# Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors #src
# This Source Code Form is subject to the terms of the Mozilla Public License #src
# v.2.0. If a copy of the MPL was not distributed with this file, You can #src
# obtain one at https://mozilla.org/MPL/2.0/. #src
# # Dualization
# The purpose of this tutorial is to explain how to use [Dualization.jl](@ref) to
# improve the performance of some conic optimization models. There are two
# important takeaways:
#
# 1. JuMP reformulates problems to meet the input requirements of the
# solver, potentially increasing the problem size by adding slack variables
# and constraints.
# 2. Solving the dual of a conic model can be more efficient than solving the
# primal.
#
# [Dualization.jl](@ref) is a package which fixes these problems, allowing you
# to solve the dual instead of the primal with a one-line change to your code.
# This tutorial uses the following packages
using JuMP
import Dualization
import SCS
# ## Background
# Conic optimization solvers typically accept one of two input formulations.
# The first is the _standard_ conic form:
# ```math
# \begin{align}
# \min_{x \in \mathbb{R}^n} \; & c^\top x \\
# \;\;\text{s.t.} \; & A x = b \\
# & x \in \mathcal{K}
# \end{align}
# ```
# in which we have a set of linear equality constraints $Ax = b$ and the
# variables belong to a cone $\mathcal{K}$.
#
# The second is the _geometric_ conic form:
# ```math
# \begin{align}
# \min_{x \in \mathbb{R}^n} \; & c^\top x \\
# \;\;\text{s.t.} \; & A x - b \in \mathcal{K}
# \end{align}
# ```
# in which an affine function $Ax - b$ belongs to a cone $\mathcal{K}$ and the
# variables are free.
# It is trivial to convert between these two representations, for example, to go
# from the geometric conic form to the standard conic form we introduce slack
# variables $y$:
# ```math
# \begin{align}
# \min_{x \in \mathbb{R}^n} \; & c^\top x \\
# \;\;\text{s.t.} \; & \begin{bmatrix}A & -I\end{bmatrix} \begin{bmatrix}x\\y\end{bmatrix} = b \\
# & \begin{bmatrix}x\\y\end{bmatrix} \in \mathbb{R}^n \times \mathcal{K}
# \end{align}
# ```
# and to go from the standard conic form to the geometric conic form, we can
# rewrite the equality constraint as a function belonging to the `{0}` cone:
# ```math
# \begin{align}
# \min_{x \in \mathbb{R}^n} \; & c^\top x \\
# \;\;\text{s.t.} & \begin{bmatrix}A\\I\end{bmatrix} x - \begin{bmatrix}b\\0\end{bmatrix} \in \{0\} \times \mathcal{K}
# \end{align}
# ```
# From a theoretical perspective, the two formulations are equivalent, and if
# you implement a model in the standard conic form and pass it to a geometric
# conic form solver (or vice versa), then JuMP will automatically reformulate
# the problem into the correct formulation.
# From a practical perspective though, the reformulations are problematic because the
# additional slack variables and constraints can make the problem much larger
# and therefore harder to solve.
# You should also note many problems contain a mix of conic constraints and
# variables, and so they do not neatly fall into one of the two formulations. In
# these cases, JuMP reformulates only the variables and constraints as necessary
# to convert the problem into the desired form.
# ## Primal and dual formulations
# Duality plays a large role in conic optimization. For a detailed description
# of conic duality, see [Duality](@ref).
# A useful observation is that if the primal problem is in standard conic form,
# then the dual problem is in geometric conic form, and vice versa. Moreover, the
# primal and dual may have a different number of variables and constraints,
# although which one is smaller depends on the problem. Therefore, instead of
# reformulating the problem from one form to the other, it can be more
# efficient to solve the dual instead of the primal.
# To demonstrate, we use a variation of the [Maximum cut via SDP](@ref) example.
# The primal formulation (in standard conic form) is:
model_primal = Model()
@variable(model_primal, X[1:2, 1:2], PSD)
@objective(model_primal, Max, sum([1 -1; -1 1] .* X))
@constraint(model_primal, primal_c[i = 1:2], 1 - X[i, i] == 0)
print(model_primal)
# This problem has three scalar decision variables (the matrix `X` is symmetric),
# two scalar equality constraints, and a constraint that `X` is positive
# semidefinite.
# The dual of `model_primal` is:
model_dual = Model()
@variable(model_dual, y[1:2])
@objective(model_dual, Min, sum(y))
@constraint(model_dual, dual_c, [y[1]-1 1; 1 y[2]-1] in PSDCone())
print(model_dual)
