# Manual

Note

As of now, this package only works for optimization models that can be written either in convex conic form or convex quadratic form.

## Supported objectives & constraints - scheme 1

For QPTH/OPTNET style backend, the package supports following Function-in-Set constraints:

MOI FunctionMOI Set
SingleVariableGreaterThan
SingleVariableLessThan
SingleVariableEqualTo
ScalarAffineFunctionGreaterThan
ScalarAffineFunctionLessThan
ScalarAffineFunctionEqualTo

and the following objective types:

MOI Function
SingleVariable
ScalarAffineFunction
ScalarQuadraticFunction

## Supported objectives & constraints - scheme 2

For DiffCP/CVXPY style backend, the package supports following Function-in-Set constraints:

MOI FunctionMOI Set
VectorOfVariablesNonnegatives
VectorOfVariablesNonpositives
VectorOfVariablesZeros
VectorOfVariablesSecondOrderCone
VectorOfVariablesPositiveSemidefiniteConeTriangle
VectorAffineFunctionNonnegatives
VectorAffineFunctionNonpositives
VectorAffineFunctionZeros
VectorAffineFunctionSecondOrderCone
VectorAffineFunctionPositiveSemidefiniteConeTriangle

and the following objective types:

MOI Function
SingleVariable
ScalarAffineFunction

## Creating a differentiable optimizer

You can create a differentiable optimizer over an existing MOI solver by using the diff_optimizer utility.

DiffOpt.diff_optimizerFunction
diff_optimizer(optimizer_constructor)::Optimizer

Creates a DiffOpt.Optimizer, which is an MOI layer with an internal optimizer and other utility methods. Results (primal, dual and slack values) are obtained by querying the internal optimizer instantiated using the optimizer_constructor. These values are required for find jacobians with respect to problem data.

One define a differentiable model by using any solver of choice. Example:

julia> using DiffOpt, GLPK

julia> model = diff_optimizer(GLPK.Optimizer)

julia> _backward_quad(model)  # for convex conic models
source

## Adding new sets and constraints

The DiffOpt Optimizer behaves similarly to other MOI Optimizers and implements the MOI.AbstractOptimizer API.

## Projections on cone sets

DiffOpt requires taking projections and finding projection gradients of vectors while computing the jacobians. For this purpose, we use MathOptSetDistances.jl, which is a dedicated package for computing set distances, projections and projection gradients.

## Conic problem formulation

Note

As of now, the package is using SCS geometric form for affine expressions in cones.

Consider a convex conic optimization problem in its primal (P) and dual (D) forms:

$$$\begin{split} \begin{array} {llcc} \textbf{Primal Problem} & & \textbf{Dual Problem} & \\ \mbox{minimize} & c^T x \quad \quad & \mbox{minimize} & b^T y \\ \mbox{subject to} & A x + s = b \quad \quad & \mbox{subject to} & A^T y + c = 0 \\ & s \in \mathcal{K} & & y \in \mathcal{K}^* \end{array} \end{split}$$$

where

• $x \in R^n$ is the primal variable, $y \in R^m$ is the dual variable, and $s \in R^m$ is the primal slack

variable

• $\mathcal{K} \subseteq R^m$ is a closed convex cone and $\mathcal{K}^* \subseteq R^m$ is the corresponding dual cone

variable

• $A \in R^{m \times n}$, $b \in R^m$, $c \in R^n$ are problem data

In the light of above, DiffOpt differentiates program variables $x$, $s$, $y$ w.r.t pertubations/sensivities in problem data i.e. $dA$, $db$, $dc$. This is achieved via implicit differentiation and matrix differential calculus.

Note that the primal (P) and dual (D) are self-duals of each other. Similarly for the constraints we support, $\mathcal{K}$ is same in format as $\mathcal{K}^*$.

### Reference articles

• Differentiating Through a Cone Program - Akshay Agrawal, Shane Barratt, Stephen Boyd, Enzo Busseti, Walaa M. Moursi, 2019
• A fast and differentiable QP solver for PyTorch. Crafted by Brandon Amos and J. Zico Kolter.
• OptNet: Differentiable Optimization as a Layer in Neural Networks

### Backward Pass vector

One possible point of confusion in finding jacobians is the role of the backward pass vector - above eqn (7), OptNet: Differentiable Optimization as a Layer in Neural Networks. While differentiating convex programs, it is often the case that we dont't want to find the acutal derivatives, rather we might be interested in computing the product of jacobians with a backward pass vector, often used in backprop in machine learing/automatic differentiation. This is what happens in scheme 1 of DiffOpt backend.

But, for the conic system (scheme 2), we provide perturbations in conic data (dA, db, dc) to compute pertubations (dx, dy, dz) in input variables. Unlike the quadratic case, these perturbations are actual derivatives, not the product with a backward pass vector. This is an important distinction between the two schemes of differential optimization.