Benders Decomposition Methods for Structured Optimization, including Stochastic Optimization

Size: px
Start display at page:

Download "Benders Decomposition Methods for Structured Optimization, including Stochastic Optimization"

Transcription

1 Benders Decomposition Methods for Structured Optimization, including Stochastic Optimization Robert M. Freund May 2, 2001

2 Block Ladder Structure Basic Model minimize x;y c T x + f T y s:t: Ax = b Bx + Dy = d x 0 y 0 : cfl2002 Massachusetts Institute of Technology. All rights reserved

3 c T x + ff1f T 1 y 1 + ff2f T 2 y ff K f T K y K y1;:::;y K x; Ax = b s:t: B K x + D K y K = d K Block Ladder Structure More Complex Structure minimize B1x + D1y1 = d1 + D2y2 = d2 B2x..... y1;y2;:::;y K 0 : x; cfl2002 Massachusetts Institute of Technology. All rights reserved

4 Block Ladder Structure More Complex Structure Stage-1 Variables Stage-2 Variables Objectives RHS State 1 State 2 State 3 State k cfl2002 Massachusetts Institute of Technology. All rights reserved

5 Basic Model Reformulation VAL = minimum x;y c T x + f T y s:t: Ax = b Bx + Dy = d x 0 y 0 ; cfl2002 Massachusetts Institute of Technology. All rights reserved

6 Basic Model Reformulation = minimum x c T x + z(x) VAL s:t: Ax = b where: x 0 ; P2 : z(x) = minimum y f T y s:t: Dy = d Bx y 0 : cfl2002 Massachusetts Institute of Technology. All rights reserved

7 Basic Model Reformulation P2 : z(x) = minimum y f T y s:t: Dy = d Bx The dual is: y 0 : D2 : z(x) = maximum p p T (d Bx) s:t: D T p» f : cfl2002 Massachusetts Institute of Technology. All rights reserved

8 Basic Model Reformulation = maximum p p T (d Bx) D2 : z(x) s:t: D T p» f : The feasible region of D2 is the set: D2 := Φ p j D T p» f Ψ : cfl2002 Massachusetts Institute of Technology. All rights reserved

9 p 1 ; : : : ; p I r 1 ; : : : ; p J Basic Model Reformulation The feasible region of D2 is the set: Extreme points are: D 2 := Φ p j D T p» f Ψ ; Extreme rays are: cfl2002 Massachusetts Institute of Technology. All rights reserved

10 Basic Model Reformulation = maximum p p T (d Bx) D2 : z(x) s:t: D T p» f : If we solve D2, two case arise: D2 is unbounded from above, or D2 has an optimal solution. cfl2002 Massachusetts Institute of Technology. All rights reserved

11 Basic Model Reformulation = maximum p p T (d Bx) D2 : z(x) s:t: D T p» f : If D2 is unbounded from above, the algorithm will return one of the extreme rays μr = r j for some j. (r j ) T (d Bx) > 0 z(x) = +1 : cfl2002 Massachusetts Institute of Technology. All rights reserved

12 Basic Model Reformulation D2 : z(x) = maximum p p T (d Bx) s:t: D T p» f : If D2 has an optimal solution, the algorithm will return one of the extreme points μp = p i for some i. = (p i ) T (d Bx) = max z(x) (pk ) T (d Bx) : k=1;:::;i cfl2002 Massachusetts Institute of Technology. All rights reserved

13 Basic Model Reformulation = maximum p p T (d Bx) D2 : z(x) s:t: D T p» f : D2 : z(x) = minimum z z s:t: (p i ) T (d Bx)» z i = 1; : : : ; I (r j ) T (d Bx)» 0 j = 1; : : : ; J : cfl2002 Massachusetts Institute of Technology. All rights reserved

14 Basic Model Reformulation Full Master Problem... = minimum x c T x + z(x) VAL s:t: Ax = b x 0 ; FMP : VAL = minimum x;z c T x + z s:t: Ax = b x 0 (p i ) T (d Bx)» z i = 1;:::;I j ) T (d Bx)» 0 j = 1;:::;J : (r cfl2002 Massachusetts Institute of Technology. All rights reserved

15 Basic Model Reformulation...Full Master Problem FMP : VAL = minimum x;z c T x + z s:t: Ax = b x 0 (p i ) T (d Bx)» z i = 1;:::;I j ) T (d Bx)» 0 j = 1;:::;J : (r We have eliminated the y variables from the problem. We have added a single scalar variable z. We have added a generically extremely large number of constraints. cfl2002 Massachusetts Institute of Technology. All rights reserved

16 Delayed Constraint Generation Restricted Master Problem Consider problem RMP k composed of only k of the extreme point/extreme ray constraints from FMP: RMP k : VAL k = minimum x;z c T x + z s:t: Ax = b x 0 (p i ) T (d Bx)» z i = 1;:::;k l (r j ) T (d Bx)» 0 j = 1;:::;l : cfl2002 Massachusetts Institute of Technology. All rights reserved

