# 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)

### Taxonomy of Optimization Problems

More formally, the general form of a constrained optimization problem is:

\displaystyle{\min_{x} \lbrace f(x) | g_i(x) \leq b, h_i(x) = c.\rbrace}

Where f(x) is the objective function to be optimized while g(x) and h(x) constrain the validity of f(x) to regions in the state space satisfying the respective equality and inequality constraints. The vector 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 f, g, and h:

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`

.

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

### Quadratic Uncosntrained Binary Optimization

## 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).

```
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

```
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]

```
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]

```
# 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:

```
conda install lava-optimization -c conda-forge
```

Alternatively with intel numpy and scipy:

```
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
```