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/*****************************************************************************
* *
* ************************************************************************* *
* *** *** *
* *** ~*~ SIMPLEX ~*~ *** *
* *** *** *
* *** A simple implementation of the simplex *** *
* *** algorithm for Linear Programming for Maxima. *** *
* *** *** *
* *** This file provides functions minimize_lp and maximize_lp. This *** *
* *** file is part of the simplex package for Maxima. *** *
* *** *** *
* *** *** *
* *** Copyright: Andrej Vodopivec <andrejv@users.sourceforge.net> *** *
* *** Version: 1.01 *** *
* *** License: GPL *** *
* *** *** *
* ************************************************************************* *
* *
* Demo *
* ===== *
* *
* 1) We want to minimize x with constraints y>=x-1, y>=-x-1, y<=x+1, y<=1-x *
* and y=x/2 *
* *
* Solution: *
* *
* load("simplex"); *
* minimize_lp(x, [y>=x-1, y>=-x-1, y<=x+1, y<=1-x, y=x/2]); *
* => [-2/3, [x=-2/3, y=-1/3]] *
* *
* *
* 2) If any variable is known to be positive, you should add an optional *
* argument to minimize_lp/maximize_lp. *
* We want to maximize x+y subject to x>=0, y>=0, y<=-x/2+3, y<=-x+4. *
* *
* Solution: *
* *
* maximize_lp(x+y, [y<=-x/2+3, y<=-x+4], [x, y]) *
* => [4, [x = 2, y = 2]] *
* *
*****************************************************************************/
define_variable(nonnegative_lp, false, boolean,
"Assume all variables are non-negative!")$
alias(nonegative_lp, nonnegative_lp)$
define_variable(return_input_lp, false, boolean,
"Return the input to linear program, not solution!")$
/*****************************************************************************
* *
* The minimize_lp function. *
* *
*****************************************************************************/
minimize_lp(target, constraints, [pozitive]) := block(
[var : [], A, b, c, inequalities:0, count, eq, t:expand(target), sol, s,
nonpozitive : 0, j, tmpvar, keepfloat:true],
/*******************************************************************
* Get the list of variables *
*******************************************************************/
for e in constraints do (
tmpvar : listofvars(e),
for v in tmpvar do
if not(member(v, var)) then var : cons(v, var),
if op(e)#"=" then inequalities : inequalities+1
),
if length(pozitive)>0 then (
if listp(pozitive[1]) then pozitive : pozitive[1]
else if pozitive[1]='all then pozitive : copylist(var)
)
else if nonnegative_lp=true then
pozitive : copylist(var),
for v in var do
if not(member(v,pozitive)) then nonpozitive : nonpozitive+1,
/*******************************************************************
* Setup A and b for linear program *
*******************************************************************/
b : makelist(0, i, 1, length(constraints)),
A : zeromatrix(length(constraints), length(var)+nonpozitive+inequalities),
count : 0,
for i:1 thru length(constraints) do (
eq : lhs(part(constraints,i))-rhs(part(constraints,i)),
j : 1,
for v in var do (
A[i,j] : ratcoeff(eq, v),
if not(constantp(A[i,j])) then
error("Error: constraint not linear (1).",A[i,j]),
eq : subst(v=0,eq),
j : j+1,
if not(member(v,pozitive)) then (
A[i,j] : -A[i,j-1],
j : j+1
)
),
if op(constraints[i])="<=" or op(constraints[i])="<" then (
count : count+1,
A[i, length(var)+nonpozitive+count] : 1
)
else if op(constraints[i])=">=" or op(constraints[i])=">" then (
count : count+1,
A[i, length(var)+nonpozitive+count] : -1
)
else if op(constraints[i])#"=" then
error("Error: not a proper constraint:", constraints[i]),
b[i] : -eq,
if not(constantp(b[i])) then
error("Error: constraint not linear (2).",b[i])
),
/*******************************************************************
* Setup c for linear program *
