polytope

• the program at b₂ = 2 has at least one feasible vertex ×33
• the program at b₂ = 2 is bounded — an objective that increases without limit has no optimal vertex to draw ×33
• the program at b₂ = 5/2 has at least one feasible vertex ×32
• the program at b₂ = 5/2 is bounded — an objective that increases without limit has no optimal vertex to draw ×32
• the program at b₁ = 2 has at least one feasible vertex ×31
• the program at b₁ = 2 is bounded — an objective that increases without limit has no optimal vertex to draw ×31
• the program at b₁ = 5/2 has at least one feasible vertex ×30
• the program at b₁ = 5/2 is bounded — an objective that increases without limit has no optimal vertex to draw ×30
• constraint row 1 constrains at least one variable ×3
• constraint row 1 has one coefficient per variable ×3
• the right-hand side is a list of 2 to 2 numbers ×2
• a program here has between two and five constraint rows ×1
• a rational is a whole numerator over a non-zero whole denominator ×1
• a rational is never divided by zero ×1
• an intersection is kept exactly when it satisfies every constraint ×1
• each entry of the right-hand side is a whole number between 0 and 400 ×1
• each piece's slope is the dual variable on that piece ×1
• every complementary product is exactly zero ×1
• every constraint coefficient is a whole number of size at most 40 ×1
• every dual certificate at this optimum gives the same price for the swept constraint ×1
• every feasible primal value is at most every feasible dual value ×1
• every objective coefficient is a whole number of size at most 40 ×1
• every pair of constraints was formed, not a selection of them ×1
• no dual variable and no reduced cost is negative ×1
• no feasible lattice point beats the best vertex ×1
• no slack and no variable is negative at the optimum ×1
• the constraint whose right-hand side is swept is a whole number between 1 and 2 ×1
• the drawn dual region is convex ×1
• the drawn primal region is convex ×1
• the dual certificate is worth exactly what the primal optimum is worth ×1
• the dual is drawn only for a program with two constraints — with more, the dual polytope has more than two variables and is not a polygon in the plane ×1
• the dual optimum is unique, so each constraint has one price — a degenerate optimum, with more than two constraints through one point, has a whole set of them and no single table of products ×1
• the dual program has at least one feasible vertex ×1
• the dual program is bounded — an objective that increases without limit has no optimal vertex to draw ×1
• the exact arithmetic stays inside the safe integer range ×1
• the exact optimum and the decimal one agree ×1
• the exact optimum at the top of the sweep and the decimal one agree ×1
• the feasible vertices are in convex position ×1
• the high end of the sweep is a whole number between 1 and 400 ×1
• the low end of the sweep is a whole number between 0 and 400 ×1
• the multipliers reproduce the objective exactly ×1
• the objective has one coefficient per variable ×1
• the objective is not identically zero ×1
• the optimal vertex has a dual certificate — a non-negative multiplier on each binding constraint ×1
• the optimal vertex's coordinates are labelled clear of every dot, every value and the axis ticks ×1
• the optimum has a dual certificate ×1
• the optimum is linear across this interval, so the drawn segment is exact ×1
• the polytope figure's mode is one of primal, dual, slack, shadow ×1
• the primal maximum and the dual minimum are the same number ×1
• the primal program has at least one feasible vertex ×1
• the primal program is bounded — an objective that increases without limit has no optimal vertex to draw ×1
• the reduced cost is the multiplier on the variable's own non-negativity ×1
• the slope of the optimum equals the dual variable for the swept constraint ×1
• the sweep produced at least one linear piece ×1
• the sweep runs over between two and sixty units of the right-hand side ×1
• the sweep solved the program once at every whole right-hand side ×1
• the two optima agree as decimals too ×1
• there is one breakpoint between consecutive pieces ×1
• there is one product per constraint and one per variable ×1
• weak duality was checked on the whole cross product of the two vertex lists ×1

The whole library · What the figures prove