From e5cc3e2ffec97c518e282cb833df7764ee4672e0 Mon Sep 17 00:00:00 2001 From: odow Date: Thu, 8 Dec 2022 09:03:40 +1300 Subject: [PATCH] [docs] add more tips and tricks for linear programs --- docs/make.jl | 1 + docs/src/changelog.md | 2 + docs/src/tutorials/linear/tips_and_tricks.jl | 112 ++++++++++++++++--- 3 files changed, 101 insertions(+), 14 deletions(-) diff --git a/docs/make.jl b/docs/make.jl index 59e972e9c20..38be7846813 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -88,6 +88,7 @@ if !_FAST for file in [ joinpath("getting_started", "getting_started_with_julia.md"), joinpath("getting_started", "getting_started_with_JuMP.md"), + joinpath("linear", "tips_and_tricks.md"), ] filename = joinpath(@__DIR__, "src", "tutorials", file) content = read(filename, String) diff --git a/docs/src/changelog.md b/docs/src/changelog.md index e69ed72163c..122380a116f 100644 --- a/docs/src/changelog.md +++ b/docs/src/changelog.md @@ -12,6 +12,8 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0 ### Other - Added Benders tutorial with in-place resolves (#3145) + - Added more [Tips and tricks](@id linear_tips_and_tricks) for linear programs + (#3144) ## Version 1.5.0 (December 8, 2022) diff --git a/docs/src/tutorials/linear/tips_and_tricks.jl b/docs/src/tutorials/linear/tips_and_tricks.jl index 84bcb3ed982..3028e1237a3 100644 --- a/docs/src/tutorials/linear/tips_and_tricks.jl +++ b/docs/src/tutorials/linear/tips_and_tricks.jl @@ -30,6 +30,90 @@ using JuMP +# ## Absolute value + +# To model the absolute value function ``t \ge |x|``, there are a few options. +# In all cases, these reformulations only work if you are minimizing ``t`` +# "down" into ``|x|``. They do not work if you are trying to maximize ``|x|``. + +# ### Option 1 + +# This option adds two linear inequality constraints: + +model = Model(); +@variable(model, x) +@variable(model, t) +@constraint(model, t >= x) +@constraint(model, t >= -x) + +# ### Option 2 + +# This option uses two non-negative variables and forms expressions for ``x`` +# and ``t``: + +model = Model(); +@variable(model, z[1:2] >= 0) +@expression(model, t, z[1] + z[2]) +@expression(model, x, z[1] - z[2]) + +# ### Option 3 + +# This option uses [`MOI.NormOneCone`](@ref) and lets JuMP choose the +# reformulation: + +model = Model(); +@variable(model, x) +@variable(model, t) +@constraint(model, [t; x] in MOI.NormOneCone(2)) + +# ## L1-norm + +# To model ``\min ||x||_1``, that is, ``\min \sum\limits_i |x_i|``, use the +# [`MOI.NormOneCone`](@ref): + +model = Model(); +@variable(model, x[1:3]) +@variable(model, t) +@constraint(model, [t; x] in MOI.NormOneCone(1 + length(x))) +@objective(model, Min, t) + +# ## Infinity-norm + +# To model ``\min ||x||_\infty``, that is, ``\min \max\limits_i |x_i|``, use the +# [`MOI.NormInfinityCone`](@ref): + +model = Model(); +@variable(model, x[1:3]) +@variable(model, t) +@constraint(model, [t; x] in MOI.NormInfinityCone(1 + length(x))) +@objective(model, Min, t) + +# ## Max + +# To model ``t \ge \max\{x, y\}``, do: + +model = Model(); +@variable(model, t) +@variable(model, x) +@variable(model, y) +@constraint(model, t >= x) +@constraint(model, t >= y) + +# This reformulation does not work for ``t \ge \min\{x, y\}``. + +# ## Max + +# To model ``t \le \min\{x, y\}``, do: + +model = Model(); +@variable(model, t) +@variable(model, x) +@variable(model, y) +@constraint(model, t <= x) +@constraint(model, t <= y) + +# This reformulation does not work for ``t \le \max\{x, y\}``. + # ## Boolean operators # Binary variables can be used to construct logical operators. Here are some @@ -39,7 +123,7 @@ using JuMP # $$x_3 = x_1 \lor x_2$$ -model = Model() +model = Model(); @variable(model, x[1:3], Bin) @constraints(model, begin x[1] <= x[3] @@ -51,7 +135,7 @@ end) # $$x_3 = x_1 \land x_2$$ -model = Model() +model = Model(); @variable(model, x[1:3], Bin) @constraints(model, begin x[3] <= x[1] @@ -63,7 +147,7 @@ end) # $$x_1 \neg x_2$$ -model = Model() +model = Model(); @variable(model, x[1:2], Bin) @constraint(model, x[1] == 1 - x[2]) @@ -71,7 +155,7 @@ model = Model() # $$x_1 \implies x_2$$ -model = Model() +model = Model(); @variable(model, x[1:2], Bin) @constraint(model, x[1] <= x[2]) @@ -89,7 +173,7 @@ model = Model() # **Example** Either $x_1 \leq 1$ or $x_2 \leq 2$. -model = Model() +model = Model(); @variable(model, x[1:2]) @variable(model, y, Bin) M = 100 @@ -118,14 +202,14 @@ M = 100 # **Example** $x_1 + x_2 \leq 1$ if $z = 1$. -model = Model() +model = Model(); @variable(model, x[1:2]) @variable(model, z, Bin) @constraint(model, z => {sum(x) <= 1}) # **Example** $x_1 + x_2 \leq 1$ if $z = 0$. -model = Model() +model = Model(); @variable(model, x[1:2]) @variable(model, z, Bin) @constraint(model, !z => {sum(x) <= 1}) @@ -137,7 +221,7 @@ model = Model() # **Example** $x_1 + x_2 \leq 1$ if $z = 1$. -model = Model() +model = Model(); @variable(model, x[1:2]) @variable(model, z, Bin) M = 100 @@ -145,7 +229,7 @@ M = 100 # **Example** $x_1 + x_2 \leq 1$ if $z = 0$. -model = Model() +model = Model(); @variable(model, x[1:2]) @variable(model, z, Bin) M = 100 @@ -169,7 +253,7 @@ M = 100 # **Example** $$x \in \{0\}\cup [1, 2]$$ -model = Model() +model = Model(); @variable(model, x in MOI.Semicontinuous(1.0, 2.0)) # ## Semi-integer variables @@ -178,7 +262,7 @@ model = Model() # bounds $[l,u]$ and can also assume the value zero: # $$x \in \{0\} \cup [l, u] \cap \mathbb{Z}.$$ -model = Model() +model = Model(); @variable(model, x in MOI.Semiinteger(5.0, 10.0)) # ## Special Ordered Sets of Type I @@ -190,7 +274,7 @@ model = Model() # variables. In other words, we have to choose at most one from a set of # possibilities. -model = Model() +model = Model(); @variable(model, x[1:3], Bin) @constraint(model, x in SOS1()) @@ -208,7 +292,7 @@ model = Model() # at most two can be non-zero, and if two are non-zero these must be consecutive # in their ordering. -model = Model() +model = Model(); @variable(model, x[1:3]) @constraint(model, x in SOS2([3.0, 1.0, 2.0])) @@ -234,7 +318,7 @@ ŷ = x̂ .^ 2 # as convex combinations of `x̂` and `ŷ`. N = length(x̂) -model = Model() +model = Model(); @variable(model, -1 <= x <= 2) @variable(model, y) @variable(model, 0 <= λ[1:N] <= 1)