Part II
Linear Constrained Optimization
Chapter 16
SIMPLEX METHOD
Chia-Hui Chang (張嘉惠)
National Central
University
2008.11.10
Solving Linear System
Augmented matrix
Perform a sequence of elementary row operations
to obtain
Let be a solution to Ax=b, xBRm, xDRn-m
Any solution to Ax=b has the form
where is a solution to Ax=0.
Canonical Form of Ax=b
Canonical representation of the system with respect
to the basis a1, a2, …, am:
We call the canonical augmented matrix of the
system w.r.t. the basis a1, a2, …, am.
Canonical Augmented Matrix
Updating the Augmented Matrix
Replacing a basic variable with a nonbasic variable,
e.g. pivoting about the (p,q)th element, 1pm, m<q
n.
Old basis {a1, a2, …, am}
New basis {a1, …,ap-1, aq, ap+1 …, am}
The Simplex Algorithm - Idea
Idea: move from one basic feasible solution to another.
Assume we are given the basic feasible solution
Suppose we decide to make aq, q>m, a basic column
Multiplying by >0
Combine with
Selecting Entering Variable
Solution to Ax=b:
Original objective function
New objective function
Thus if then the new objective function can
be decreasing.
Selecting Entering Variable
(Cont.)
Theorem 16.2 A basic feasible solution is optimal if
and only if the corresponding reduced cost
coefficients are all nonnegative.
Proof: General solution to Ax=b:
objective function
Since xi 0 for a feasible solution, we have
If ci-zi 0 for all i=m+1, …, n, then any feasible solution x
will have objective function no smaller than z0.
ri=ci-zi is called the reduced cost coefficient
Entering
variable
Leaving
variable
Matrix Form of the Simplex Method
Objective function
Matrix Form of the Simplex Method (Cont.)
Elementary row operations on A
Elementary row operations on cT
Degenerate cases
Two Phases Simplex Method
Phase I: find initial basic feasible solution
Phase II: find optimal basic feasible solution
Phase I
Phase
II
Revised Simplex Method
Reasoning
Computation of current solution B-1b does
not require computation of B-1D (see slide
12).
Keep track of [B-1, B-1b]
Let yq=B-1aq and p=argmini {yio/yiq: yiq>0}
Updating [B-1, y0, yq] by pivoting at the p th
element of the last column
Phase I
Phase
II