17 Delayed Constraint Generation Restricted Master Problem Lower Bound on VAL RMP k : VAL k = minimum x;z c T x + z s:t: Ax = b x 0 (p i ) T (d Bx)» z i = 1;:::;k l (r j ) T (d Bx)» 0 j = 1;:::;l : We solve this problem, obtaining VAL k, μx; μz k» VAL : VAL cfl2002 Massachusetts Institute of Technology. All rights reserved

18 Delayed Constraint Generation Restricted Master Problem RMP k : VAL k = minimum x;z c T x + z s:t: Ax = b x 0 (p i ) T (d Bx)» z i = 1;:::;k l (r j ) T (d Bx)» 0 j = 1;:::;l : Check whether the μx; μz solution is optimal for the full master problem. Check μx; μz whether violates any of the non-included constraints. cfl2002 Massachusetts Institute of Technology. All rights reserved

19 Delayed Constraint Generation Solve the Subproblem Finding a Violated Constraint Solve: Q(μx) : q Λ = maximum p p T (d Bμx) = minimum y f T y s:t: D T p» f s:t: Dy = d Bμx y 0 : If Q(μx) is unbounded, return an extreme ray μr = r j for some j, where μx will satisfy: j ) T (d Bμx) > 0 : (r μx Therefore has violated the constraint: (r j ) T (d Bx)» 0 ; Add this constraint to RMP and re-solve RMP. cfl2002 Massachusetts Institute of Technology. All rights reserved

20 c T μx + f T μy» UB; then BESTSOL ψ (μx; μy) : If ψ minfub;c T μx + f T μyg : UB Delayed Constraint Generation Solve the Subproblem Upper Bound on VAL Q(μx) : q Λ = maximum p p T (d Bμx) = minimum y f T y s:t: D T p» f s:t: Dy = d Bμx y 0 : If Q(μx) has an optimal solution, return an optimal extreme point μp = p i for some i, and the optimal solution μy to the minimization problem of Q(μx). (μx; μy) is feasible for the original problem. Suppose that UB is an upper bound on VAL. Update UB: cfl2002 Massachusetts Institute of Technology. All rights reserved

21 Delayed Constraint Generation Solve the Subproblem Finding a Violated Constraint Q(μx) : q Λ = maximum p p T (d Bμx) = minimum y f T y s:t: D T p» f s:t: Dy = d Bμx If q Λ > μz, then y 0 : (p i ) T (d Bμx) = q Λ > μz : We therefore have found that μx; μz has violated the constraint: (p i ) T (d Bx)» z ; and so we add this constraint to RMP and re-solve RMP. cfl2002 Massachusetts Institute of Technology. All rights reserved

22 Delayed Constraint Generation Solve the Subproblem Checking for Full Optimality Q(μx) : q Λ = maximum p p T (d Bμx) = minimum y f T y s:t: D T p» f s:t: Dy = d Bμx If q Λ» μz, then: y 0 : max (pk ) T (d B μx) = (p i ) T (d B μx) = q Λ» μz ; k=1;:::;i μx; μz is feasible and hence optimal for FMP (μx; μy) is optimal for the original problem. cfl2002 Massachusetts Institute of Technology. All rights reserved

23 Delayed Constraint Generation Solve the Subproblem Termination Criterion Otherwise terminate if: UB VAL k» " for a pre-specified tolerance ". cfl2002 Massachusetts Institute of Technology. All rights reserved

24 Delayed Constraint Generation Algorithm 0. LB= 1 Set UB= and A typical iteration starts with the relaxed master problem RMP, in which only k constraints of the full master problem FMP are included. An optimal k μx; μz solution to the relaxed master problem is computed. We update the lower bound on VAL: LB ψ VAL k : 2. Solve the subproblem Q(μx): Q(μx) : q Λ = maximum p p T (d Bμx) = minimum y f T y s:t: D T p» f s:t: Dy = d Bμx 0 : y cfl2002 Massachusetts Institute of Technology. All rights reserved

25 Delayed Constraint Generation Algorithm 3. If Q(μx) has an optimal solution, let μp and μy be the primal and dual solutions of Q(μx). Ifq Λ» μz, then μx; μz satisfies all constraints of FMP, and so μx; μz is optimal for FMP. Furthermore, this implies that (μx; μy) is optimal for the original problem. If q Λ > μz, then we add the constraint: μp T (d Bx)» z to the restricted master problem RMP. cfl2002 Massachusetts Institute of Technology. All rights reserved

26 Delayed Constraint Generation Algorithm Update the upper bound and the best solution: If c T μx + f T μy» UB; then BESTSOL ψ (μx; μy) UB ψ minfub; c T μx + f T μyg If UB LB» ", then terminate. Otherwise, return to step 1. cfl2002 Massachusetts Institute of Technology. All rights reserved

27 Delayed Constraint Generation Algorithm 4. If Q(μx) is unbounded from above, let μr be the extreme ray generated by the algorithm that solves Q(μx). Add the constraint: μr T (d Bx)» 0 to the restricted master problem RMP, and return to step 1. cfl2002 Massachusetts Institute of Technology. All rights reserved

