From 7f624986f9756b60b910467df2290bdce9b760ec Mon Sep 17 00:00:00 2001
From: mforets <mforets@gmail.com>
Date: Mon, 1 Oct 2018 14:06:38 -0300
Subject: [PATCH 1/6] add overapporimate cap

---
 src/Approximations/overapproximate.jl | 21 +++++++++++++++++++++
 src/Intersection.jl                   |  4 +++-
 2 files changed, 24 insertions(+), 1 deletion(-)

diff --git a/src/Approximations/overapproximate.jl b/src/Approximations/overapproximate.jl
index 256fa2ecdf..9f4542bd7a 100644
--- a/src/Approximations/overapproximate.jl
+++ b/src/Approximations/overapproximate.jl
@@ -166,3 +166,24 @@ function overapproximate(S::LazySet{N}, ::Type{Interval}) where {N<:Real}
     hi = σ([one(N)], S)[1]
     return Interval(lo, hi)
 end
+
+# overapproximate an intersection with a polytope in H-representation given a
+# set of template directions.
+function overapproximate(cap::Intersection{N, <:LazySet, S},
+                         dir::AbstractDirections{N};
+                         kwargs...) where {N<:Real, S<:AbstractPolytope{N}}
+
+    X = cap.X    # compact set
+    P = cap.Y    # polytope
+
+    Q = HPolytope{N}()
+
+    for Hi in constraints_list(P)
+        for di in dir
+            # can overwrite defaults through kwargs
+            ρ_X_Hi = ρ(di, X∩Hi, kwargs...)
+            addconstraint!(Q, HalfSpace(Hi.a, ρ_X_Hi))
+        end
+    end
+    return Q
+end
diff --git a/src/Intersection.jl b/src/Intersection.jl
index 005767f6f3..51ca312393 100644
--- a/src/Intersection.jl
+++ b/src/Intersection.jl
@@ -319,6 +319,8 @@ The scalar value of the support function of the set `cap` in the given direction
 
 ### Notes
 
+It is assumed that the set `cap.X` is compact.
+
 The `check_intersection` flag can be useful if you know in advance that the
 intersection is non-empty.
 
@@ -349,7 +351,7 @@ function ρ(d::AbstractVector{N},
            algorithm::String="line_search",
            check_intersection::Bool=true,
            kwargs...) where {N<:AbstractFloat, S<:Union{HalfSpace, Hyperplane}}
-    
+
     X = cap.X    # compact set
     H = cap.Y    # halfspace or hyperplane
 

From a5a2ede6da780bd002fc62d1b82b2dad255f4703 Mon Sep 17 00:00:00 2001
From: mforets <mforets@gmail.com>
Date: Mon, 1 Oct 2018 14:23:13 -0300
Subject: [PATCH 2/6] swap dir loop

---
 src/Approximations/overapproximate.jl | 13 ++++++++-----
 1 file changed, 8 insertions(+), 5 deletions(-)

diff --git a/src/Approximations/overapproximate.jl b/src/Approximations/overapproximate.jl
index 9f4542bd7a..9d46e94422 100644
--- a/src/Approximations/overapproximate.jl
+++ b/src/Approximations/overapproximate.jl
@@ -176,14 +176,17 @@ function overapproximate(cap::Intersection{N, <:LazySet, S},
     X = cap.X    # compact set
     P = cap.Y    # polytope
 
+    clist = constraints_list(P)
+    m = length(clist)
     Q = HPolytope{N}()
+    ρ_X_Hi = Vector{N}(undef, m)
 
-    for Hi in constraints_list(P)
-        for di in dir
-            # can overwrite defaults through kwargs
-            ρ_X_Hi = ρ(di, X∩Hi, kwargs...)
-            addconstraint!(Q, HalfSpace(Hi.a, ρ_X_Hi))
+    for di in dir
+        for (i, Hi) in enumerate(clist)
+            ρ_X_Hi[i] = ρ(di, X∩Hi, kwargs...))
         end
+        min_ρ_X_Hi = minimum(ρ_X_Hi)
+        addconstraint!(Q, HalfSpace(Hi.a, min_ρ_X_Hi))
     end
     return Q
 end

