QBUS2310 Chap.3 Feasible Geometry and Basic Solutions
Feasible Geometry and Basic Solutions
Each linear inequality cuts space into a halfspace, and their intersection forms a polyhedron. Geometry makes feasibility, binding constraints, and candidate optima visible. Algebra makes the same ideas scale beyond two dimensions through active-row rank and basic feasible solutions. The qualified extreme-point theorem is powerful, but it must not be overstated when a feasible region has no extreme points.
Knowing these terms lets a reader confirm which constraints are active at a candidate vertex. Halfspace is one side of a hyperplane, including its boundary, described by a linear inequality. Active Constraint is a constraint that holds with equality at the point being examined. Slack Variable is a nonnegative variable that converts a less-than inequality into an equality.
Basic Solution is a solution obtained by selecting enough independent active equations to determine the variables. Extreme Point is a feasible point that is not a proper convex combination of two distinct feasible points. Degeneracy is a basic feasible solution with more active restrictions than are needed to determine the point.
The candidate vertex above is verified by enumeration: The optimum is x=3 and y=2 with objective value 13. The result is certified by enumerating all feasible vertices of this bounded polygon and evaluating the linear objective at each.
What this chapter covers
- 01
Halfspaces and polyhedra
- 02
Active and inactive constraints
- 03
Slack and surplus
- 04
Vertices and extreme points
- 05
Basic solutions and rank
- 06
Basic feasible solutions
- 07
Degeneracy
- 08
Standard-form conversion
Feasible Geometry and Basic Solutions worked example
- +1Find axis intercepts and intersect the two resource boundaries. Retain only points satisfying every inequality.
- +1The feasible vertices are (0,0), (4,0), (3,2), and (0,5). Check each directly rather than trusting the sketch.
- +1Evaluate the objective: zero, twelve, thirteen, and ten. The largest value is thirteen at (3,2).
- +1At (3,2), both resource constraints are active and their normals are independent, which gives a basic feasible solution.
Key terms
- Halfspace
- One side of a hyperplane, including its boundary, described by a linear inequality.
- Active Constraint
- A constraint that holds with equality at the point being examined.
- Slack Variable
- A nonnegative variable that converts a less-than inequality into an equality.
- Basic Solution
- A solution obtained by selecting enough independent active equations to determine the variables.
- Extreme Point
- A feasible point that is not a proper convex combination of two distinct feasible points.
- Degeneracy
- A basic feasible solution with more active restrictions than are needed to determine the point.
Feasible Geometry and Basic Solutions FAQ
Why does halfspaces and polyhedra matter?
Halfspace: One side of a hyperplane, including its boundary, described by a linear inequality. Active Constraint: A constraint that holds with equality at the point being examined. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
How can I check vertices and extreme points?
Active Constraint: A constraint that holds with equality at the point being examined. Slack Variable: A nonnegative variable that converts a less-than inequality into an equality. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
What separates halfspace from active constraint?
Slack Variable: A nonnegative variable that converts a less-than inequality into an equality. Basic Solution: A solution obtained by selecting enough independent active equations to determine the variables. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
Which error is most likely around basic feasible solutions?
Basic Solution: A solution obtained by selecting enough independent active equations to determine the variables. Extreme Point: A feasible point that is not a proper convex combination of two distinct feasible points. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
How should I practise standard-form conversion?
Extreme Point: A feasible point that is not a proper convex combination of two distinct feasible points. Degeneracy: A basic feasible solution with more active restrictions than are needed to determine the point. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
Exam move
For a two-variable model, build a vertex table with coordinates, feasibility checks, active constraints, and objective value. For higher dimensions, replace the sketch with rank reasoning. Rehearse the chapter method in this order: Find axis intercepts and intersect the two resource boundaries. Retain only points satisfying every inequality. The feasible vertices are (0,0), (4,0), (3,2), and (0,5).
Check each directly rather than trusting the sketch. Evaluate the objective: zero, twelve, thirteen, and ten. The largest value is thirteen at (3,2). At (3,2), both resource constraints are active and their normals are independent, which gives a basic feasible solution.
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