# This problem has two scalar decision variables, and a 2x2 positive
# semidefinite matrix constraint.
# !!! tip
# If you haven't seen conic duality before, try deriving the dual problem
# based on the description in [Duality](@ref). You'll need to know that the
# dual cone of [`PSDCone`](@ref) is the [`PSDCone`](@ref).
# When we solve `model_primal` with `SCS.Optimizer`, SCS reports three variables
# (`variables n: 3`), five rows in the constraint matrix (`constraints m: 5`),
# and five non-zeros in the matrix (`nnz(A): 5`):
set_optimizer(model_primal, SCS.Optimizer)
optimize!(model_primal)
# (There are five rows in the constraint matrix because SCS expects problems in
# geometric conic form, and so JuMP has reformulated the `X, PSD` variable
# constraint into the affine constraint `X .+ 0 in PSDCone()`.)
# The solution we obtain is:
value.(X)
#-
dual.(primal_c)
#-
objective_value(model_primal)
# When we solve `model_dual` with `SCS.Optimizer`, SCS reports two variables
# (`variables n: 2`), three rows in the constraint matrix (`constraints m: 3`),
# and two non-zeros in the matrix (`nnz(A): 2`):
set_optimizer(model_dual, SCS.Optimizer)
optimize!(model_dual)
# and the solution we obtain is:
dual.(dual_c)
#-
value.(y)
#-
objective_value(model_dual)
# This particular problem is small enough that it isn't meaningful to compare
# the solve times, but in general, we should expect `model_dual` to solve faster
# than `model_primal` because it contains fewer variables and constraints. The
# difference is particularly noticeable on large-scale optimization problems.
# ## `dual_optimizer`
# Manually deriving the conic dual is difficult and error-prone. The package
# [Dualization.jl](@ref) provides the `Dualization.dual_optimizer` meta-solver,
# which wraps any MathOptInterface-compatible solver in an interface that
# automatically formulates and solves the dual of an input problem.
# To demonstrate, we use `Dualization.dual_optimizer` to solve `model_primal`:
set_optimizer(model_primal, Dualization.dual_optimizer(SCS.Optimizer))
optimize!(model_primal)
# The performance is the same as if we solved `model_dual`, and the correct
# solution is returned to `X`:
value.(X)
#-
dual.(primal_c)
# Moreover, if we use `dual_optimizer` on `model_dual`, then we get the same
# performance as if we had solved `model_primal`:
set_optimizer(model_dual, Dualization.dual_optimizer(SCS.Optimizer))
optimize!(model_dual)
#-
dual.(dual_c)
#-
value.(y)
# ## A mixed example
# The [Maximum cut via SDP](@ref) example is nicely defined because the primal
# is in standard conic form and the dual is in geometric conic form. However,
# many practical models contain a mix of the two formulations. One example is
# [The minimum distortion problem](@ref):
D = [0 1 1 1; 1 0 2 2; 1 2 0 2; 1 2 2 0]
model = Model()
@variable(model, c²)
@variable(model, Q[1:4, 1:4], PSD)
@objective(model, Min, c²)
for i in 1:4, j in (i+1):4
@constraint(model, D[i, j]^2 <= Q[i, i] + Q[j, j] - 2 * Q[i, j])
@constraint(model, Q[i, i] + Q[j, j] - 2 * Q[i, j] <= c² * D[i, j]^2)
end
@constraint(model, Q[1, 1] == 0)
@constraint(model, c² >= 1)
# In this formulation, the `Q` variable is of the form $x\in\mathcal{K}$, but
# there is also a free variable, `c²`, a linear equality constraint,
# `Q[1, 1] == 0`, and some linear inequality constraints. Rather than attempting
# to derive the formulation that JuMP would pass to SCS and its dual, the
# simplest solution is to try solving the problem with and without
# `dual_optimizer` to see which formulation is most efficient.
set_optimizer(model, SCS.Optimizer)
optimize!(model)
#-
set_optimizer(model, Dualization.dual_optimizer(SCS.Optimizer))
optimize!(model)
# For this problem, SCS reports that the primal has
# `variables n: 11, constraints m: 24` and that the dual has
# `variables n: 14, constraints m: 24`. Therefore, we should probably use the
# primal formulation because it has fewer variables and the same number of
# constraints.
# ## When to use `dual_optimizer`
# Because it can make the problem larger or smaller, depending on the problem
# and the choice of solver, there is no definitive rule on when you should
# use `dual_optimizer`. However, you should try `dual_optimizer` if your conic
# optimization problem takes a long time to solve, or if you need to repeatedly
# solve similarly structured problems with different data. In some cases solving
# the dual instead of the primal can make a large difference.