Feasibility Pump Heuristics for Column Generation Approaches

Transcription

1 1 / 29 Feasibility Pump Heuristics for Column Generation Approaches Ruslan Sadykov 2 Pierre Pesneau 1,2 Francois Vanderbeck 1,2 1 University Bordeaux I 2 INRIA Bordeaux Sud-Ouest SEA 2012 Bordeaux, France, June 9, 2012

2 2 / 29 Outline Generic Primal Heuristics Generic Primal Heuristics for Branch-and-Price Column Generation based Feasibility Pump heuristic Numerical tests Conclusion

3 3 / 29 Generic Primal Heuristics for MIPs good feasible solutions using the tools of exact optimization

4 4 / 29 Generic Primal Heuristics for MIPs good feasible solutions using the tools of exact optimization Truncating an exact method Building from the relaxation used for the exact approach Defining a target based on the relaxation Using dual information to price choices in greedy heuristics Exact approach used to explore a neighborhood

5 5 / 29 Generic Primal Heuristics for MIPs good feasible solutions using the tools of exact optimization Truncating an exact method Building from the relaxation used for the exact approach Defining a target based on the relaxation Using dual information to price choices in greedy heuristics Exact approach used to explore a neighborhood Examples: [Berthold 06] 1. Large Scale Neighborhood Search [Ahuja al 02] 2. Relaxation Induced Neighborhood Search [Dana al 05] 3. Local Branching [Fischetti al 03] 4. Feasibility Pump [Fischetti al 05]

6 6 / 29 Generic LP based heuristics min{ j c jx j : j a ijx j b i i, l j x j u j j}

7 7 / 29 Generic LP based heuristics min{ j c jx j : j a ijx j b i i, l j x j u j j} Rounding: Iteratively select a var x j and bound/fix it least fractional: argmin j {min{x j x j, x j x j }} guided search: argmin j { x j x inc j }

8 8 / 29 Generic LP based heuristics min{ j c jx j : j a ijx j b i i, l j x j u j j} Rounding: Iteratively select a var x j and bound/fix it least fractional: argmin j {min{x j x j, x j x j }} guided search: argmin j { x j x inc j } Diving: rounding + LP resolve + reiterate heuristic depth search in branch-and-bound tree branching rule that of exact branch-and-bound

9 9 / 29 Generic LP based heuristics min{ j c jx j : j a ijx j b i i, l j x j u j j} Rounding: Iteratively select a var x j and bound/fix it least fractional: argmin j {min{x j x j, x j x j }} guided search: argmin j { x j x inc j } Diving: rounding + LP resolve + reiterate heuristic depth search in branch-and-bound tree branching rule that of exact branch-and-bound sub-miping: rounding/diving + MIP sol of the residual prob.

10 10 / 29 Heuristic search in branch-and-bound tree Diving sub-miping

11 11 / 29 Feasibility Pump heuristic Target solution x is obtained by rounding LP solution x LP to the closest integer solution. If x is not feasible, the problem is modified: 0 1 integer program { ( min c x +ɛ x j + j: x j =0 j: x j =1 general integer program (l j x j u j ) { ( min c x + ɛ (x j l j ) + j: x j =l j ) (1 x j ) : A x a, x [0, 1] n} j: x j =u j (u j x j ) + j:l j < x j <u j d j ) : A x a, d j x j x j d j + x j j, x R n}

12 12 / 29 The Branch-and-Price Approach min c 1 x 1 + c 2 x c K x K D x 1 + D x D x K d B x 1 b B x 2 b B x K x 1 N n, x 2 N n,... x K N n. b

13 13 / 29 The Branch-and-Price Approach min c 1 x 1 + c 2 x c K x K D x 1 + D x D x K d B x 1 b B x 2 b B x K x 1 N n, x 2 N n,... x K N n. Relax Dx d = decomposition: subproblem {B x b, x N n } and a reformulation solved by Branch-and-Price: b min g G cx g λ g g G Dx g λ g d g G λg = K λ N G Solve Master LP Pricing Problem Solve Master LP Pricing Problem Solve Master LP Pricing Problem

14 14 / 29 The Branch-and-Price Approach min c 1 x 1 + c 2 x c K x K D x 1 + D x D x K d B x 1 b B x 2 b B x K x 1 N n, x 2 N n,... x K N n. Relax Dx d = decomposition: subproblem {B x b, x N n } and a reformulation solved by Branch-and-Price: b min g G cx g λ g g G Dx g λ g d g G λg = K λ N G Solve Master LP Pricing Problem y := k x k = g G x g λ g Solve Master LP Solve Master LP Pricing Problem Pricing Problem

15 15 / 29 Difficulties in a B-a-P context for generic heuristics Heuristic paradigm in original space or the reformulation?