28 Delayed Constraint Generation Algorithm Comments This algorithm is known formally as Benders decomposition. Benders decomposition method was developed in Benders decomposition method is the same as Dantzig-Wolfe decomposition applied to the dual problem. cfl2002 Massachusetts Institute of Technology. All rights reserved

29 B K x + D K y K = d K Benders Decomposition Block-Ladder Structure = minimize c T x + ff1f T 1 y 1 + ff2f T 2 y ff K f T K y K VAL y1;:::;y K x; s:t: Ax = b B1x + D1y1 = d1 + D2y2 = d2 B2x..... y1;y2;:::;y K 0 : x; cfl2002 Massachusetts Institute of Technology. All rights reserved

30 minimum c K P x + = ff!z T (x)! x!=1 P2! : z! (x) = minimum y! f T! y! Benders Decomposition Block-Ladder Structure Reformulation... VAL s:t: Ax = b x 0 ; where: s:t: D! y! = d! B! x y! 0 : cfl2002 Massachusetts Institute of Technology. All rights reserved

31 P2! : z! (x) = minimum y! f T! y! Benders Decomposition Block-Ladder Structure...Reformulation... s:t: D! y! = d! B! x y! 0 : The dual is: D2! : z! (x) = maximum p! p T!(d! B! x) s:t: D T! p!» f! : cfl2002 Massachusetts Institute of Technology. All rights reserved

32 1 I!! ;:::;p! p 1 J!! ;:::;p! r Benders Decomposition Block-Ladder Structure...Reformulation... D2! : z! (x) = maximum p! p T!(d! B! x) The feasible region of D2! is the set: s:t: D T! p!» f! : Extreme points of D! 2 : D! 2 := Φ p! j D T! p!» f! Ψ ; Extreme rays of D! 2 : cfl2002 Massachusetts Institute of Technology. All rights reserved

33 Benders Decomposition Block-Ladder Structure...Reformulation... D2! : z! (x) = maximum p! p T! (d! B! x) s:t: D T! p!» f! : If we solve D2!, two case arise: D2! is unbounded from above, or D2! has an optimal solution. cfl2002 Massachusetts Institute of Technology. All rights reserved

34 Benders Decomposition Block-Ladder Structure...Reformulation... D2! : z! (x) = maximum p! p T! (d! B! x) s:t: D T! p!» f! : If D2! is unbounded from above, return one of the extreme rays μr! = r j! for some j, with the property that in which case (r j! )T (d! B! x) > 0 z! (x) = +1 : cfl2002 Massachusetts Institute of Technology. All rights reserved

35 Benders Decomposition Block-Ladder Structure...Reformulation... D2! : z! (x) = maximum p! p T! (d! B! x) s:t: D T! p!» f! : If D2! has an optimal solution, return one of the extreme points μp! = p i! for some i as well as the optimal objective function value z! (x) z! (x) = (p i! )T (d! B! x) = max (p k! )T (d! B! x) :! k=1;:::;i cfl2002 Massachusetts Institute of Technology. All rights reserved

36 D2! : z! (x) = minimum z! z! s:t: (p i!) T (d! B! x)» z! i = 1;:::;I! Benders Decomposition Block-Ladder Structure...Reformulation... D2! : z! (x) = maximum p! p T!(d! B! x) s:t: D T! p!» f! : (r j!) T (d! B! x)» 0 j = 1;:::;J! : cfl2002 Massachusetts Institute of Technology. All rights reserved

37 = x c T K x + P VAL ff! z! (x) minimum!=1 x; z 1 ;:::;z K Benders Decomposition Block-Ladder Structure...Reformulation s:t: Ax = b x 0 ; FMP : VAL = minimum T K + P c ff! z! x!=1 s:t: Ax = b x 0 (p i!) T (d! B! x)» z! i = 1;:::;I! ;! = 1;:::;K (r j!) T (d! B! x)» 0 j = 1;:::;J! ;! = 1;:::;K : cfl2002 Massachusetts Institute of Technology. All rights reserved

38 T K P c ff!z! + x!=1 z1;:::;z K x; Ax = b s:t: Block-Ladder Structure Restricted Master Problem Consider problem RMP k composed of only k of the extreme point/extreme ray constraints from FMP: RMP k : VAL k = minimum x 0 (p i!) T (d! B! x)» z! for some i and! j!) T (d! B! x)» 0 for some i and! ; (r where there are a total k of of the inequality constraints. cfl2002 Massachusetts Institute of Technology. All rights reserved

39 T K P c ff!z! + x!=1 z1;:::;z K x; Ax = b s:t: Block-Ladder Structure Restricted Master Problem Lower Bound on VAL RMP k : VAL k = minimum x 0 (p i!) T (d! B! x)» z! for some i and! (r j!) T (d! B! x)» 0 for some i and! ; Solve for VAL k and μx; μz 1 ; : : : ; μz K cfl2002 Massachusetts Institute of Technology. All rights reserved