From 575dfcab6f2440d3d62cf0cce23e614b5342881c Mon Sep 17 00:00:00 2001
From: mforets <mforets@gmail.com>
Date: Mon, 1 Oct 2018 14:27:10 -0300
Subject: [PATCH 3/6] typo

---
 src/Approximations/overapproximate.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Approximations/overapproximate.jl b/src/Approximations/overapproximate.jl
index 9d46e94422..860adb88c3 100644
--- a/src/Approximations/overapproximate.jl
+++ b/src/Approximations/overapproximate.jl
@@ -183,7 +183,7 @@ function overapproximate(cap::Intersection{N, <:LazySet, S},
 
     for di in dir
         for (i, Hi) in enumerate(clist)
-            ρ_X_Hi[i] = ρ(di, X∩Hi, kwargs...))
+            ρ_X_Hi[i] = ρ(di, X∩Hi, kwargs...)
         end
         min_ρ_X_Hi = minimum(ρ_X_Hi)
         addconstraint!(Q, HalfSpace(Hi.a, min_ρ_X_Hi))

From 6c56da5c91b99de0fe07661ec53d751230b1a706 Mon Sep 17 00:00:00 2001
From: mforets <mforets@gmail.com>
Date: Mon, 1 Oct 2018 14:44:03 -0300
Subject: [PATCH 4/6] more fixes

---
 src/Approximations/overapproximate.jl | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/Approximations/overapproximate.jl b/src/Approximations/overapproximate.jl
index 860adb88c3..3fa3adf683 100644
--- a/src/Approximations/overapproximate.jl
+++ b/src/Approximations/overapproximate.jl
@@ -185,8 +185,8 @@ function overapproximate(cap::Intersection{N, <:LazySet, S},
         for (i, Hi) in enumerate(clist)
             ρ_X_Hi[i] = ρ(di, X∩Hi, kwargs...)
         end
-        min_ρ_X_Hi = minimum(ρ_X_Hi)
-        addconstraint!(Q, HalfSpace(Hi.a, min_ρ_X_Hi))
+        i_min = indmin(ρ_X_Hi)
+        addconstraint!(Q, HalfSpace(clist[i_min].a, ρ_X_Hi[i_min]))
     end
     return Q
 end

From b9c9aa455c1c97c7a670ffc223f25616898fea1a Mon Sep 17 00:00:00 2001
From: mforets <mforets@gmail.com>
Date: Tue, 2 Oct 2018 04:22:08 -0300
Subject: [PATCH 5/6] fix di, docstring, comparison inside loop

---
 src/Approximations/overapproximate.jl | 59 +++++++++++++++++++++++----
 1 file changed, 50 insertions(+), 9 deletions(-)

diff --git a/src/Approximations/overapproximate.jl b/src/Approximations/overapproximate.jl
index 3fa3adf683..8bbdfac5ea 100644
--- a/src/Approximations/overapproximate.jl
+++ b/src/Approximations/overapproximate.jl
@@ -167,8 +167,47 @@ function overapproximate(S::LazySet{N}, ::Type{Interval}) where {N<:Real}
     return Interval(lo, hi)
 end
 
