3.4 Relaxations and bounds

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1 3.4 Relaxations and bounds Consider a generic Discrete Optimization problem z = min{c(x) : x X} with an optimal solution x X. In general, the algorithms generate not only a decreasing sequence of upper bounds u 1 >... > u k z but also an increasing sequence of lower bounds l 1 <... < l k z. They terminate when (u k l k ) ε. Primal bounds For minimization problems, any feasible solution x X yields an upper bound u = c(x) on the optimal value, namely u z. In some cases, even finding a feasible solution may be challenging (NP-hard). Dual bounds To obtain lower bounds for minimization problems, we consider a relaxation of the problem. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

2 Quality guarantee: If x k is the best feasible solution found so far and l k the best dual bound, the termination criterion (c(x k ) l k ) ε, for a given ε > 0, guarantees that (c(x k ) z ) ε. For maximization problems, the primal (dual) bounds are lower (upper) bounds. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

3 Definition: Given a problem (P) z = min{c(x) : x X R n }, a problem (RP) z = min{ c(x) : x X R n } is a relaxation of P if X X c(x) c(x) for each x X. Proposition: If RP is a relaxation of P, z z. Proof: Let x be an optimal solution of P, then x X X and c(x ) c(x ) = z. Since x X, we have z c(x ). Proposition: Let x RP be an optimal solution of RP. If x RP is feasible for P (x RP X) and c(x RP ) = c(x RP ), then x RP is also optimal for P. We aim at a tradeoff between the bound quality (how tight is the upper/lower bound w.r.t. the optimal value z ) and the computational load needed to solve the relaxation. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

4 3.4.1 Different types of relaxations 1) Linear programming relaxation Definition: Given an arbitrary MILP (ILP) problem z ILP = min c t 1 x +ct 2 y A 1x +A 2y b x 0,y 0, integer its linear (programming) relaxation is the following LP problem: z LP = min c t 1 x +ct 2 y A 1x +A 2y b x 0,y 0 where the integrality constraints on the variables y j are omitted. Recall that the definition of the best formulation for a MILP (ILP) is closely related to that of linear relaxation: the stronger the formulation, the tighter the dual bound z LP provided by its linear relaxation. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

5 2) Relaxation by elimination A straightforward way to obtain a relaxation of a MILP problem is just to delete one ore more constraints. Examples: 1) Asymmetric TSP Delete the subtour elimination (cut-set) constraints and just keep the assignment constraints. 2) Multi-dimensional binary knapsack problem max s.t. n j=1 p jx j n j=1 w ijx j W i i {1,2,...,m} (1) x j {0,1} j {1,2,...,n} (2) If we delete all but one constraints, we obtain a standard binary knapsack problem. (Very) weak relaxations. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

6 3) Surrogate relaxation Idea: Take a linear combination of some constraints with multipliers λ i 0, and replace these constraints with the resulting (surrogate) constraint. Example: Multiple binary knapsack problem z mkp = max m n i=1 j=1 p jx ij s.t. n j=1 w jx ij W i i {1,2,...,m} (3) m i=1 x ij 1 j {1,2,...,n} (4) x ij {0,1} i, j (5) Given m knapsacks of capacities W i, select m disjoint subsets of items (one for each knapsack) so as to mazimize the total profit, while satisfying the capacity constraints. A surrogate relaxation: z S(λ) = max s.t. m n i=1 j=1 p jx ij m i=1 λ n i j=1 w jx ij m i=1 λ iw i (6) m i=1 x ij 1 j {1,2,...,n} (7) x ij {0,1} i, j (8) a knapsack problem with m copies of each item j (i-th copy has weight λ i w j and profit p j ) and at most one copy of each item can be selected. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

7 z S(λ) = max s.t. m n i=1 j=1 p jx ij m n i=1 j=1 (λ iw j )x ij m i=1 λ iw i (9) m i=1 x ij 1 j {1,2,...,n} (10) x ij {0,1} i, j (11) Since for each item j a copy i with smallest λ i is more convenient, the problem is a standard binary knapsack problem with capacity W = m i=1 λ iw i. To look for a multiplier vector λ 0 providing the tightest (smallest) upper bound, we may solve the surrogate dual problem: min λ 0 z S(λ). Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

8 4) Lagrangian relaxation Often the linear relaxation and the relaxation by elimination provide weak bounds (e.g., when deleting the connectivity constraints in TSP, and demand constraints in UFL). Idea: Eliminate the difficult constraints and add, for each one of them, a term in the objective function with a multiplier u which penalizes its violation and that is 0 (max problem) for all feasible solutions. Example: Multiple binary knapsack problem z mkp = max s.t. m n i=1 j=1 p jx ij n j=1 w jx ij W i i {1,2,...,m} (12) m i=1 x ij 1 j {1,2,...,n} (13) x ij {0,1} i, j (14) Lagrangian relaxation of the cardinality constraints (13): z L(u) = max m n i=1 j=1 p jx ij + n j=1 u j(1 m i=1 x ij) s.t. n j=1 w jx ij W i i {1,2,...,m} (15) x ij {0,1} i, j (16) with multipliers u j 0 for all i, so that z L(u) is an upper bound for z mkp, for every u 0. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

