Neuromorphic Constrained Optimization Library ============================================= **A library of solvers that leverage neuromorphic hardware for constrained optimization.** About the Project ----------------- Constrained optimization searches for the values of input variables that minimize or maximize a given objective function, while the variables are subject to constraints. This kind of problem is ubiquitous throughout scientific domains and industries. Constrained optimization is a promising application for neuromorphic computing as it `naturally aligns with the dynamics of spiking neural networks `__. When individual neurons represent states of variables, the neuronal connections can directly encode constraints between the variables: in its simplest form, recurrent inhibitory synapses connect neurons that represent mutually exclusive variable states, while recurrent excitatory synapses link neurons representing reinforcing states. Implemented on massively parallel neuromorphic hardware, such a spiking neural network can simultaneously evaluate conflicts and cost functions involving many variables, and update all variables accordingly. This allows a quick convergence towards an optimal state. In addition, the fine-scale timing dynamics of SNNs allow them to readily escape from local minima. This Lava repository currently supports solvers for the following constrained optimization problems: - Quadratic Programming (QP) - Quadratic Unconstrained Binary Optimization (QUBO) As we continue development, the library will support more constrained optimization problems that are relevant for robotics and operations research. We currently plan the following development order in such a way that new solvers build on the capabilities of existing ones: - Constraint Satisfaction Problems (CSP) [problem interface already available] - Integer Linear Programming (ILP) - Mixed-Integer Linear Programming (MILP) - Mixed-Integer Quadratic Programming (MIQP) - Linear Programming (LP) .. figure:: https://user-images.githubusercontent.com/83413252/135428779-d128aaaa-54ed-4ae1-a5b1-8e0fcc08c96e.png?raw=true :alt: Lava features a growing suite of constrained optimization solvers Overview_Solvers Taxonomy of Optimization Problems ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ More formally, the general form of a constrained optimization problem is: .. math:: \displaystyle{\min_{x} \lbrace f(x) | g_i(x) \leq b, h_i(x) = c.\rbrace} Where :math:`f(x)` is the objective function to be optimized while :math:`g(x)` and :math:`h(x)` constrain the validity of :math:`f(x)` to regions in the state space satisfying the respective equality and inequality constraints. The vector :math:`x` can be continuous, discrete or a mixture of both. We can then construct the following taxonomy of optimization problems according to the characteristics of the variable domain and of :math:`f`, :math:`g`, and :math:`h`: .. figure:: https://user-images.githubusercontent.com/83413252/192852018-dbc08018-ddda-4571-8494-cd1fbfa8405f.png :alt: image image In the long run, lava-optimization aims to offer support to solve all of the problems in the figure with a neuromorphic backend. OptimizationSolver and OptimizationProblem Classes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The figure below shows the general architecture of the library. We harness the general definition of constraint optimization problems to create ``OptimizationProblem`` instances by composing ``Constraints``, ``Variables``, and ``Cost`` classes which describe the characteristics of every subproblem class. Note that while a quadratic problem (QP) will be described by linear equality and inequality constraints with variables on the continuous domain and a quadratic function. A constraint satisfaction problem (CSP) will be described by discrete constraints, defined by variable subsets and a binary relation describing the mutually allowed values for such discrete variables and will have a constant cost function with the pure goal of satisfying constraints. An API for every problem class can be created by inheriting from ``OptimizationSolver`` and composing particular flavors of ``Constraints``, ``Variables``, and ``Cost``. .. figure:: https://user-images.githubusercontent.com/83413252/192851930-919035a7-122d-4a82-8032-f1acc6da717b.png :alt: image image The instance of an ``Optimization problem`` is the valid input for instantiating the generic ``OptimizationSolver`` class. In this way, the ``OptimizationSolver`` interface is left fixed and the ``OptimizationProblem`` allows the greatest flexibility for creating new APIs. Under the hood, the ``OptimizationSolver`` understands the composite structure of the ``OptimizationProblem`` and will in turn compose the required solver components and Lava processes. Tutorials --------- Quadratic Programming ~~~~~~~~~~~~~~~~~~~~~ - `Solving LASSO. `__ Quadratic Uncosntrained Binary Optimization ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - `Solving Maximum Independent Set. `__ Examples -------- Solving QP problems ~~~~~~~~~~~~~~~~~~~ Currently, QP problems can be solved using the specific ``QPSolver``. In future releases, this will be merged with the generic API of ``OptimizationSolver`` (used in the next example). .. code:: python import numpy as np from lava.lib.optimization.problems.problems import QP from lava.lib.optimization.solvers.qp.solver import QPSolver # Define QP problem Q = np.array([[100, 0, 0], [0, 15, 0], [0, 0, 5]]) p = np.array([[1, 2, 1]]).T A = -np.array([[1, 2, 2], [2, 100, 3]]) k = -np.array([[-50, 50]]).T problem = QP(Q, p, A, k) # Define hyper-parameters alpha, beta = 0.001, 1 alpha_d, beta_g = 10000, 10000 iterations = 400 # Solve using QPSolver solver = QPSolver(alpha=alpha, beta=beta, alpha_decay_schedule=alpha_d, beta_growth_schedule=beta_g) solver.solve(problem, iterations=iterations) Solving QUBO ~~~~~~~~~~~~ .. code:: python import numpy as np from lava.lib.optimization.problems.problems import QUBO from lava.lib.optimization.solvers.generic.solver import OptimizationSolver # Define QUBO problem q = np.array([[-5, 2, 4, 0], [ 2,-3, 1, 0], [ 4, 1,-8, 5], [ 0, 0, 5,-6]])) qubo = QUBO(q) # Solve using generic OptimizationSolver solver = OptimizationSolver(problem=qubo1) solution = solver.solve(timeout=3000, target_cost=-50, backend=“Loihi2”) Getting Started --------------- Requirements ~~~~~~~~~~~~ - Working installation of Lava, installed automatically with poetry below. `For custom installs see Lava installation tutorial. `__ Installation ~~~~~~~~~~~~ [Linux/MacOS] ^^^^^^^^^^^^^ .. code:: bash cd $HOME git clone git@github.com:lava-nc/lava-optimization.git cd lava-optimization curl -sSL https://install.python-poetry.org | python3 - poetry config virtualenvs.in-project true poetry install source .venv/bin/activate pytest [Windows] ^^^^^^^^^ .. code:: powershell # Commands using PowerShell cd $HOME git clone git@github.com:lava-nc/lava-optimization.git cd lava-optimization python3 -m venv .venv .venv\Scripts\activate pip install -U pip curl -sSL https://install.python-poetry.org | python3 - poetry config virtualenvs.in-project true poetry install pytest [Alternative] Installing Lava via Conda ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you use the Conda package manager, you can simply install the Lava package via: .. code:: bash conda install lava-optimization -c conda-forge Alternatively with intel numpy and scipy: .. code:: bash conda create -n lava-optimization python=3.9 -c intel conda activate lava-optimization conda install -n lava-optimization -c intel numpy scipy conda install -n lava-optimization -c conda-forge lava-optimization --freeze-installed