QBUS2310 Chap.2 Linear Algebra, Convexity and Model Structure
Linear Algebra, Convexity and Model Structure
Optimisation models compress many variables and constraints into vector and matrix notation. That notation is not cosmetic: dimensions determine whether an expression is legal, rank explains whether active constraints identify a point, and convexity explains why local and geometric arguments can certify global behaviour. This chapter reviews the algebra and set language needed to read later formulations safely.
These definitions let a reader check dimension agreement and convexity before trusting a geometric argument. Matrix Product is an array operation defined only when the inner dimensions of the factors agree. Linear Independence is a property meaning no vector in a set can be formed as a nontrivial combination of the others. Convex Combination is a weighted average with nonnegative weights that sum to one.
Convex Set is a set containing the entire line segment between any two of its points. Convex Function is a function whose value at a convex combination does not exceed the same combination of endpoint values. Epigraph is the set of points lying on or above a function's graph.
The convexity argument above is confirmed as follows: Because linear inequalities are preserved by convex combinations, every point on the segment between two feasible plans is feasible. The argument requires weights that are nonnegative and sum to one.
What this chapter covers
- 01
Sets, membership and quantifiers
- 02
Vector and matrix dimensions
- 03
Transpose and matrix products
- 04
Linear independence and rank
- 05
Convex combinations
- 06
Convex sets and line segments
- 07
Convex and concave functions
- 08
Epigraphs and affine functions
Linear Algebra, Convexity and Model Structure worked example
- +1Check that the two plans have the same dimension and each satisfies every constraint.
- +1Form theta times the first plan plus one minus theta times the second, with theta between zero and one.
- +1Apply each linear inequality to the blend. The left side becomes the same weighted average of the two feasible left sides.
- +1Conclude that the blend remains within every bound. This is the line-segment test for a convex feasible set.
Key terms
- Matrix Product
- An array operation defined only when the inner dimensions of the factors agree.
- Linear Independence
- A property meaning no vector in a set can be formed as a nontrivial combination of the others.
- Convex Combination
- A weighted average with nonnegative weights that sum to one.
- Convex Set
- A set containing the entire line segment between any two of its points.
- Convex Function
- A function whose value at a convex combination does not exceed the same combination of endpoint values.
- Epigraph
- The set of points lying on or above a function's graph.
Linear Algebra, Convexity and Model Structure FAQ
Why does sets, membership and quantifiers matter?
Matrix Product: An array operation defined only when the inner dimensions of the factors agree. Linear Independence: A property meaning no vector in a set can be formed as a nontrivial combination of the others. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
How can I check linear independence and rank?
Linear Independence: A property meaning no vector in a set can be formed as a nontrivial combination of the others. Convex Combination: A weighted average with nonnegative weights that sum to one. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
What separates matrix product from linear independence?
Convex Combination: A weighted average with nonnegative weights that sum to one. Convex Set: A set containing the entire line segment between any two of its points. 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 convex sets and line segments?
Convex Set: A set containing the entire line segment between any two of its points. Convex Function: A function whose value at a convex combination does not exceed the same combination of endpoint values. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
How should I practise epigraphs and affine functions?
Convex Function: A function whose value at a convex combination does not exceed the same combination of endpoint values. Epigraph: The set of points lying on or above a function's graph. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
Exam move
Check dimensions before arithmetic. Sketch the line segment between two points whenever a convexity definition feels abstract, then translate the picture back into the weighted algebra. Rehearse the chapter method in this order: Check that the two plans have the same dimension and each satisfies every constraint. Form theta times the first plan plus one minus theta times the second, with theta between zero and one.
Apply each linear inequality to the blend. The left side becomes the same weighted average of the two feasible left sides. Conclude that the blend remains within every bound. This is the line-segment test for a convex feasible set.
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