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

P a r t 2 2 I n t e g e r p r o g r a m m i n g p r o b l e m s 1 4 0

Terminology Discrete optimization problems: If some or all variables can only take on certain values. Example: min x f ( x ) subject to Ax ≥ b x 1 ∈{ 1,2.5,3.75, 17 } x 2 ∈{ 2, 4,6,8 } 1 4 1

Terminology Integer optimization problems: If some or all variables can only take on integer values. Example: min x f ( x ) subject to Ax ≥ b x 1 ∈{ 1,2,3, 7 } x 2 ∈{ 2, 4,6,8 } 1 4 2

Terminology Note: We can typically convert discrete optimization problems into integer optimization problems. Example: min x f ( x ) subject to Ax ≥ b x 1 ∈{ 1,2.5,3.75, 17 } x 2 ∈{ 2, 4,6,8 } is equivalent to min x , y f ( x ) subject to Ax ≥ b x 2 ∈{ 2, 4,6,8 } y 1 , y 2 , y 3 , y 4 ∈{ 0,1 } y 1 + y 2 + y 3 + y 4 = 1 x 1 = 1 y 1 + 2.5 y 2 + 3.75 y 3 + 17 y 4 1 4 3

Terminology Integer linear program (ILP): If all variables can only take on integer values and the objective function is linear. Example: T x min x 1 , x 2 f ( x )= c subject to Ax ≥ b x 1 ∈{ 1,2,3, 7 } x 2 ∈{ 2, 4,6,8 } 1 4 4

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: T x min x 1 , x 2 , x 3 f ( x )= c subject to Ax ≥ b x 1 ∈{ 1,2,3, 7 } x 2 ∈{ 2, 4,6,8 } 1 4 5

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 x 1 ∈{ 1,2,3, 7 } x 2 ∈{ 2, 4,6,8 } 1 4 6

Terminology Zero-one linear program (ZOIP): If all variables can only take on zero or one values and the objective function is linear. T x Example: min x , y , z c subject to Ax ≥ b y 1 , y 2 , y 3 , y 4 ∈{ 0,1 } z 1 , z 2 , z 3 , z 4 ∈{ 0,1 } y 1 + y 2 + y 3 + y 4 = 1 z 1 + z 2 + z 3 + z 4 = 1 x 1 = 1 y 1 + 2.5 y 2 + 3.75 y 3 + 17 y 4 x 2 = 2 z 1 + 4 z 2 + 6 z 3 + 8 z 4 Note: ILPs can typically be converted into ZOIPs. 1 4 7

Examples: The knapsack problem Setting: Objects 1...N with values c i and weights w i . Goal: Pack maximal value subject to a weight constraint. Formulation: T x max x c subject to x i ∈{ 0,1 } T x = ∑ i w i x i ≤ M w 1 4 8

Examples: The facility location problem Setting: N potential facility locations to serve M clients. Fixed cost c i to open a facility at i . Cost to serve client j from i is d ij . Goal: Select facility locations and assign clients to facilities while minimizing cost. Formulation: min x i , y ij ∑ i c i x i + ∑ i ∑ j d ij y ij subject to x i ∈{ 0,1 } y ij ∈{ 0,1 } y ij ≤ x i Note: The last constraint implies that we can only serve client j from location i if location i has actually been selected. 1 4 9

Examples: The traveling salesman problem (TSP) Setting: N cities. Time to travel from i to j is t ij . Goal: Find a tour through all cities that takes the shortest amount of time. Formulation: min x ij ∑ i ∑ j t ij x ij subject to x ij ∈{ 0,1 } ∑ j x ij = 2 for every i ∑ i ∈ S ∑ j ∉ S x ij ≥ 2 for every S ⊂{ 1... N } ,S ≠∅ 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. 1 5 0

The geometry of ILP problems T x min x i c Consider the problem subject to x i ∈ℕ Ax ≥ b 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! 1 5 1

Solution strategies: Branch and bound T x min x i c Consider the problem subject to x i ∈{ 0,1 } Ax ≥ b Think of this as a decision tree that we can enumerate: ● Pick x 1 =0 ● Are constraints satisfiable? If yes, pick x 2 =0 ● Are constraints satisfiable? If yes, pick x 3 =0 ● … ● Choose x 2 =1 ● Are constraints satisfiable? If yes, pick x 3 =0 ● … We visit every node of the tree. Keep track of the best one seen so far. 1 5 2

Solution strategies: Branch and bound T x min x i c Consider the problem subject to x i ∈{ 0,1 } Ax ≥ b 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. 1 5 3

Solution strategies: Branch and bound T x min x i c Consider the problem subject to x i ∈{ 0,1 } Ax ≥ b 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. 1 5 4

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. 1 5 5