The assumed global optimum solution obtained in linear programming is not an assumed characteristic of separable linear programming. Separable programming is non-linear programming and must possess certain sufficient conditions for a global optimum to be obtained. The global optimum conditions for separable programming are set forth.Research Methods/ Statistical Methods,

Kilmer, Richard L.

English

Research Papers in Economics

Optimality and Separable  Linear
Programming: An
Additional  Reminder
Richard L.  Kilmer
The  assumed  global optimum  solution  obtained  in  linear  programming  is not  an
assumed characteristic  of separable linear programming.  Separable programming is non-
linear programming and must possess certain sufficient conditions  for a global optimum
to be obtained.  The global optimum  conditions for separable programming  are setforth.
Separable linear  programming is  a method
for  solving  nonlinear  problems  by  using the
simplex  algorithm  employed  in  linear  pro-
gramming.  Its  use  in agricultural  economics
is  illustrated  by the  Blakley  and Kloth  study
of  plant  location  and  the  Holland  and
Baritelle study of school location.  However,  a
shortcoming of separable linear programming
is  the risk of not obtaining  a global optimum
solution.  Neither  of  the  above  studies  re-
ported  information  on the  likelihood  of hav-
ing obtained non-global  solutions.  While this
problem  is  reasonably  well  documented  in
literature  on  quantitative  methods,  it  is  re-
examined and illustrated in the following dis-
cussion  to  help  assure  the  proper  use  of
separable programming in applied research.
Sufficient conditions  for a global  optimum
solution  to  a  mathematical  programming
problem  are  predicated  on a particular  com-
bination  of convex and concave  functions.  In
a mathematical programming  problem where
the objective  is  to:
(1)  Maximize f(xl,...  ,xn)
Richard  L.  Kilmer  is  Assistant  Professor  of Food  and
Resource  Economics,  University  of Florida.
Florida  Agricultural  Experiment  Station  Journal  Series
No.  921.
The  author has benefited  from  helpful  comments  by
Takashi  Takayama,  Thomas  Spreen,  Ronald  Ward,  the
Editor and two anonymous Journal reviewers.
subject  to:
(2)  gi(X)  bi for i =  ,...,  n;
X = (X,  ...  , Xn)
(3)  X>0
the sufficient  conditions for a global optimum
are:
(4)  f(xl,...  ,xn)  is  a  concave  function
(convex  for minimization)
and
(5)  X = {xlgi >  bi for i=  1, 2,..., n, X>O
is  a convex  set.
The  Hessian  matrix  can  be used  to deter-
mine the  concavity  (convexity)  of the  objec-
tive function.  If the function  is  concave  (con-
vex)  the  Hessian  will  be  negative  (positive)
definite  or  semi-definite.
This  test  becomes  unwieldly  for  large
problems.  An  alternative  approach  is  to
determine  if  each  term  in  the  objective
function  is  concave  (convex)  by  using  the
Hessian  matrix.  For  example,  assume  the
n
functional  form  f(x) =  hi(X).  If  all  hi
i=l
functions  are  concave  (convex),  the  summa-
tion of all terms in the objective  function  f(X)
results  in  a  concave  (convex)  function.  The
functions  must  not  be  quasi-concave  or
quasi-convex.  Quasi functions  do  not  neces-
sarily preserve their quasi-concave  or convex
properties under addition  [Simmons,  p.  151].
81Western Journal of Agricultural  Economics
A  linear  programming  package  that  does
not contain  a separable programming  routine
may be  used to  approximate  a  mathematical
programming problem if the sufficient  condi-
tions  for  a  global  optimum  are  met.  The
characteristics  of a separable  nonlinear  func-
tion  are  introduced  into  a  linear  program
through  a  series of linear segments.'  The  re-
sulting  optimum  is  global  for  the  linearized
problem  that  approximates  the original  non-
linear problem.  By increasing  the number  of
linear  segments for  each function,  the global
optimum  of the  approximating  problem  will
converge  on the global  optimum of the origi-
nal nonlinear  problem.
A  linear  programming  package  that  con-
tains  a  separable  programming  routine  may
be  used  to  approximate  the  mathematical
programming problem with or without meet-
ing the  sufficient  conditions  for  a  global  op-
timum. 2 However,  a  global  optimum  is  not
assured.  If the sufficient conditions  are met, a
global  optimum  will be obtained  if the prob-
lem  contains  "a  relatively  small  number  of
nonlinear  constraints"  [IBM,  p.  238].  A
"small number" is undefined.  If the sufficient
conditions  are  not  met,  the global  optimum
will  remain  unidentified  because  of the  un-
known  number  of local  optimums  that  com-
prise the set from  which a global optimum  is
chosen.
The following problem illustrates  the diffi-
culties  encountered  when  sufficient  condi-
tions  for a global  optimum  are not met.
(6)  Maximize  z = y - 3x
'Data preparation  instructions  are  found in  Simmons.
2Special  variables  are  used  to  approximate  a  nonlinear
function  by using  a series of linear segments.  The basis