40 T K P c ff!z! + x!=1 z1;:::;z K x; Ax = b s:t: Block-Ladder Structure Restricted Master Problem RMP k : VAL k = minimum x 0 (p i!) T (d! B! x)» z! for some i and! (r j!) T (d! B! x)» 0 for some i and! ; Solve for VAL and μx; μz1;:::;μz k K Check whether the μx; μz1;:::;μz solution K is optimal for FMP Check μx; μz1;:::;μz whether K violates any of the non-included constraints. cfl2002 Massachusetts Institute of Technology. All rights reserved

41 Q! (μx) : q Λ! = maximum p! pt!(d! B! μx) = minimum y! f T! y! s:t: D T! p!» f! Block-Ladder Structure Restricted Master Problem Solve the Subproblems s:t: D! y! = d! B! μx y! 0 : If Q! (μx) is unbounded, return an extreme ray μr! = r j! for some j, where μx will satisfy: j!) T (d! B! μx) > 0 : (r μx Therefore has violated the constraint: j!) T (d! B! x)» 0 ; (r and so we add this constraint to RMP. cfl2002 Massachusetts Institute of Technology. All rights reserved

42 Q! (μx) : q Λ! = maximum p! pt!(d! B! μx) = minimum y! f T! y! s:t: D T! p!» f! Block-Ladder Structure Restricted Master Problem Finding a Violated Constraint s:t: D! y! = d! B! μx! 0 y : (μx) If Q! has an optimal solution, return an μp optimal = p extreme! point i! for some i, as well as μy the optimal solution! of the minimization problem of Q! (μx). If!, then > μz Λ q! i!) T (d! B! μx) = q Λ! > μz! : (p μx; μz Therefore! has violated the constraint: i!) T (d! B! x)» z! ; (p and so we add this constraint to RMP. cfl2002 Massachusetts Institute of Technology. All rights reserved

43 Q! (μx) : q Λ! = maximum p! pt!(d! B! μx) = minimum y! f T! y! s:t: D T! p!» f! KX! f T! μy!» UB; then BESTSOL ψ (μx; μy 1 ;:::;μy K ) : ff!=1 KX! f T! μy! g : ff!=1 Block-Ladder Structure Restricted Master Problem Upper Bound on VAL s:t: D! y! = d! B! μx y! 0 : If q Λ! is finite for all! = 1;:::;K, then μx; μy1;:::;μy K satisfies all of the constraints of the original problem. Suppose that UB is an upper bound on VAL, we update UB as follows: If c T μx + UB ψ minfub;c T μx + cfl2002 Massachusetts Institute of Technology. All rights reserved

44 Q! (μx) : q Λ! = maximum p! pt!(d! B! μx) = minimum y! f T! y! s:t: D T! p!» f! max (p l!) T (d! B! μx) = (p i!) T (d! B! μx) = q Λ!» μz!! l=1;:::;i Block-Ladder Structure Restricted Master Problem Checking for Optimality s:t: D! y! = d! B! μx If q Λ!» μz! for all! = 1;:::;K, then we have: y! 0 : for all! = 1;:::;K; μz1;:::;μz K satisfies all of the constraints in FMP. μx; μz1;:::;μz K is feasible and hence optimal for FMP μx; μy1;:::;μy K ) is optimal for the original problem, and we can terminate the (μx; algorithm. cfl2002 Massachusetts Institute of Technology. All rights reserved

45 Block-Ladder Structure Restricted Master Problem Termination Criterion Else, we can also terminate the algorithm whenever: UB VAL k» " for a pre-specified tolerance ". cfl2002 Massachusetts Institute of Technology. All rights reserved

46 Q! (μx) : q Λ! = maximum p! pt!(d! B! μx) = minimum y! f T! y! s:t: D T! p!» f! Benders Decomposition Block-Ladder Structure Algorithm LB= 1 Set UB= and A typical iteration starts with the relaxed master problem RMP, in which only k constraints of the full master problem FMP are included. An optimal k μx; μz1;:::;μz solution K to the relaxed master problem is computed. We update the lower bound on VAL: LB ψ VAL k : 2. For! = 1;:::;K, solve the subproblem Q! (μx): s:t: D! y! = d! B! μx! 0 : y cfl2002 Massachusetts Institute of Technology. All rights reserved

47 Benders Decomposition Block-Ladder Structure...Algorithm... If Q! (μx) is unbounded from above, let μr! be the ffl extreme ray generated by the algorithm that solves Q(μx). Add the constraint: μr T! (d! B! x)» 0 to the restricted master problem RMP. cfl2002 Massachusetts Institute of Technology. All rights reserved

48 μp T! (d! B! x)» z! Benders Decomposition Block-Ladder Structure...Algorithm... If Q (μx)! has an optimal solution, μp let! μy and! ffl be the primal and dual solutions of Q Ifq!! (μx).!, μz then we add the constraint: > Λ to the restricted master problem RMP. cfl2002 Massachusetts Institute of Technology. All rights reserved

49 KX KX Benders Decomposition Block-Ladder Structure...Algorithm 3. If q Λ!» μz! for all! = 1;:::;K, then (μx; μy1;:::;μy K ) is optimal for the original problem, and the algorithm terminates. 4. If Q! (μx) has an optimal solution for all! = 1;:::;K, then update the upper bound on VAL as follows: If c T μx + ff! f! T μy!» UB; then BESTSOL ψ (μx; μy1;:::;μy K ) :!=1 ff f T! μy! g :!!=1 If» UB LB ", then terminate. Otherwise, add all of the new constraints to RMP and return to step 1. UB ψ minfub;c T μx + cfl2002 Massachusetts Institute of Technology. All rights reserved

