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Convex hull of two triangles

Description
Determine the convex hull for two given triangles.

Further explanation of this example: This model is discussed in Section 10.3 of the book 'J. Kallrath: Business Optimization Using Mathematical Programming - An Introduction with Case Studies and Solutions in Various Algebraic Modeling Languages' (2nd edition, Springer, Cham, 2021, DOI 10.1007/978-3-030-73237-0).


Source Files
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ch-2tri.mos[download]





ch-2tri.mos

(!*********************************************************************
   Mosel Example Problems
   ======================

   file ch-2tri.mos
   ````````````````
   Find the convex hull of two triangles T1 and T2.

     Example discussed in section 10.3 of
     J. Kallrath: Business Optimization Using Mathematical Programming -
     An Introduction with Case Studies and Solutions in Various Algebraic 
     Modeling Languages. 2nd edition, Springer Nature, Cham, 2021 

   Mosel version of CH-twoTriangles.gms (T.Romanova & J.Kallrath):
   For the following example of two triangles with vertices (A,B,C)
       T1: A_1=(0,0), B_1=(14,0), C_1=(10,-5)
       T2: A_2=(0,0), B_2=( 8,0), C_2=( 6, 4)
   the analytic solution is u*=7, alpha_2=33.6900675, x=1, and the minimal 
   length of the convex hull perimeter: 33.7079796215289.

   Notations
   ---------
   S  is the number of sides of the convex hull
   A is the set of sides of the first triangle (called triangle A)
   B is the set of sides of the second triangle (called triangle B)
   VXA,VYA,VXB,VYB  are sets of the appropriate coordinates (x,y) of the
                    vertices of  two  triangles
   aA,bA,cA  are sets of the appropriate coefficients of the side equations
             of triangle A
   xB,yB,fB are variable placement parameters of triangle B
   xS,yS,fS,lS are sets of variable parameters of sides of the convex hull
   lPCH is the perimeter of the convex hull


   author: S. Heipcke, June 2020

   (c) Copyright 2020 Fair Isaac Corporation
  
    Licensed under the Apache License, Version 2.0 (the "License");
    you may not use this file except in compliance with the License.
    You may obtain a copy of the License at
 
       http://www.apache.org/licenses/LICENSE-2.0
 
    Unless required by applicable law or agreed to in writing, software
    distributed under the License is distributed on an "AS IS" BASIS,
    WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
    See the License for the specific language governing permissions and
    limitations under the License.

*********************************************************************!)

model "ch_2tri"
  uses "math", "mmxnlp"
  uses "mmsvg"

!**** Input data ****
 declarations
   S: range                         ! Sides of the convex hull S1,...,S4
   A = 1..3                         ! Vertices of triangle A   A1,...,A3
   B = 1..3                         ! Vertices of triangle B   B1,...,B3
   VXA, VYA: array(A) of real       ! x/y-coordinates of vertices of triangle A
   VXB, VYB: array(B) of real       ! x/y-coordinates of vertices of triangle B