16 Difficulties in a B-a-P context for generic heuristics Heuristic paradigm in original space or the reformulation? On master variables: λ (aggregated decisions) Cannot fix bounds (as in rounding) Cannot modify costs (as in feasibility pump) 16 / 29

17 Difficulties in a B-a-P context for generic heuristics Heuristic paradigm in original space or the reformulation? On master variables: λ (aggregated decisions) Cannot fix bounds (as in rounding) Cannot modify costs (as in feasibility pump) On original variables: x (disaggregated decisions) Cannot grasp individual SP var. after aggregation in the common case of identical SPs Cannot modify the SP structure required by the oracle 17 / 29

18 Difficulties in a B-a-P context for generic heuristics Heuristic paradigm in original space or the reformulation? On master variables: λ (aggregated decisions) Cannot fix bounds (as in rounding) Cannot modify costs (as in feasibility pump) On original variables: x (disaggregated decisions) Cannot grasp individual SP var. after aggregation in the common case of identical SPs Cannot modify the SP structure required by the oracle Differences Acting on master λ variables results in a more macroscopic decision. Faster progress to an integer solution, but you can quickly paint yourself in a corner 18 / 29

19 19 / 29 Generic modifications of the master Setting a lower bound of a column: λ g l g Decreasing cost c g of a column λ g

20 20 / 29 Generic modifications of the master Setting a lower bound of a column: λ g l g Decreasing cost c g of a column λ g In both cases, pricing oracle overestimates the reduced cost of column λ g already included in the master.

21 21 / 29 Generic modifications of the master Setting a lower bound of a column: λ g l g Decreasing cost c g of a column λ g In both cases, pricing oracle overestimates the reduced cost of column λ g already included in the master. Preprocessing Lower bound setting is done by fixing a partial ( rounded-down ) solution After that, the residual master problem is defined by preprocessing: updating RHS of the master; updating bounds for subproblem variables.

22 22 / 29 Pure Diving Heuristic Depth-First Search select least fractional col: λ s λ s update master and SP apply preprocessing

23 23 / 29 Generic Feasibility Pump algorithm I Solution λ is defined by rounding the LP solution λ LP. If λ is feasible, stop. Otherwise, we use λ as a target point. We decrease the cost of rounded-up columns and increase the cost of rounded-down ones (but not beyond the original cost).

24 24 / 29 Generic Feasibility Pump algorithm I Solution λ is defined by rounding the LP solution λ LP. If λ is feasible, stop. Otherwise, we use λ as a target point. We decrease the cost of rounded-up columns and increase the cost of rounded-down ones (but not beyond the original cost). Cost modification factor functions f 1 (λ, α) f 2 (λ, α) α 1 λ α 1 λ f 1 (λ, α) = { 0.1 λ α 0.1 (1 λ) (1 α) if λ α if λ > α f 2 (λ, α) = { 0.1 (1 λ α ) 0.1 (λ α) (1 α) if λ α if λ > α

25 25 / 29 Embedding Feasibility Pump in a Diving heuristic At iteration t, the modified master becomes { min cgλ t g : (Ax g )λ g a t ; λ g = K t ; λ g N g G t g G t g G t g G Before defining target solution λ t, the rounded-down integer part of λ t LP is fixed and removed: λt g λ t g λ t g (this way the residual master is close to a 0 1 problem). Cycling can occur if no columns are rounded up in λ t. In this case, we decrease fractionality threshold parameter α (initially α 0.5).

26 26 / 29 Cutting Stock Problem n = 50, 100 d i [1, 50] W = w i [500, 2500] 50 instances for each n n function found opt gap time 50 Pure Div. 50/50 43/ f 1 50/50 45/ f 2 50/50 41/ Pure Div. 50/50 35/ f 1 50/50 43/ f 2 50/50 40/

27 Generalized Assignment Tasks assignment Machines cost cost cost cost cost Instances from OR-Library (type D) 50 instances for each (n, m) m n function found gap time Pur Div. 34/ % f 1 36/ % f 2 48/ % Pur Div. 35/ % f 1 36/ % f 2 42/ % / 29

28 28 / 29 Conclusions Summary Feasibility Pump heuristic can be extended to the column generation context The key is to restrict problem modifications to setting lower bound and cost reduction. Compared with the generic diving heuristic, feasibility pump heuristic produced more feasible primal solutions without loosing on the quality.

29 Conclusions Summary Feasibility Pump heuristic can be extended to the column generation context The key is to restrict problem modifications to setting lower bound and cost reduction. Compared with the generic diving heuristic, feasibility pump heuristic produced more feasible primal solutions without loosing on the quality. Future work Adaptation of diversification mechanisms for the Feasibility Pump heuristic Numerical tests on a larger scope of applications Compare Feasibility Pump heuristic on aggregated variables λ versus Feasibility Pump in the space of original variables x. 29 / 29