50 Benders Decomposition Block-Ladder Structure Comment In this version of Benders decomposition, we might add as many as K new constraints per major iteration. cfl2002 Massachusetts Institute of Technology. All rights reserved

51 min x;y 125 x i + i P c ff! c i x i» 10; 000 i=1 4P 0:01f i (!)h j y ijk! y ijk! D jk! for all j; k;! (Demand constraints) i=1 5P The Power Plant Investment Model 15 P i=1 4P!=1 i=1 5P j=1 5P k=1 (Budget constraint) s.t. x 4» 5:0 (Hydroelectric constraint) y ijk!» x i for i = 1;:::;4; all j; k;! (Capacity constraints) x 0; y 0! is the probability of scenario! ff jk! is the power demand in block j and year k under scenario!. D cfl2002 Massachusetts Institute of Technology. All rights reserved

52 The Power Plant Investment Model Size of Model 4 stage-1 variables 46; 875 stage-2 variables 2 stage-1 constraints constraints for each of the 125 scenarios in 375 stage-2. Total of 46; 877 constraints. cfl2002 Massachusetts Institute of Technology. All rights reserved

53 P 5 i=1 y ijk! D jk! for all j; k (Demand constraints) The Power Plant Investment Model Subproblem Consider a fixed x which is feasible (i.e., x 0 and satisfies the budget constraint and hydroelectric constraint). z! (x) := min y 5 j=1 i=1p 5P 15P 0:01f i (!)h j y ijk! k=1 s.t. y ijk!» x i for i = 1;:::;4; all j; k; (Capacity constraints) y 0: Each subproblem has only 375 variables and 375 constraints. (Furthermore, the structure of this problem lends itself to a simple greedy optimal solution.) cfl2002 Massachusetts Institute of Technology. All rights reserved

54 z! (x) := max p;q 15P 5 i p ijk + P x j=1 k=1 D jk! q jk The Power Plant Investment Model Dual of Subproblem The dual of the above problem is: 15 P k=1 i=1 4P j=1 5P s.t. p ijk + q jk» 0:01f i (!)h j i = 1;:::;4; for all j; k p ijk» 0 q jk 0: cfl2002 Massachusetts Institute of Technology. All rights reserved

55 Computational Results Original Model Iterations All models were solved using CPLEX, called from the AMPL programming environment on a 450 MHz Pentium II processor. Original Model Iterations Simplex 41,339 Interior-point 51 cfl2002 Massachusetts Institute of Technology. All rights reserved

56 Computational Results Benders Decomposition Iterations and Constraints Benders duality gap tolerance is " = Benders Decomposition Basic (1 Block) Block-Ladder (125 Blocks) Master Iterations Generated Constraints 42 1,726 cfl2002 Massachusetts Institute of Technology. All rights reserved

57 Computational Results CPU Time Benders Decomposition Original Model Basic (1 Block) Block-Ladder (125 Blocks) Simplex 7:50 4:55 2:35 Interior-point 1:00 6:20 3:00 cfl2002 Massachusetts Institute of Technology. All rights reserved

58 Computational Results Upper and Lower Bounds 50 Block Ladder 125 Blocks Basic 1 Block 40 Bounds (in billion $) Iteration cfl2002 Massachusetts Institute of Technology. All rights reserved

59 Computational Results Duality Gap 10 5 Block Ladder 125 Blocks Basic 1 Block 10 4 Upper bound lower bound Iteration cfl2002 Massachusetts Institute of Technology. All rights reserved

60 References D. Anderson. Models for determining least-cost investments in electricity supply. The Bell Journal of Economics, 3: , J. Benders. Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik, 4: , D. Bertsimas and J. Tsitisklis. Introduction to Linear Optimization. Athena Scientific, J. Birge and F. Louveaux. Introduction to Stochastic Programming. Springer-Verlag, cfl2002 Massachusetts Institute of Technology. All rights reserved

Benders Decomposition Methods for Structured Optimization, including Stochastic Optimization

Benders Decomposition Methods for Structured Optimization, including Stochastic Optimization Benders Decomposition Methods for Structured Optimization, including Stochastic Optimization Robert M. Freund April 29, 2004 c 2004 Massachusetts Institute of echnology. 1 1 Block Ladder Structure We consider

More information

Benders Decomposition

Benders Decomposition Benders Decomposition Yuping Huang, Dr. Qipeng Phil Zheng Department of Industrial and Management Systems Engineering West Virginia University IENG 593G Nonlinear Programg, Spring 2012 Yuping Huang (IMSE@WVU)

More information

Decomposition methods in optimization

Decomposition methods in optimization Decomposition methods in optimization I Approach I: I Partition problem constraints into two groups: explicit and implicit I Approach II: I Partition decision variables into two groups: primary and secondary

