The Simplex Algorithm. An approach to optimization problems for linear real arithmetic constraints. Content. Optimization – what is that? The Simplex Algorithm – background Simplex form Basic feasible solved form / basic feasible solution The algorithm Initial basic feasible solved form.
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The Simplex Algorithm
An approach to optimization problems for linear real arithmetic constraints
The Simplex Algorithm
The Simplex Algorithm
e.g.
The Simplex Algorithm
Objective function f:
The Simplex Algorithm
a valuation θ (substituting variables by values):
solution of objective function using θ:
The Simplex Algorithm
preferred valuations:
if f(θ) < f(θ')
optimal solution:
if f(θ) < f(θ') for all solutions θ' θ
(there is no solution that is preferred to θ)
The Simplex Algorithm
Do all problems have an optimal solution?
The Simplex Algorithm
An optimization problem
Find the closest point to the origin satisfying the C. Some solutions and f value
Optimal solution
The Simplex Algorithm
The Simplex Algorithm
The Simplex Algorithm
Eugene Lawler (1980):
[Linear programming] is used to allocate resources, plan production, schedule workers, plan investment portfolios and formulate marketing (and military) strategies. The versatility and economic impact of linear programming in today's industrial world is truly awesome.
The Simplex Algorithm
Dantzig I:
The tremendous power of the simplex method is a constant surprise to me.
Dantzig II:
... it is interesting to note that the original problem that started my research is still outstanding - namely the problem of planning or scheduling dynamically over time, particularly planning dynamically under uncertainty. If such a problem could be successfully solved it could eventually through better planning contribute to the well- being and stability of the world.
The Simplex Algorithm
The Simplex Algorithm
An optimization problem showing contours of the objective function
if C has the form
The Simplex Algorithm
allowed conversions to get simplex form:
The Simplex Algorithm
An equivalent simplex form is:
An optimization problem showing contours of the objective function
The Simplex Algorithm
feasible = practicable, able to be carried out (durchführbar, anwendbar)
if all equations in CE (of the simplex form) have the form:
The Simplex Algorithm
The Simplex Algorithm
The Simplex Algorithm
Basic feasible solved form:
An equivalent simplex form is:
An optimization problem showing contours of the objective function
The Simplex Algorithm
The Simplex Algorithm
in other words:
The Simplex Algorithm
The Simplex Algorithm
Problem:
Which variables should be exiting resp. entering?
The Simplex Algorithm
a Yj with negative dj decreases f
The Simplex Algorithm
Exiting variable:
all bi's have to be 0
choose a Xi so that –bI/aIJ is a minimum of:
The Simplex Algorithm
Choose variable Z, the 2nd eqn is only one with neg. coeff
Choose variable Y, the first eqn is only one with neg. coeff
No variable can be chosen, optimal value 2 is found
starting from a problem in bfs form
repeat
Choose a variable y with negative coefficient in the obj. func.
Find the equation x = b + cy + ... where c<0 and -b/c is minimal
Rewrite this equation with y the subject y = -b/c + 1/c x + ...
Substitute -b/c + 1/c x + ... for y in all other eqns and obj. func.
until no such variable y exists or no such equation exists
if no such y exists optimum is found
else there is no optimal solution
The Simplex Algorithm
Basic feasible solution form: circle
Choose S3, replace using 2nd eq
Optimal solution: box
The Simplex Algorithm
idea:
this optimization problem should have an initial basic feasible solved form, which:
The Simplex Algorithm
add artificial variables and minimize on them:
The Simplex Algorithm
to get basic feasible solved form:
solve this problem
The Simplex Algorithm
possible outcomes:
The Simplex Algorithm
all z become parametric
The Simplex Algorithm
An equivalent simplex form is:
An optimization problem showing contours of the objective function
Original simplex form equations
With artificial vars in bfs form:
Objective function: minimize
Problem after minimization of objective function
Removing the artificial variables, the original problem
finding a basic feasible solution is exactly a constraint satisfaction problem
efficient constraint solver for linear inequalities
The Simplex Algorithm
Problem:
danger of pivoting back
Solution:
The Simplex Algorithm
We have seen that optimisations of linear real arithmetic constraints play an important role in many applications.
The Simplex Method which was introduced here provides a very efficient algorithm to determine whether there exists an optimal solution to linear real arithmetic constraints and if there exists one, to compute it.
The Simplex Algorithm
George B. Dantzig, Mukund N. Thapa
"Linear Programming I: Introduction"
Springer Verlag
Kim Marriott & Peter J. Stuckey
"Programming with Constraints: An Introduction"
MIT Press
The Simplex Algorithm
http://www.cs.mu.oz.au/~pjs/book/course.html
http://www-lehre.inf.uos.de/~sbitzer/clp
The Simplex Algorithm