*******************************************************************/
c : makelist(0, i, 1, length(var)+nonpozitive+count),
j : 1,
for v in var do (
c[j] : ratcoeff(t, v),
if not(constantp(c[j])) then
error("Error: cost function not linear."),
t : subst(v=0,t),
j : j+1,
if not(member(v,pozitive)) then (
c[j] : -c[j-1],
j : j+1
)
),
if not(constantp(t)) then
error("Error: cost function not linear in constrained variables."),
if return_input_lp then return([A, b, c]),
/*******************************************************************
* Solve the linear program *
*******************************************************************/
sol : linear_program(A, b, c),
if not(listp(sol)) then sol
else (
if sol[2]=-inf then [-inf, []]
else (
s : [],
j : 1,
for v in var do (
if member(v,pozitive) then (
s : append(s, [v=sol[1][j]]),
j : j+1
)
else (
s : append(s, [v=sol[1][j]-sol[1][j+1]]),
j : j+2
)
),
[sol[2]+t, s]
)
)
)$
maximize_lp(target, constraints, [pozitive]) := block(
[sol : apply(minimize_lp, append([-target], [constraints], pozitive))],
if not(listp(sol)) then sol
else [-sol[1], sol[2]]
)$
/*****************************************************************************
* *
* %functions are for debugging purposes. *
* *
*****************************************************************************/
%prepare_standard_form_lp(target,constraints,pozitive) := block([listconstvars : false,
vars, slack_vars, nonpozitive_vars, nonpozitive_vars0, nonpozitive_subs, slack_vars0, nonpozitive_subs0, vars0],
vars : listofvars(append([target],constraints)),
slack_vars : map(lambda([e],gensym()), constraints),
nonpozitive_vars : sublist(vars,lambda([e], not(member(e,pozitive)))),
nonpozitive_subs : block([mgensym : lambda([x], gensym(printf(false,"~a_",x)))],
map(lambda([x], x=mgensym(x)-mgensym(x)), nonpozitive_vars)),
constraints : block([ltoreq : lambda([l,r], -l >= -r)],
subst(["<"=ltoreq, "<="=ltoreq], constraints)),
constraints : map(lambda([e,s], if op(e)="=" then e else lhs(e)-s-rhs(e)),constraints,slack_vars),
slack_vars0 : block([cv:listofvars(constraints)],
sublist(slack_vars, lambda([x],member(x,cv)))),
nonpozitive_vars0 : listofvars(map('rhs,nonpozitive_subs)),
vars0 : block([vars:vars,nonpozitive_subs1:map('lhs,nonpozitive_subs)],
for v in nonpozitive_subs1 do vars:delete(v,vars), vars),
[target, constraints] : subst("="="-", subst(nonpozitive_subs, [target,constraints])),
[target, constraints, vars0, nonpozitive_subs, nonpozitive_vars0, slack_vars0]
)$
%minimize_lp(target, constraints, [pozitive]) := block([A, b, c, vars0, vars, tgt, vals, lp,
nonpozitive_subs, nonpozitive_vars0, slack_vars0, mcoefmatrix],
local(mcoefmatrix),
mcoefmatrix(l,v) := apply('matrix,outermap(coeff,l,v)),
pozitive : if pozitive=[] then [] else if listp(part(pozitive,1)) then part(pozitive,1) else if part(pozitive,1)='all or nonnegative_lp then copylist(block([listconstvars:false], listofvars(constraints))) else error("Unrecognized input."),
[target, constraints, vars0, nonpozitive_subs, nonpozitive_vars0, slack_vars0] : %prepare_standard_form_lp(target, constraints, pozitive),
vars : append(vars0, nonpozitive_vars0, slack_vars0),
lp : linear_program(mcoefmatrix(constraints,vars), -subst(map(lambda([x],x=0),vars), constraints), first(args(mcoefmatrix([target],vars)))),
if atom(lp) then lp else ([vals,tgt] : lp,
block([maperror:false, mapprint:false],
[tgt, append(map("=",vars0,vals),subst(map("=",nonpozitive_vars0,rest(vals,length(vars0))),nonpozitive_subs))])));
%maximize_lp(target, constraints, [pozitive]) := block([tgt, vals, lp],
lp : %minimize_lp(-target, constraints,first(pozitive)),
if atom(lp) then lp else ([tgt,vals] : lp,
[-tgt, vals]))$
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