More information

An Adaptive Partition-based Approach for Solving Two-stage Stochastic Programs with Fixed Recourse

An Adaptive Partition-based Approach for Solving Two-stage Stochastic Programs with Fixed Recourse An Adaptive Partition-based Approach for Solving Two-stage Stochastic Programs with Fixed Recourse Yongjia Song, James Luedtke Virginia Commonwealth University, Richmond, VA, ysong3@vcu.edu University

More information

Applications. Stephen J. Stoyan, Maged M. Dessouky*, and Xiaoqing Wang

Applications. Stephen J. Stoyan, Maged M. Dessouky*, and Xiaoqing Wang Introduction to Large-Scale Linear Programming and Applications Stephen J. Stoyan, Maged M. Dessouky*, and Xiaoqing Wang Daniel J. Epstein Department of Industrial and Systems Engineering, University of

More information

Applied Lagrange Duality for Constrained Optimization

Applied Lagrange Duality for Constrained Optimization Applied Lagrange Duality for Constrained Optimization February 12, 2002 Overview The Practical Importance of Duality ffl Review of Convexity ffl A Separating Hyperplane Theorem ffl Definition of the Dual

More information

An Integer Cutting-Plane Procedure for the Dantzig-Wolfe Decomposition: Theory

An Integer Cutting-Plane Procedure for the Dantzig-Wolfe Decomposition: Theory An Integer Cutting-Plane Procedure for the Dantzig-Wolfe Decomposition: Theory by Troels Martin Range Discussion Papers on Business and Economics No. 10/2006 FURTHER INFORMATION Department of Business

More information

The L-Shaped Method. Operations Research. Anthony Papavasiliou 1 / 44

The L-Shaped Method. Operations Research. Anthony Papavasiliou 1 / 44 1 / 44 The L-Shaped Method Operations Research Anthony Papavasiliou Contents 2 / 44 1 The L-Shaped Method [ 5.1 of BL] 2 Optimality Cuts [ 5.1a of BL] 3 Feasibility Cuts [ 5.1b of BL] 4 Proof of Convergence

More information

Lagrangian Relaxation in MIP

Lagrangian Relaxation in MIP Lagrangian Relaxation in MIP Bernard Gendron May 28, 2016 Master Class on Decomposition, CPAIOR2016, Banff, Canada CIRRELT and Département d informatique et de recherche opérationnelle, Université de Montréal,

More information

Interior-Point versus Simplex methods for Integer Programming Branch-and-Bound

Interior-Point versus Simplex methods for Integer Programming Branch-and-Bound Interior-Point versus Simplex methods for Integer Programming Branch-and-Bound Samir Elhedhli elhedhli@uwaterloo.ca Department of Management Sciences, University of Waterloo, Canada Page of 4 McMaster

More information

i.e., into a monomial, using the Arithmetic-Geometric Mean Inequality, the result will be a posynomial approximation!

i.e., into a monomial, using the Arithmetic-Geometric Mean Inequality, the result will be a posynomial approximation! Dennis L. Bricker Dept of Mechanical & Industrial Engineering The University of Iowa i.e., 1 1 1 Minimize X X X subject to XX 4 X 1 0.5X 1 Minimize X X X X 1X X s.t. 4 1 1 1 1 4X X 1 1 1 1 0.5X X X 1 1

More information

Capacity Planning with uncertainty in Industrial Gas Markets

Capacity Planning with uncertainty in Industrial Gas Markets Capacity Planning with uncertainty in Industrial Gas Markets A. Kandiraju, P. Garcia Herreros, E. Arslan, P. Misra, S. Mehta & I.E. Grossmann EWO meeting September, 2015 1 Motivation Industrial gas markets

More information

Benders decomposition [7, 17] uses a problem-solving strategy that can be generalized to a larger context. It assigns some of the variables trial valu

Benders decomposition [7, 17] uses a problem-solving strategy that can be generalized to a larger context. It assigns some of the variables trial valu Logic-Based Benders Decomposition Λ J. N. Hooker Graduate School of Industrial Administration Carnegie Mellon University, Pittsburgh, PA 15213 USA G. Ottosson Department of Information Technology Computing

More information

INEXACT CUTS IN BENDERS' DECOMPOSITION GOLBON ZAKERI, ANDREW B. PHILPOTT AND DAVID M. RYAN y Abstract. Benders' decomposition is a well-known techniqu

INEXACT CUTS IN BENDERS' DECOMPOSITION GOLBON ZAKERI, ANDREW B. PHILPOTT AND DAVID M. RYAN y Abstract. Benders' decomposition is a well-known techniqu INEXACT CUTS IN BENDERS' DECOMPOSITION GOLBON ZAKERI, ANDREW B. PHILPOTT AND DAVID M. RYAN y Abstract. Benders' decomposition is a well-known technique for solving large linear programs with a special

More information

Disconnecting Networks via Node Deletions

Disconnecting Networks via Node Deletions 1 / 27 Disconnecting Networks via Node Deletions Exact Interdiction Models and Algorithms Siqian Shen 1 J. Cole Smith 2 R. Goli 2 1 IOE, University of Michigan 2 ISE, University of Florida 2012 INFORMS

