Linear Program: Optimization Techniques and Geometric Interpretation
Explore linear programming concepts including optimizing objectives, linear relationships, geometric interpretations, and canonical forms. Understand how to convert constraints and solve problems efficiently using linear programming techniques.
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Presentation Transcript
Recap Shortest path Minimum spanning tree Maximum matching Common: Optimizing some objective (length of paths, weight of the tree, number of matches) Very different techniques Hope: A common technique that can solve all problems
Linear Relationships Inequalities that are linear in all parameters. Example. If d[u] = shortest path distance from s to u, then for any edge (u,v) ? ? ? ? + ?(?,?)
Linear Relationships Example 2: Let xi,j= 1 if course i is matched to classroom j, and 0 otherwise. Each classroom is matched to at most one course. ? ?=1 ??,? 1 ?
Linear Program Optimize a linear function (objective), under a set of linear inequality constraints. max2? + ? ? 0 ? 0 ? + ? 1
Geometric Interpretation Linear inequality Half planes y y y x x x ? + ? 1 ? 0 ? 0
Geometric Interpretation System of linear inequalities intersections y x Green: Feasible set. Point in Green: feasible solution.
Geometric Interpretation Objective function Direction of gravity y x max2? + ?
Geometric Interpretation Optimal Point Lowest point Optimal solution: (x, y) = (1, 0), value = 2.
Canonical Form min ?,? ?? ? ? 0 c x x b A
Converting to Canonical Form Equality constraints (e.g. x+y = 3) Solution: Split into two constraints ? + ? 3 ? + ? 3
Converting to Canonical Form Free variable: x may or may not be nonnegative. Solution: Split x into x1 and x2 ? = ?1 ?2 ?1 0 ?2 0
Using LP to solve graph problems Edge (i, j): Course i can be scheduled into classroom j Courses Classrooms Solution: A set of edges that do not share any vertices. (a matching)
Using LP to solve graph problems 1 2 1 2 3 3 1 2 4 5 6 4 1 1 3 2 7 8 9 1 1
