An Introduction to Dual Ascent Heuristics

Transcription

1 An Introduction to Dual Ascent Heuristics

2 Introduction A substantial proportion of Combinatorial Optimisation Problems (COPs) are essentially pure or mixed integer linear programming. COPs are in general NP-hard. Hence, algorithms guaranteeing optimality need most often to be used on some kind of implicit enumeration, typically within the framework of branch-and-bound (BB). A successful BB-algorithm can be viewed as a lucky compromise between the quality of the bounds and the computational effort invested in obtaining them.

3 Bounding Techniques The most common bounding techniques are: Lagrangian Relaxation Linear programming relaxation Dual based procedures They are computationally more efficient than both Lagrangian and LP-relaxation techniques.

4 Primal min z p =cx s.t Ax b LP-Relaxation min z LP p =cx s.t Ax b Dual max z LP D =wb s.t wa c x {0,1} x 0 w 0 In dual ascent heuristics, an approximate solution to the primal problem and a feasible solution to the dual of an LP relaxation are constructed simultaneously; the performance guarantee is proved by comparing the values of both solutions.

5 Characteristics of a dual ascent technique Generating good lower bounds for the dual problem (maximization) relatively fast by solving the linear programming dual problem approximately. Identifying feasible network designs that serve as starting solutions for local improvement heuristics. Reducing the problem by eliminating some design variables.

6 General Discription Dual ascent heuristics solve the dual of the LPrelaxation approximately. In fact: They start with an initial solution. The solution will then iteratively improve according to an ascent strategy. In every iteration, the obtained solution is at least as good as the former one.

7 Researchers have successfully applied dual ascent to several network design related models including: Uncapacitated facility location problem (Bilde and Krarup, 1977; Erlenkotter, 1978; Van Roy and Erlenkotter, 1982). The generalized assignment problem (Fisher et al, 1986). The Steiner tree problem (Wong, 1984). The set covering problem (Kedia and Fisher, 1986). The set partitioning problem (Fisher and Kedia, 1986). In 1989, Balakrishnan et al, introduced a network design model formulation, together with a general dual-ascent framework, which generalized the former proposed dualascent methods.

8 In 2001, Rosenberg used the improved formulation by Balakrishnan et al, and introduced a different dual ascent strategy. In 2007, Canovas et al, proposed a dual ascent technique to solve a four indexed dual formulation of the uncapacitated multiple allocation hup location problem. In 2008, Boschetti et al, improved the former obtained lower bounds for the set partitionning problem, on the basis of a new dual ascent heuristic. In 2010, Bardossy and Raghavan proposed a heuristic that combines dual ascent and local search, which together yield strong lower and upper bounds for facility location problem.

9 Problem P min Problem DP max s.t (1) s.t (5) (2) (6) (3a) (3b) (4) (7) (8)

10 Dual Ascent Framework For any given vector w={w k ij, w k ji} that satisfies constraints (7) of DP, the best v-values are obtained by solving (5)-(6). Doing so, assume the subproblem SP k (w) corresponding to commodity k: max (9) s.t (10) Where The optimal value of this subproblem is the length of the shortest path from origin O(k) to destination D(k).

11 The ascent strategy Iteratively increase one or more w-values, so that: Constraints (7) remain feasible ( ) The shortest O-D path length v k D(k) increases for at least one commodity k K at each stage. Therefore, the dual ascent procedure seeks to selectively allocate the unabsorbed fixed charges s ij in order to increase the length of the shortest O-D path for one or more commodities at each iteration.

12 Question How to select the arc(s) whose slack must be allocated and how to allocate this slack to the various commodities? Different arc selection and slack allocation schemes give rise to different implementations of the dualascent method.

13 Alternative Implementation (1) Path Diversion: changes a single w-value at each iteration. Initialization: all w-values are set to zero, hence v k D(k) for all k K is the length of the shortest path from O(k) to D(k), using c k ij and c k ji as arc lengths. Ascent Iterations: Select a directed arc (i, j) and commodity k satisfying s ij > 0 & l k ij > v k D(k) Increase w k ij & v k D(k) by min{s ij, [l k ij-v k D(k)]}.

14 A point Selecting more arcs offers the potential for reducing the operating (or routing) costs at the expense of higher fixed costs. On the other hand, with fewer arcs in the design, the fixed costs are lower but the routing costs increase. Let l k ij denote the length of the shortest O-D path excluding arc (i, j) for each directed arc (i, j) and commodity k. Then l k ij > v k D(k)

15 Alternative Implementation (2) Labeling Method: simultaneously increases w-values that correspond to several arcs. Initialization: {i, j} A and k K, w k ij=0 & w k ji=0. {i, j} A, s ij =F ij. i N and k K, v k i=shortest path length from O(k) to node i. k K, N 1 (k)=n \{D(k)}, N 2 (k)={d(k)}. z D = k v k D(k). Set CANDIDATES = {k K :O(k) N 1 (k)}.

16 Ascent Iterations A. calculate the amount of w-increase (Step 1) Select a commodity k CANDIDATES Set, A(k)={(i, j) A: i N 1 (k), j N 2 (k)} A (k)={(i, j): c k ij + w k ij - (v k j-v k i) = 0, (i, j) A(k)} δ 1 =min{s ij : (i, j) A (k)} δ 2 =min{c k ij + w k ij v k j + v k i: (i, j) A (k) =A(k)\A (k)} δ=min{δ 1, δ 2 }

17 Ascent Iterations B. update relevant w-values, slacks and shortest path lengths: (i, j) A (k), w k ij=w k ij+δ & s ij = s ij - δ l N 2 (k), z D =z D + δ. v k l = v k l + δ C. label a new node: If δ=δ 1, for some (i*, j*) A (k) satisfying s i*j* = 0, set N 1 (k)=n 1 (k)\{i*} N 2 (k)=n 2 (k) {i*}. Remove commodity k from CANDIDATES and repeat Step 1.

18 Stopping Rule k K, if O(k) N 2 (k), then STOP. Otherwise, set CANDIDATES={k K: O(k) N 1 (k)}, and return to Step 1. When the algorithms terminates, that is, when all origins are labeled, any further increases in w-values, with respect to the final dual solution, cannot improve the final lower bound z D.

19 Does Dual Ascent Heuristic Cover Every Problem? Dual-based procedures are applicable only if the original problem has a special structure. Hence, other lower bounding techniques can tackle a wider spectrum of problems. How can we modify the method to solve a wider range of problems? By modifying the mathematical formulation of the considered problems. By revising the ascent strategy.