P a r t 2 2 I n t e g e r p r o g r a m m i n - - PowerPoint PPT Presentation

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r a m m i n g p r

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Terminology

Discrete optimization problems: If some or all variables can only take

• n certain values.

Example:

min x f (x) subject to Ax≥b x1∈{1,2.5,3.75, 17} x2∈{2, 4,6,8}

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Terminology

Integer optimization problems: If some or all variables can only take

• n integer values.

Example:

min x f (x) subject to Ax≥b x1∈{1,2,3, 7} x2∈{2, 4,6,8}

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Terminology

Note: We can typically convert discrete optimization problems into integer optimization problems. Example: is equivalent to

min x f (x) subject to Ax≥b x1∈{1,2.5,3.75, 17} x2∈{2, 4,6,8} min x , y f (x) subject to Ax≥b x2∈{2, 4,6,8} y1 , y2 , y3 , y4∈{0,1} y1+ y2+ y3+ y4=1 x1=1 y1+2.5 y2+3.75 y3+17 y4

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Terminology

Integer linear program (ILP): If all variables can only take on integer values and the objective function is linear. Example:

min x 1, x 2 f (x)=c

T x

subject to Ax≥b x1∈{1,2,3, 7} x2∈{2, 4,6,8}

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Terminology

Mixed integer linear program (MILP): If some but not all variables can only take on integer values and the objective function is linear. Example:

min x 1, x 2, x 3 f ( x)=c

T x

subject to Ax≥b x1∈{1,2,3, 7} x2∈{2, 4,6,8}

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Terminology

Mixed integer nonlinear program (MINP): If some but not all variables can only take on integer values and the objective function is nonlinear. Example:

min x 1, x 2, x 3 f ( x) subject to Ax≥b x1∈{1,2,3, 7} x2∈{2, 4,6,8}

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Terminology

Zero-one linear program (ZOIP): If all variables can only take on zero or one values and the objective function is linear. Example: Note: ILPs can typically be converted into ZOIPs.

min x , y , z c

T x

subject to Ax≥b y1 , y2 , y3 , y4∈{0,1} z1 , z2, z 3, z4∈{0,1} y1+ y2+ y3+ y4=1 z1+ z2+ z3+ z4=1 x1=1 y1+2.5 y2+3.75 y3+17 y4 x2=2 z1+4 z2+6 z3+8 z 4

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Examples: The knapsack problem

Setting: Objects 1...N with values ci and weights wi. Goal: Pack maximal value subject to a weight constraint. Formulation:

maxx c

T x

subject to xi∈{0,1} w

T x = ∑i wi xi ≤ M

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Examples: The facility location problem

Setting: N potential facility locations to serve M clients. Fixed cost ci to

• pen a facility at i. Cost to serve client j from i is dij.

Goal: Select facility locations and assign clients to facilities while minimizing cost. Formulation: Note: The last constraint implies that we can only serve client j from location i if location i has actually been selected.

min xi, yij ∑i ci xi+∑i ∑ j dij yij subject to xi∈{0,1} yij∈{0,1} yij ≤ xi

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Examples: The traveling salesman problem (TSP)

Setting: N cities. Time to travel from i to j is tij. Goal: Find a tour through all cities that takes the shortest amount of time. Formulation: Note: The second constraint implies that each city has exactly two selected edges. The third that each proper subset of cities must have at least two edges selected to the rest of the cities.

min xij ∑i∑ j tij xij subject to xij∈{0,1} ∑ j xij=2 for every i ∑i∈S∑ j∉S xij≥2 for every S⊂{1...N },S≠∅

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The geometry of ILP problems

Consider the problem Feasible points form a lattice inside the polyhedron described by the linear inequalities: In general, we will have to try a substantial fraction of points to find the optimum!

min xi c

T x

subject to xi∈ℕ Ax≥b

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Solution strategies: Branch and bound

Consider the problem Think of this as a decision tree that we can enumerate:

• Pick x1=0
• Are constraints satisfiable? If yes, pick x2=0
• Are constraints satisfiable? If yes, pick x3=0
• …
• Choose x2=1
• Are constraints satisfiable? If yes, pick x3=0
• …

We visit every node of the tree. Keep track of the best one seen so far.

min xi c

T x

subject to xi∈{0,1} Ax≥b

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Solution strategies: Branch and bound

Consider the problem Think of this as a decision tree that we can enumerate. Improvement 1: If the cost so far exceeds the cost of the best path encountered, we need not continue searching in a sub-tree.

min xi c

T x

subject to xi∈{0,1} Ax≥b

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Solution strategies: Branch and bound

Consider the problem Think of this as a decision tree that we can enumerate. Improvement 2: If the cost so far plus a lower bound for the cost of the rest of the path exceeds the cost of the best path encountered, we need not continue searching in a sub-tree.

min xi c

T x

subject to xi∈{0,1} Ax≥b

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Solution strategies: Branch and bound

Think this through for the Traveling Salesman Problem: Consider four cities in a square around a desert, with travel times

• 1 – 2: 1 hour
• 2 – 3: 1 hour
• 3 – 4: 1 hour
• 1 – 4: 1 hour
• 1 – 3: 4 hours
• 1 – 4: 3 hours

The shortest tour is clearly 4 hours.