entry  of special  variables  is restricted in order to incor-
porate the  characteristics  of the nonlinear  function.  For
example,  assume  a  nonlinear  function  with  10  special
variables.  The  special  variables  have  the following  re-
strictions:  (1) each variable  must be  bounded between
zero  and a finite  number,  (2) only  one special  variable
may be in  the basis with a value between zero and the
upper bound,  and (3) the special variables preceding the
basis variable  equal the upper bound and those  follow-
ing  the basis  variable  equal zero.
82
subject  to:
(7)
(8)
(9)
y - 8x < -4
y + 2x - 5x2 +  3 < 0
y,x >  O.
Equation eight is nonlinear and has both con-
cave  and  convex  arcs  (Figure  1);  therefore,
the  constraints  do  not  form  a  convex  set  as
required in equation  5.  Two  local optimums
exist  at  points  L  and  G  with  G  being  the
global  optimum.  Once  the  separable  pro-
gramming  algorithm  has  found  point  L,  a
point immediately to the right or left of point
L  will  result  in  a  lower  objective  function
value.  The algorithm  will indicate that an op-
timum has  been  reached.
The dual should be run to verify the results
of a separable programming  problem when  a
restricted  basis  entry  routine  is  used.  This
holds whether or not the sufficient conditions
for a global optimum  are  met. Further verifi-
cation  can  be  accomplished  by  solving  the
problem  with the  special  variables  set at the
upper (lower) bound, depending on the value
used  to obtain  the  initial solution.
If the  results  are  not verified,  parametric
variations  in  the  constraints,  the  objective
function,  or both may be used to determine if
a better local  optimum  exists.  If another area
of the feasible  region  increases the objective
function value,  the basis should be saved and
the  coefficients  of the  original  problem used
to solve the problem from the new basis. The
resulting optimum  may or  may  not  improve
the  previously  identified  local  optimums.
The  search  process  may  resume  again  until
an  "acceptable"  solution  is  obtained.  The
global  optimum  will  remain  unidentified
even  if it  is  the  "acceptable"  solution.  The
search  process  may  or  may  not  exhaust  all
local  optimums,  one  of  which  may  be  the
global  optimum.
In  conclusion,  separable  programming  is a
technique  through  which  nonlinear  pro-
gramming problems  may be solved using the
simplex  method.  The  assumed  global  op-
timum  characteristics  of linear programming
are  not  necessarily  preserved  in  separable
programming.  A  separable  programming
July 1978Separable Programming
y-8x  + 4  < 0
.
{  .'~~~~~~~
x
Figure 1.  Global and  local optima
83
Y
12.
11.5
11.
10.5
10.
9.5
9.
8.5
8.
7.5
7.
6.5
6.
5.5
5.
4.5
4.
3.5
3.
2.5
2.
1.5
1.
.5
KilmerWestern Journal of Agricultural  Economics
problem  must meet  specific  sufficient  condi-
tions in order  to  obtain  a  global  optimum.  If
the  sufficient  conditions  are  met,  separable
programs  should  be  solved  without  using  a
restricted  basis  separable  programming  op-
tion unless "a relatively  small number of non-
linear  constraints"  [IBM,  p.  238]  can  be de-
fined.  Finally,  articles dealing with separable
programming  must  establish  the  optimality
characteristics  of the solution  obtained.
References
Blakley,  Leo.  V.  and  Donald  W.  Kloth,  "Price  Align-
ment  and Movements  of  Class  1 Milk  Between  Mar-
kets",  American Journal of Agricultural Economics,
54(1972):  496-502.
Cooper,  L.  and  D.  Steinberg,  Introduction to Methods
of  Optimization,  W.  B.  Saunders  Co.:Philadelphia,
1970.
Holland,  David W.  and John  L.  Baritelle,  "School  Con-
solidation in Sparsely Populated Rural Areas:  A Separ-
able  Programming  Approach",  American Journal of
Agricultural  Economics,  57(1975):  567-575.
International  Business  Machines  Corporation,
Mathematical Programming System  - Extended
(MPSX),  and Generalized  Upper Bounding (CUB)
Program Description, White  Plains,  New  York,  Sec-
ond  Edition  (September  1972,  Revised  August  30,
1973),  1972.
Kloth,  Donald  W.  and  Leo.  V.  Blakely,  "Optimum
Dairy  Plant  Location  with  Economies  of  Size  and
Market-Share  Restrictions",  American Journal of Ag-
ricultural  Economics, 53(1971):  461-466.
Simmons,  Donald  M., Nonlinear Programming  For Op-
erations Research,  Prentice-Hall  Inc.:Englewood
Cliffs,  New  Jersey,  1975.
84
July 1978

Introduction to Methods of

Nonlinear Programming For Operations Research, Prentice-Hall Inc.:Englewood Cliffs,

Optimum Dairy Plant Location with Economies of Size and Market-Share Restrictions&quot;,

Price Alignment and Movements of Class 1 Milk Between Markets&quot;,

Programming System - Extended (MPSX), and Generalized Upper Bounding (CUB) Program Description, White Plains,

School Consolidation in Sparsely Populated Rural Areas: A Separable Programming Approach&quot;,

OPTIMALITY AND SEPARABLE LINEAR PROGRAMMING: AN ADDITIONAL REMINDER

OPTIMALITY AND SEPARABLE LINEAR PROGRAMMING: AN ADDITIONAL REMINDER

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