-# overapproximate an intersection with a polytope in H-representation given a
-# set of template directions.
+"""
+    overapproximate(cap::Intersection{N, <:LazySet, S},
+                    dir::AbstractDirections{N};
+                    kwargs...) where {N<:Real, S<:AbstractPolytope{N}}
+
+Return the overapproximation of the intersection between a compact set and a
+polytope given a set of template directions.
+
+### Input
+
+- `cap`    -- intersection of a compact set and a polytope
+- `dir`    -- template directions
+- `kwargs` -- additional arguents that are passed to the support fumction algorithm
+
+### Output
+
+A polytope in H-representation such that the normal direction of each half-space
+is given by an element of `dir`.
+
+### Algorithm
+
+Let `di` be a direction drawn from the set of template directions `dir`.
+Let `X` be the compact set and let `P` be the polytope; that we would like
+to overapproximate the set `X ∩ H`.
+
+Solving the univariate optimization problem `ρ(di, X ∩ Hi)` for each
+half-space in the set `P` and then taking the minimum gives an overapproximation
+of the exact support function.
+
+This algorithm is inspired  from [G. Frehse, R. Ray. Flowpipe-Guard Intersection
+for Reachability Computations with Support
+Functions](https://www.sciencedirect.com/science/article/pii/S1474667015371809).
+
+### Notes
+
+This method relies on having available the `constraints_list` of the polytope
+`P`.
+
+This method of overapproximations can return a non-empty set even if the original
+intersection is empty.
+"""
 function overapproximate(cap::Intersection{N, <:LazySet, S},
                          dir::AbstractDirections{N};
                          kwargs...) where {N<:Real, S<:AbstractPolytope{N}}
@@ -176,17 +215,19 @@ function overapproximate(cap::Intersection{N, <:LazySet, S},
     X = cap.X    # compact set
     P = cap.Y    # polytope
 
-    clist = constraints_list(P)
-    m = length(clist)
+    Hi = constraints_list(P)
+    m = length(Hi)
     Q = HPolytope{N}()
-    ρ_X_Hi = Vector{N}(undef, m)
 
     for di in dir
-        for (i, Hi) in enumerate(clist)
-            ρ_X_Hi[i] = ρ(di, X∩Hi, kwargs...)
+        ρ_X_Hi_min = ρ(di, X ∩ Hi[1], kwargs...) 
+        for i in 2:m
+            ρ_X_Hi = ρ(di, X ∩ Hi[i], kwargs...)
+            if ρ_X_Hi < ρ_X_Hi_min
+                ρ_X_Hi_min = ρ_X_Hi
+            end
         end
-        i_min = indmin(ρ_X_Hi)
-        addconstraint!(Q, HalfSpace(clist[i_min].a, ρ_X_Hi[i_min]))
+        addconstraint!(Q, HalfSpace(di, ρ_X_Hi_min))
     end
     return Q
 end

From 1fa4e2407138dcc3362b37ebee25f859c0a8c129 Mon Sep 17 00:00:00 2001
From: mforets <mforets@gmail.com>
Date: Tue, 2 Oct 2018 09:34:13 -0300
Subject: [PATCH 6/6] revise docstring

---
 src/Approximations/overapproximate.jl | 13 +++++++------
 1 file changed, 7 insertions(+), 6 deletions(-)

diff --git a/src/Approximations/overapproximate.jl b/src/Approximations/overapproximate.jl
index 8bbdfac5ea..1b7fe24860 100644
--- a/src/Approximations/overapproximate.jl
+++ b/src/Approximations/overapproximate.jl
@@ -179,7 +179,7 @@ polytope given a set of template directions.
 
 - `cap`    -- intersection of a compact set and a polytope
 - `dir`    -- template directions
-- `kwargs` -- additional arguents that are passed to the support fumction algorithm
+- `kwargs` -- additional arguents that are passed to the support function algorithm
 
 ### Output
 
@@ -189,14 +189,15 @@ is given by an element of `dir`.
 ### Algorithm
 
 Let `di` be a direction drawn from the set of template directions `dir`.
-Let `X` be the compact set and let `P` be the polytope; that we would like
-to overapproximate the set `X ∩ H`.
+Let `X` be the compact set and let `P` be the polytope. We overapproximate the
+set `X ∩ H` with a polytope in constraint representation using a given set of
+template directions `dir`.
 
-Solving the univariate optimization problem `ρ(di, X ∩ Hi)` for each
-half-space in the set `P` and then taking the minimum gives an overapproximation
+The idea is to solve the univariate optimization problem `ρ(di, X ∩ Hi)` for each
+half-space in the set `P` and then take the minimum. This gives an overapproximation
 of the exact support function.
 
-This algorithm is inspired  from [G. Frehse, R. Ray. Flowpipe-Guard Intersection
+This algorithm is inspired from [G. Frehse, R. Ray. Flowpipe-Guard Intersection
 for Reachability Computations with Support
 Functions](https://www.sciencedirect.com/science/article/pii/S1474667015371809).