9 Since m n n m p j x ij + u j (1 x ij ) = i=1 j=1 j=1 i=1 m n n (p j u j )x ij + u j, i=1 j=1 j=1 in the Lagrangian subproblem each item j has profit p j = p j u j, weight w j and can be inserted in several knapsacks. The relaxed problem is thus equivalent to m independent binary knapsack problems. For i = 1,2,...,m, we have z i = max n j=1 p jx j s.t. n j=1 w jx j W i (17) and z L(u) = m i=1 z i + n j=1 u j x j {0,1} j {1,2,...,n} (18) To find a multiplier vector u providing the tightest Lagrangian bound, we can solve the Lagrangian dual problem: min u 0 z L(u). Lagrangian relaxation will be discussed in detail in Section 3.6. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

10 Comparison between the previous four types of relaxations (linear, elimination, surrogate and Lagrangian)? Do some of them dominate others? Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

11 5) Combinatorial relaxations i) Asymmetric TSP Given a directed graph G = (V,A) with V = {1,2,...,n} and a cost c ij R associated to each arc (i,j) A, determine a Hamiltonian circuit of minimum total cost. The set of all the assignments, namely X = {x {0,1} A : i:(i,j) A x ij = 1 j, is a relaxation of the set of all the Hamiltonian circuits. j:(i,j) A x ij = 1 i}, ii) Knapsack problem The set n X = {x Z n + : a j x j b } j=1 is a relaxation of X = {x Z n + : n j=1 a jx j b}. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

12 iii) Symmetric TSP Definition: Given an undirected graph G = (V,E) with V = {1,...,n}, a 1-tree di G is a subgraph which contains two edges incident to node 1, together with the edges of a spanning tree with respect to the nodes {2,...,n}. Example: Observation: The set of all the 1-trees is a relaxation of the set of all the Hamiltonian cycles. To find a minimum cost 1-tree: determine a minimum cost spanning tree on nodes {2,...,n} by using a greedy algorithm (Kruskal, Prim,...), select two edges incident in node 1 with smallest cost. Kruskal algorithm for minimum spanning tree: Consider the edges in the order of non decreasing cost. At each step, discard the edge if it creates a cycle with the previously selected edges. Stop when the selected edges cover all the nodes. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

13 3.4.2 Heuristics for primal bounds 1) Greedy methods A feasible solution is constructed piece by piece from scratch. At each step, a piece is selected among the available ones that yield the best local profit, and previous choices are not reconsidered. Example 1: Knapsack problem z ILP = max 16x 1 +22x 2 +12x 3 +8x 4 s.t. 5x 1 +7x 2 +4x 3 +3x 4 14 x 1,...,x 4 {0,1} The variables (items) are ordered by non-increasing profit/volume ratios (p j /a j ) : x 1 x 2 x 3 x 4 p j a j p j /a j Consider items in that order and select (x j = 1) those that do not violate the (residual) capacity constraint, skip the others (x j = 0). Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

14 Feasible solution obtained with the greedy procedure: x = (1,1,0,0) with z greedy = 38. Optimal integer solution: x = (0,1,1,1) with z ILP = 42. Clearly z greedy z ILP. Worst case? Example 2: Symmetric TSP with complete graph Nearest neighbor heuristic: Start from any node, at each step insert the closest node not yet visited, come back to the starting node. Complexity: O(n 2 ), where n is the number of nodes. See TSP algorithms in action at Empirical performance: on TSPLIB(rary) instances it yields tours whose average cost is about 1.26 times that of optimal tours. Worst-case performance: there are instances for which it finds tours that are arbitrarily worse than the optimal tours. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

15 2) Local search methods Consider a generic optimization problem min c(x) x X and try to iteratively improve a current feasible solution. Define, for any feasible solution x, a neighborhood N(x), i.e., a subset of nearby feasible solutions. Start from an initial feasible solution x 0. At each iteration: - Find a best solution x in the neighborhood N(x k ) of the current solution x k. - If c(x ) < c(x k ) then x k+1 = x, otherwise return x k which is a local minimum w.r.t. the neighborhood N(x k ) under consideration. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

16 Example: 2-opt heuristic for Symmetric TSP Given G = (V,E) and a current tour (Hamiltonian cycle) H E. For any given pair of nonadjacent edges e 1 and e 2, try to delete them and replace them by the two unique alternative edges that recombining the two paths into a new tour H. The 2-opt neighborhood N(H) contains all the tours that can be obtained by such a 2-interchange. If c(h ) < c(h) for such a 2-interchange then H becomes the current solution, otherwise H is a local minimum w.r.t the 2-opt neighborhood. Complexity: O(n 2 ) with n = V. Also k-opt for k = 3, with complexity O(n 3 ). Empirical performance: on TSPLIB instances 2-opt provides tours about 1.06 times the optimum, while 3-opt tours 1.04 times the optimum. For step-by-step application see Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

17 Metaheuristics To try to escape from local optima and improve upon local search heuristics, there are metaheuristics such as Simulated Annealing, Tabu Search or Genetic algorithms. Simulated Annealing: Allow occasional moves that yield feasible solutions with worse objective function values. The probability of performing such worsening moves depends on the difference in objective function value. Tabu Search: Allow moves to the best neighboring solution of the current feasible solution even if it has a worse value. Cycling (reconsidering previously examined feasible solutions) is avoided by forbidding to undo recent moves (by making them tabu) for a certain number of iterations. Edoardo Amaldi (PoliMI) Ottimizzazione A.A / 17

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