More information

Column Generation. MTech Seminar Report. Soumitra Pal Roll No: under the guidance of

Column Generation. MTech Seminar Report. Soumitra Pal Roll No: under the guidance of Column Generation MTech Seminar Report by Soumitra Pal Roll No: 05305015 under the guidance of Prof. A. G. Ranade Computer Science and Engineering IIT-Bombay a Department of Computer Science and Engineering

More information

Inverse Stochastic Linear Programming

Inverse Stochastic Linear Programming Inverse Stochastic Linear Programming Görkem Saka, Andrew J. Schaefer Department of Industrial Engineering University of Pittsburgh Pittsburgh, PA USA 15261 Lewis Ntaimo Department of Industrial and Systems

More information

Introduction to Integer Linear Programming

Introduction to Integer Linear Programming Lecture 7/12/2006 p. 1/30 Introduction to Integer Linear Programming Leo Liberti, Ruslan Sadykov LIX, École Polytechnique liberti@lix.polytechnique.fr sadykov@lix.polytechnique.fr Lecture 7/12/2006 p.

More information

Column Generation. ORLAB - Operations Research Laboratory. Stefano Gualandi. June 14, Politecnico di Milano, Italy

Column Generation. ORLAB - Operations Research Laboratory. Stefano Gualandi. June 14, Politecnico di Milano, Italy ORLAB - Operations Research Laboratory Politecnico di Milano, Italy June 14, 2011 Cutting Stock Problem (from wikipedia) Imagine that you work in a paper mill and you have a number of rolls of paper of

More information

Research Article A New Global Optimization Algorithm for Solving Generalized Geometric Programming

Research Article A New Global Optimization Algorithm for Solving Generalized Geometric Programming Mathematical Problems in Engineering Volume 2010, Article ID 346965, 12 pages doi:10.1155/2010/346965 Research Article A New Global Optimization Algorithm for Solving Generalized Geometric Programming

More information

Decomposition Techniques in Mathematical Programming

Decomposition Techniques in Mathematical Programming Antonio J. Conejo Enrique Castillo Roberto Minguez Raquel Garcia-Bertrand Decomposition Techniques in Mathematical Programming Engineering and Science Applications Springer Contents Part I Motivation and

More information

Computations with Disjunctive Cuts for Two-Stage Stochastic Mixed 0-1 Integer Programs

Computations with Disjunctive Cuts for Two-Stage Stochastic Mixed 0-1 Integer Programs Computations with Disjunctive Cuts for Two-Stage Stochastic Mixed 0-1 Integer Programs Lewis Ntaimo and Matthew W. Tanner Department of Industrial and Systems Engineering, Texas A&M University, 3131 TAMU,

More information

Convex Analysis 2013 Let f : Q R be a strongly convex function with convexity parameter µ>0, where Q R n is a bounded, closed, convex set, which contains the origin. Let Q =conv(q, Q) andconsiderthefunction

More information

Introduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs

Introduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs Introduction to Mathematical Programming IE406 Lecture 10 Dr. Ted Ralphs IE406 Lecture 10 1 Reading for This Lecture Bertsimas 4.1-4.3 IE406 Lecture 10 2 Duality Theory: Motivation Consider the following

More information

Benders' Method Paul A Jensen

Benders' Method Paul A Jensen Benders' Method Paul A Jensen The mixed integer programming model has some variables, x, identified as real variables and some variables, y, identified as integer variables. Except for the integrality

More information

Stochastic Integer Programming

Stochastic Integer Programming IE 495 Lecture 20 Stochastic Integer Programming Prof. Jeff Linderoth April 14, 2003 April 14, 2002 Stochastic Programming Lecture 20 Slide 1 Outline Stochastic Integer Programming Integer LShaped Method

More information

Column Generation for Extended Formulations

Column Generation for Extended Formulations 1 / 28 Column Generation for Extended Formulations Ruslan Sadykov 1 François Vanderbeck 2,1 1 INRIA Bordeaux Sud-Ouest, France 2 University Bordeaux I, France ISMP 2012 Berlin, August 23 2 / 28 Contents

More information

Solving Bilevel Mixed Integer Program by Reformulations and Decomposition

Solving Bilevel Mixed Integer Program by Reformulations and Decomposition Solving Bilevel Mixed Integer Program by Reformulations and Decomposition June, 2014 Abstract In this paper, we study bilevel mixed integer programming (MIP) problem and present a novel computing scheme

More information

Lecture #21. c T x Ax b. maximize subject to

Lecture #21. c T x Ax b. maximize subject to COMPSCI 330: Design and Analysis of Algorithms 11/11/2014 Lecture #21 Lecturer: Debmalya Panigrahi Scribe: Samuel Haney 1 Overview In this lecture, we discuss linear programming. We first show that the

More information

Integer Programming ISE 418. Lecture 16. Dr. Ted Ralphs

Integer Programming ISE 418. Lecture 16. Dr. Ted Ralphs Integer Programming ISE 418 Lecture 16 Dr. Ted Ralphs ISE 418 Lecture 16 1 Reading for This Lecture Wolsey, Chapters 10 and 11 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.3.7, II.5.4 CCZ Chapter 8

