aug 2025).

Autonomous-driving formulations use the same discrete-continuous split but with vehicle-specific dynamics approximations. One paper represents free space as a finite union of convex polytopes organized as a directed graph, parameterizes the spatial path by Bézier curves and timing by a polynomial time-scaling function, and enforces approximate dynamic feasibility under small-slip, linear-tire assumptions. In CommonRoad scenarios, reported average runtimes over 500 runs are 307.6 ms versus 12.1 ms for static obstacle avoidance, 745.6 ms versus 13.9 ms for lane changing, and 1179.9 ms versus 12.4 ms for overtaking, when comparing a nonlinear discrete-time optimal control baseline against the GCS-based method. The paper also notes differences in steering and speed profiles, more aggressive longitudinal acceleration, and heuristic timing constraints for dynamic obstacles (Wagner et al., 13 May 2026).

Interactive and distributed variants push GCS into game-theoretic and multi-agent allocation settings. IBR-GCS treats highway driving as a generalized noncooperative game in which each vehicle repeatedly constructs a strategy-dependent GCS from lane-specific, time-varying, convex collision-free regions; the resulting best-response problems are shortest paths in GCS, and the sequential updates converge to an approximate generalized Nash equilibrium under the stated inexactness assumptions. In a six-vehicle, four-lane merging scenario with $H\in\mathcal H$5 and $H\in\mathcal H$6 s, the algorithm converged in 2 iterations (Käfer et al., 27 Jan 2026). In a different direction, GCS has been coupled to the Consensus-Based Bundle Algorithm in a 3D+time setting so that task-allocation bids reflect actual collision-free trajectory costs in dynamic clutter. The reported experiments include a single multirotor with three tasks completed in 27.49 s and a three-agent, six-task scenario completed in 20.89 s (Osburn et al., 16 Jun 2026).

7. Guarantees, limitations, and software ecosystem

Theoretical guarantees in GCS are highly formulation-dependent. Exact MICP formulations provide global optimality within the modeled family of admissible subgraphs and convex constraints (Marcucci et al., 2021, Marcucci, 23 Oct 2025). Search variants recover completeness, cost-optimality, or bounded suboptimality only under specific admissibility, dominance, or relaxation assumptions (Chia et al., 2024, Natarajan et al., 2024). Temporal-logic planners provide soundness, but their overall complexity still inherits the cost of logic compilation; for LTL, the automaton may have $H\in\mathcal H$7 states (Kurtz et al., 2023). Exact augmented formulations for TSP or subset-based precedence remain exponential in the number of targets or keys (Luna et al., 7 Apr 2026, You et al., 30 May 2026).

Several recurrent limitations are not merely implementation details but structural features of the framework. First, the underlying optimization problems remain NP-hard in general (Marcucci et al., 2021). Second, relaxation quality is problem-dependent: $H\in\mathcal H$8-GCS reports a random overlapping-box instance on which the new method did not significantly outperform baseline because the heuristics were weak or inconsistent (Sundar et al., 2024). Third, model fidelity may be traded for convexity. The autonomous-driving approximation is explicitly based on small-slip and linear-tire assumptions and is not as accurate as the full nonlinear optimal-control baseline in enforcing tight acceleration limits, exact vehicle dynamics, or general dynamic-obstacle safety (Wagner et al., 13 May 2026). Fourth, graph quality matters: ST-GCS studies note that better convex coverage improves path quality but at significant computational cost (Osburn et al., 13 Aug 2025). Finally, when nonlinear parametrizations distort configuration-space geometry, standard convex path-length surrogates can optimize the wrong metric. The “undistortion” approach keeps GCS constraints convex but replaces the convex surrogate objective by a nonconvex objective in the original space and refines the result with projected gradient descent; feasibility is preserved by projection, but global optimality is no longer guaranteed and the method provides local improvement around the GCS solution (Garg et al., 2024).

The software ecosystem reflects these multiple strands. The temporal-logic motion-planning implementation uses Drake for GCS tooling and kinematics, MOSEK for convex optimization, and LTLf2DFA or MONA for automaton synthesis, with open-source code provided in the cited work (Kurtz et al., 2023). More generally, GCSOPT is an open-source Python library that automatically transforms ILP models of graph problems into MICPs over graphs of convex sets and supports standard problems such as shortest path, TSP, minimum spanning arborescence, facility location, and matching (Marcucci, 23 Oct 2025). This infrastructure, together with application-specific use of CVXPY and CLARABEL in space-time planning formulations, indicates that GCS has developed into a general-purpose modeling language for optimization problems at the boundary of convex geometry and combinatorial search (Osburn et al., 13 Aug 2025, Osburn et al., 16 Jun 2026).