   AlphaA, BetaA, GammaA: array(A) of real ! Coeff.s of the side equations of A
   SideLengthsA: array(A) of real   ! Side lengths of A
   SideLengthsB: array(B) of real   ! Side lengths of B
   AnglesA, CosA: array(A) of real  ! Angles at vertices of A
   AnglesB, CosB: array(B) of real  ! Angles at vertices of B ;
   CosNA, SinNA: array(A) of real   ! Normal vector angles onto sides A
   CosNB, SinNB: array(B) of real   ! Normal vector angles onto sides B
   AnglesNA,NX,NY: array(A) of real ! Normal vector angles onto sides A
   AnglesNB: array(B) of real       ! Normal vector angles onto sides B
 end-declarations
 
 ! Input data: shape and position of triangles
 VXA::(1..3)[0,10,14]
 VYA::(1..3)[0,-5, 0]
 VXB::(1..3)[0, 8, 6]
 VYB::(1..3)[0, 0, 4]
 AlphaA::(1..3)[-0.447214,  0.780869,0]
 BetaA:: (1..3)[-0.894427, -0.624695,1]
 GammaA::(1..3)[ 0       ,-10.932163,0]
 S:= 0..3
 sz:= S.size

 ! Compute side lengths
 forall(i in A) SideLengthsA(i):= 
   sqrt( (VXA(if(i<3,i+1,1))-VXA(i))^2 + (VYA(if(i<3,i+1,1))-VYA(i))^2 )
 forall(i in B) SideLengthsB(i):= 
   sqrt( (VXB(if(i<3,i+1,1))-VXB(i))^2 + (VYB(if(i<3,i+1,1))-VYB(i))^2 )

 ! Compute angles
 forall(i in A) do
   CosA(i):= ( (VXA(if(i<3,i+1,1))-VXA(i))*(VXA(if(i>1,i-1,3))-VXA(i)) +
               (VYA(if(i<3,i+1,1))-VYA(i))*(VYA(if(i>1,i-1,3))-VYA(i)) ) / 
             SideLengthsA(i) / SideLengthsA(if(i>1,i-1,3)) 
   AnglesA(i) := arccos(CosA(i)) 
   AnglesNA(i):= 3/2*M_PI - AnglesA(i) 
   AnglesNA(i):= 3/2*M_PI - AnglesA(i) 
   CosNA(i):= cos(AnglesNA(i))
   SinNA(i):= sin(AnglesNA(i))
 end-do

! Construct the normal vector onto sides of A
 forall(i in A) do
   NX(i):= ( VYA(if(i<3,i+1,1))-VYA(i) ) / SideLengthsA(i) 
   NY(i):= - ( VXA(if(i<3,i+1,1))-VXA(i) ) / SideLengthsA(i)

   AlphaA(i):= NX(i) 
   BetaA(i) := NY(i) 
   GammaA(i):= - ( NX(i) * VXA(i) + NY(i) * VYA(i) )
 end-do

 ! writeln("Normal vector,d:", NX,NY,array(i in A)(NX(i)*VXA(i)+NY(i)*VYA(i)))

! **** Optimization problem ****
 declarations
   xB: mpvar             ! x-position of triangle B point belonging to convex hull
   yB: mpvar             ! y-position of triangle B point belonging to convex hull
   xS: array(S) of mpvar ! Vertex S of the convex hull (x-coordinate)
   yS: array(S) of mpvar ! Vertex S of the convex hull (y-coordinate)
   lPCH: mpvar           ! Length of the convex hull perimeter (objective function)
   lS: array(S) of mpvar ! Side length of convex hull edge S
   fS: array(S) of mpvar ! Orientation angle of convex hull edge S
   fB: mpvar             ! Angle of triangle B (=convex hull vertex) with x-axis
   cfB,sfB: mpvar
   cfS,sfS: array(S) of mpvar 
   wx,wy: array(B) of mpvar ! Translated and rotated vertex B of triangle B (x/y-coordinate)
 end-declarations

 xB is_free; yB is_free; lPCH is_free
 forall(s in S) xS(s) is_free
 forall(s in S) yS(s) is_free
 cfB is_free; sfB is_free
 forall(s in S) cfS(s) is_free
 forall(s in S) sfS(s) is_free
 forall(i in B) wy(i) is_free

 cfB = cos(fB); sfB = sin(fB)
 forall(s in S) cfS(s) = cos(fS(s))
 forall(s in S) sfS(s) = sin(fS(s))

! Pythagorean theorem for cos(fB) and sin(fB)
 TrigIDfB:= sfB^2 + cfB^2 = 1

! Pythagorean theorem for cos(fS(S)) and sin(fS(S))
