From 5e7b7623db93353263508e5dd31e9f6ed721f050 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Maximilian=20K=C3=B6hler?=
 <maximilian.koehler@ruhr-uni-bochum.de>
Date: Thu, 19 Nov 2020 22:20:37 +0100
Subject: [PATCH 1/2] Add centered forms for scalar functions

---
 Project.toml                |  3 +++
 src/IntervalOptimisation.jl |  6 ++++--
 src/centered_forms.jl       | 33 +++++++++++++++++++++++++++++++++
 src/optimise.jl             |  5 ++++-
 4 files changed, 44 insertions(+), 3 deletions(-)
 create mode 100644 src/centered_forms.jl

diff --git a/Project.toml b/Project.toml
index 5fda099..d59c4fc 100644
--- a/Project.toml
+++ b/Project.toml
@@ -3,7 +3,10 @@ uuid = "c7c68f13-a4a2-5b9a-b424-07d005f8d9d2"
 version = "0.4.2"
 
 [deps]
+ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
 IntervalArithmetic = "d1acc4aa-44c8-5952-acd4-ba5d80a2a253"
+IntervalRootFinding = "d2bf35a9-74e0-55ec-b149-d360ff49b807"
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 
 [compat]
 IntervalArithmetic = "0.15, 0.16, 0.17"
diff --git a/src/IntervalOptimisation.jl b/src/IntervalOptimisation.jl
index 4fb2e93..aaf3570 100644
--- a/src/IntervalOptimisation.jl
+++ b/src/IntervalOptimisation.jl
@@ -3,6 +3,7 @@ module IntervalOptimisation
 export minimise, maximise,
        minimize, maximize
 export HeapedVector, SortedVector
+export mean_value_form_scalar, third_order_taylor_form_scalar
 
 include("StrategyBase.jl")
 using .StrategyBase
@@ -13,10 +14,11 @@ using .SortedVectors
 include("HeapedVectors.jl")
 using .HeapedVectors
 
-using IntervalArithmetic
+using IntervalArithmetic, IntervalRootFinding
+using LinearAlgebra
 
 include("optimise.jl")
-
+include("centered_forms.jl") 
 
 const minimize = minimise
 const maximize = maximise
diff --git a/src/centered_forms.jl b/src/centered_forms.jl
new file mode 100644
index 0000000..42278a7
--- /dev/null
+++ b/src/centered_forms.jl
@@ -0,0 +1,33 @@
+import ForwardDiff: gradient, jacobian, hessian
+
+gradient(f, X::IntervalBox) = gradient(f, X.v)
+# jacobian(f, X::IntervalBox) = jacobian(f, X.v)
+hessian(f, X::IntervalBox) = hessian(f, X.v)
+
+"""
+Calculate the mean-value form of a function ``f:\\mathbb{R}^n \\to \\mathbb{R}``
+using the gradient ``\nabla f``;
+this gives a second-order approximation.
+"""
+function mean_value_form_scalar(f, X)
+    m = IntervalBox(mid(X))
+
+    return f(m) + gradient(f, X.v) ⋅ (X - m)
+end
+
+mean_value_form_scalar(f) = X -> mean_value_form_scalar(f, X)
+
+
+"""
+Calculate a third-order Taylor form of ``f:\\mathbb{R}^n \\to \\mathbb{R}`` using the Hessian.
+"""
+function third_order_taylor_form_scalar(f, X)
+    m = IntervalBox(mid(X))
+
+    H = hessian(f, X)
+    δ = X - m
+
+    return f(m) + gradient(f, m) ⋅ δ + 0.5 * sum(δ[i]*H[i,j]*δ[j] for i in 1:length(X) for j in 1:length(X))
+end
+
+third_order_taylor_form_scalar(f) = X -> third_order_taylor_form_scalar(f, X)
diff --git a/src/optimise.jl b/src/optimise.jl
index 705936c..e244162 100644
--- a/src/optimise.jl
+++ b/src/optimise.jl
@@ -1,6 +1,9 @@
 numeric_type(::Interval{T}) where {T} = T
 numeric_type(::IntervalBox{N, T}) where {N, T} = T
 
+interval_mid(X::Interval) = Interval(mid(X))
+interval_mid(X::IntervalBox) = IntervalBox(mid(X))
+
 """
     minimise(f, X, structure = SortedVector, tol=1e-3)
     or
@@ -37,7 +40,7 @@ function minimise(f, X::T; structure = HeapedVector, tol=1e-3) where {T}
         end
 
         # find candidate for upper bound of global minimum by just evaluating a point in the interval:
-        m = sup(f(Interval.(mid.(X))))   # evaluate at midpoint of current interval
+        m = sup(f(interval_mid(X)))   # evaluate at midpoint of current interval
 
         if m < global_min
             global_min = m

From 3ef21e3c0e92b20ce8d0a5d240d395d25abca138 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Maximilian=20K=C3=B6hler?=
 <maximilian.koehler@ruhr-uni-bochum.de>
Date: Fri, 20 Nov 2020 10:07:48 +0100
Subject: [PATCH 2/2] updated CI Services from 0.7 testing to 1.5

---
 .travis.yml  | 3 +--
 appveyor.yml | 2 +-
 2 files changed, 2 insertions(+), 3 deletions(-)

diff --git a/.travis.yml b/.travis.yml
index a6fbab2..01b2f71 100644
--- a/.travis.yml
+++ b/.travis.yml
@@ -6,8 +6,8 @@ os:
   - osx
 
 julia:
-  - 0.7
   - 1.0
+  - 1.5
   - nightly
 
 notifications:
@@ -15,7 +15,6 @@ notifications:
 
 matrix:
   allow_failures:
-    - julia: 0.7
     - julia: nightly
 
 after_success:
diff --git a/appveyor.yml b/appveyor.yml
index c2588f1..c799c17 100644
--- a/appveyor.yml
+++ b/appveyor.yml
@@ -1,7 +1,7 @@
 environment:
   matrix:
-  - julia_version: 0.7
   - julia_version: 1
+  - julia_version: 1.5
   - julia_version: nightly
 
 platform: