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IE 2082: Simplex Algorithms
Brady Hunsaker
August 30, 2006
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Simplex algorithm
Uses the general method of the previous lecture. Only considers certain solutions, called basic feasible solutions. Algebraically: Divide variables into a set of basic variables and a set of nonbasic variables. Set each nonbasic variable to one of its bounds (lower bound or upper bound) and solve for basic variables to satisfy constraints. Geometrically: The set of feasible solutions is a polyhedron, and basic feasible solutions are extreme points of the polyhedron. Issue: Is it sufficient to consider only basic feasible solutions?
Convert to standard form
Standard form for LPs is the following: Maximization problem Only ≤ constraints and nonnegativity constraints. All variables nonnegative. We use standard form to help us understand the algorithm, but most implementations do not actually do it this way.
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Standard form with slack variables added
max c 1 x 1 + c 2 x 2 + · · · + cnxn s.t. a 11 x 1 + a 12 x 2 + · · · + a 1 nxn + w 1 = b 1 a 21 x 1 + a 22 x 2 + · · · + a 2 nxn + w 2 = b 2 .. . am 1 x 1 + am 2 x 2 + · · · + amnxn + wm = bm x 1 , x 2 ,... , xn ≥ 0 w 1 , w 2 ,... , wm ≥ 0
Bases and dictionaries
For now we will not differentiate between slack variables (sometimes called auxiliary variables) and original variables (sometimes called structural variables).
max ζ = c 1 x 1 + c 2 x 2 + · · · + cnxn s.t. xn+1 = b 1 − a 11 x 1 − a 12 x 2 − · · · − a 1 nxn xn+2 = b 2 − a 21 x 1 − a 22 x 2 − · · · − a 2 nxn .. . xn+m = bm − am 1 x 1 − am 2 x 2 − · · · − amnxn x 1 , x 2 ,... , xn+m ≥ 0
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Basic solutions
We associate each dictionary with a basic solution. The m variables we solved for are the basic variables. The other variables are nonbasic variables. Set each nonbasic variable to one of its bounds. In this case, variables have lower bounds of 0 and no upper bounds, so all nonbasic variables are set to zero. (Later in the course we will consider the more general case.) Then it is easy to determine the values of the basic variables. Is the resulting basic solution feasible? In determining the values of basic variables, we ignored any lower and upper bounds on the variables. The basic solution is a basic feasible solution if all basic variables satisfy their bounds.
General form of a dictionary
Let B be the set of basic variable indices and N be the set of nonbasic variable indices.
ζ = ζ¯ +
j∈N
¯cj xj
xi = ¯bi −
j∈N
¯aij xj i ∈ B
The bars are placed over coefficients because their values are not the same as the initial values.
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Optimality conditions
Remember the second step of the general method: Is this solution good enough? How can we tell whether a basic feasible solution is optimal? Look at the row in the dictionary with the objective value: ζ = ζ¯ +
j∈N ¯cj^ xj^. The objective value of the current bfs is ζ¯.
Simplex pivot (continued)
Punchline: choose leaving variable xl such that ¯a ¯blkl is maximal.
Equivalently: choose xl such that ¯alk > 0 and ¯bl ¯alk is minimal. This is called the ratio test. Note that the choice of entering variable and leaving variable may not be unique! So “the” simplex algorithm is really a family of algorithms, each with its own pivot rules to select an entering and leaving variable. After choosing the entering and leaving variable, solve the leaving variable’s equation for the entering variable. Then plug this value for the entering variable into all the other equations to get the new dictionary.
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Pivot practice