 forall(s in S) TrigIDfS(s):= cfS(s)^2 + sfS(s)^2 = 1

 forall(s in S) GlueX(s):= xS((s+1) MOD sz) = xS(s) + lS(s) * cfS(s) 
 forall(s in S) GlueY(s):= yS((s+1) MOD sz) = yS(s) + lS(s) * sfS(s) 

 forall(i in A, s in S) 
   BelongingA(i,s):= - (VXA(i)-xS(s)) * sfS(s) + (VYA(i)-yS(s)) * cfS(s) >= 0 

 forall(i in B, s in S) BelongingB(i,s):= - (wx(i)-xS(s)) * sfS(s) +
                     (wy(i)-yS(s)) * cfS(s) >= 0 

! Defining vertex coordinates wx and wy for new position of triangle B
 forall(i in B) DefwxB(i):= wx(i) = xB + VXB(i)*cfB - VYB(i)*sfB 
 forall(i in B) DefwyB(i):= wy(i) = yB + VXB(i)*sfB + VYB(i)*cfB 

! Objective function
 Obj:= lPCH = sum(s in S) lS(s)

! Ensure that triangle T1=A and triangle T2=B do not overlap
 forall(i in B) 
   Nonoverlaping(i):=  AlphaA(3)*wx(i) + BetaA(3)*wy(i) + GammaA(3) >= 0

 SmallEps:= 0.00125 

! Equation Convexity
 forall(s in S)
   Convexity(s):= -(xS((s+2) MOD sz)-xS(s))*sfS(s)+(yS((s+2) MOD sz)-yS(s))*cfS(s) >= SmallEps

! Bounds on the rotated vertices of triangle B
 forall(i in B) do
   -20 <= wx(i); wx(i) <= 20
   -20 <= wy(i); wy(i) <= 20
 end-do

! Bounds on the vertices of the convex hull
 forall(s in S) do
   0 <= xS(s); xS(s) <= 14 
   -5 <= yS(s); yS(s) <= 8 
 end-do

! Upper bound on the rotation angle of the convex hull vertices
 forall(s in S) fS(s) <= 2*M_PI

! Bounds on cos and sin
 forall(s in S) do
   -1 <= cfS(s); cfS(s) <= 1 
   -1 <= sfS(s); sfS(s) <= 1
 end-do
 -1 <= cfB; cfB <=1 
 -1 <= sfB; sfB <=1 

! Bounds on the vertex C of triangle B
 xB >= 0 ; xB <= 14
 yB >= 0 ; yB <=  8 

! Break symmetry by fixing one convex hull vertex
 xS(0) = 10 
 yS(0) = -5 

 setparam("XNLP_verbose",true)
 setparam("XNLP_MAXTIME", 3600)
! Use multi-start heuristic
 addmultistart("random points", XNLP_MSSET_INITIALVALUES, 50)

 minimize(lPCH)
 writeln("Solution (convex hull perimeter): ", getobjval)
 writeln("B: convex hull point: x:",xB.sol, " y:", yB.sol, "   f:", fB.sol)
 forall(i in B)
    writeln("   vertex ", i, " x:",wx(i).sol, " y:", wy(i).sol)
 forall(s in S)
   writeln("Hull vertex ", s,":  x:",xS(s).sol, " y:", yS(s).sol, "   f:", fS(s).sol,
           " l:", lS(s).sol)

 ! **** Draw result as SVG graphic ****
 svgaddgroup("A", "Triangle A")
 svgsetstyle(SVG_FILL, SVG_CURRENT)
 svgsetstyle(SVG_FILLOPACITY, 0.25)
 svgsetstyle(SVG_STROKEWIDTH,0.1)
 svgaddpolygon(sum(i in A) [VXA(i),VYA(i)])
 svgaddgroup("B", "Original triangle B", SVG_YELLOW)
 svgsetstyle(SVG_STROKEWIDTH,0.1)
 svgsetstyle(SVG_FILL, SVG_CURRENT)
 svgsetstyle(SVG_FILLOPACITY, 0.25)
 svgaddpolygon(sum(i in B) [VXB(i),VYB(i)])
 svgaddgroup("B2", "Shifted triangle B", SVG_ORANGE)
 svgsetstyle(SVG_STROKEWIDTH,0.1)
 svgsetstyle(SVG_FILL, SVG_CURRENT)
 svgsetstyle(SVG_FILLOPACITY, 0.25)
 svgaddpolygon(sum(i in B) [wx(i).sol,wy(i).sol])
 svgaddgroup("S", "Convex Hull", SVG_GREEN)
 svgsetstyle(SVG_STROKEWIDTH,0.1)
 svgaddpolygon(sum(s in S) [xS(s).sol,yS(s).sol])

 svgsetgraphviewbox(-10,-10,25,25)
 svgshowgraphaxes(true)

! Display the graph
 svgrefresh
 svgsave("ch2tri.svg")
! Wait for display window to close
 svgwaitclose("Close browser window to terminate model execution.", 1)

end-model

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