DyNECT stands for Dynamic Nash Equilibrium Control Toolbox - a Julia package for modeling and solving constrained dynamic Nash equilibrium control problems

A linear quadratic (LQ) game is a multi-agent control problem, where each decision maker influences the evolution of the shared linear dynamical system

and optimizes a quadratic performance metric

where $P_i, Q_i,, R_{i}$ are the weights of the optimization problem for agent $i$, and $u_{-i}$ is the collection of inputs that belong to agents other than $i$, namely, $u_{-i}=(u_j)_{j\neq i}$.
DyNECT helps you find the open-loop Nash equilibrium of the game, that is, the input sequences $(u_1^*,..., u_N^*)$ such that

DyNECT supports state constraints, local input constraints and shared input constraints of the kind

Many functionalities of DyNECT are based on reformulating the dynamic game into the affine variational inequality (AVI)

where $\mathcal F$ is a linear mapping parametric in the initial state of the system $x^0$ of the kind

and $\mathcal C$ is a polyhedron of the kind

You can find more details on the problem formulation, and on the LQ game-to-VI conversion, as well as a performance comparison between some implemented solution algorithms, on our paper

The explicit game-theoretic linear quadratic regulator for constrained multi-agent systems E. Benenati, G. Belgioioso, 2025

• Collection of state-of-the-art iterative and explicit solvers for open-loop dynamic games
• Utility tools for streamlined implementation of game-theoretic MPC
• Automatic reformulation of LQ games as (multi-parametric) Variational Inequalities (VIs)
• Solution of coupled Riccati equations arising in unconstrained infinite-horizon games
• Integration with Monviso for access to multiple VI solution algorithms
• Integration with pDAQP for offline precomputation of state-to-solution map, enabling fast online control
• Unified solver access through the CommonSolve.jl interface

DyNECT depends on 'ParametricDAQP.jl' and 'Monviso.jl', which are not in the general registry. From the Julia REPL:

] add https://github.com/nicomignoni/Monviso.jl.git
] add https://github.com/darnstrom/ParametricDAQP.jl
] add https://github.com/bemilio/DyNECT.git

The example in examples/solve_LQGame_as_VI.jl:

• Implements a basic LQ game
• Converts it into a multi-parametric VI
• Finds explicitely the explicit solution mapping for all initial states
• Given an initial state, converts the multi-parametric VI into a VI
• Solves the VI via several iterative algorithms

Additional examples can be found at this link.