More information

Projection in Logic, CP, and Optimization

Projection in Logic, CP, and Optimization Projection in Logic, CP, and Optimization John Hooker Carnegie Mellon University Workshop on Logic and Search Melbourne, 2017 Projection as a Unifying Concept Projection is a fundamental concept in logic,

More information

Decomposition Algorithms for Two-Stage Chance-Constrained Programs

Decomposition Algorithms for Two-Stage Chance-Constrained Programs Mathematical Programming manuscript No. (will be inserted by the editor) Decomposition Algorithms for Two-Stage Chance-Constrained Programs Xiao Liu Simge Küçükyavuz Luedtke James Received: date / Accepted:

More information

Introduction to integer programming II

Introduction to integer programming II Introduction to integer programming II Martin Branda Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Computational Aspects of Optimization

More information

Lecture 9: Dantzig-Wolfe Decomposition

Lecture 9: Dantzig-Wolfe Decomposition Lecture 9: Dantzig-Wolfe Decomposition (3 units) Outline Dantzig-Wolfe decomposition Column generation algorithm Relation to Lagrangian dual Branch-and-price method Generated assignment problem and multi-commodity

More information

arxiv: v2 [math.oc] 31 May 2014

arxiv: v2 [math.oc] 31 May 2014 Chance Constrained Mixed Integer Program: Bilinear and Linear Formulations, and Benders Decomposition Bo Zeng, Yu An and Ludwig Kuznia arxiv:1403.7875v2 [math.oc] 31 May 2014 Dept. of Industrial and Management

More information

Strengthened Benders Cuts for Stochastic Integer Programs with Continuous Recourse

Strengthened Benders Cuts for Stochastic Integer Programs with Continuous Recourse Strengthened Benders Cuts for Stochastic Integer Programs with Continuous Recourse Merve Bodur 1, Sanjeeb Dash 2, Otay Günlü 2, and James Luedte 3 1 Department of Mechanical and Industrial Engineering,

More information

Outline. Relaxation. Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING. 1. Lagrangian Relaxation. Lecture 12 Single Machine Models, Column Generation

Outline. Relaxation. Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING. 1. Lagrangian Relaxation. Lecture 12 Single Machine Models, Column Generation Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING 1. Lagrangian Relaxation Lecture 12 Single Machine Models, Column Generation 2. Dantzig-Wolfe Decomposition Dantzig-Wolfe Decomposition Delayed Column

More information

Linear programming. Saad Mneimneh. maximize x 1 + x 2 subject to 4x 1 x 2 8 2x 1 + x x 1 2x 2 2

Linear programming. Saad Mneimneh. maximize x 1 + x 2 subject to 4x 1 x 2 8 2x 1 + x x 1 2x 2 2 Linear programming Saad Mneimneh 1 Introduction Consider the following problem: x 1 + x x 1 x 8 x 1 + x 10 5x 1 x x 1, x 0 The feasible solution is a point (x 1, x ) that lies within the region defined

More information

Solution Methods for Stochastic Programs

Solution Methods for Stochastic Programs Solution Methods for Stochastic Programs Huseyin Topaloglu School of Operations Research and Information Engineering Cornell University ht88@cornell.edu August 14, 2010 1 Outline Cutting plane methods

More information

Introduction to optimization

Introduction to optimization Introduction to optimization Geir Dahl CMA, Dept. of Mathematics and Dept. of Informatics University of Oslo 1 / 24 The plan 1. The basic concepts 2. Some useful tools (linear programming = linear optimization)

More information

Solving large Semidefinite Programs - Part 1 and 2

Solving large Semidefinite Programs - Part 1 and 2 Solving large Semidefinite Programs - Part 1 and 2 Franz Rendl http://www.math.uni-klu.ac.at Alpen-Adria-Universität Klagenfurt Austria F. Rendl, Singapore workshop 2006 p.1/34 Overview Limits of Interior

More information

Decomposition Algorithms for Two-Stage Distributionally Robust Mixed Binary Programs

Decomposition Algorithms for Two-Stage Distributionally Robust Mixed Binary Programs Decomposition Algorithms for Two-Stage Distributionally Robust Mixed Binary Programs Manish Bansal Grado Department of Industrial and Systems Engineering, Virginia Tech Email: bansal@vt.edu Kuo-Ling Huang

More information

56:270 Final Exam - May

56:270  Final Exam - May @ @ 56:270 Linear Programming @ @ Final Exam - May 4, 1989 @ @ @ @ @ @ @ @ @ @ @ @ @ @ Select any 7 of the 9 problems below: (1.) ANALYSIS OF MPSX OUTPUT: Please refer to the attached materials on the

More information

Brief summary of linear programming and duality: Consider the linear program in standard form. (P ) min z = cx. x 0. (D) max yb. z = c B x B + c N x N

Brief summary of linear programming and duality: Consider the linear program in standard form. (P ) min z = cx. x 0. (D) max yb. z = c B x B + c N x N Brief summary of linear programming and duality: Consider the linear program in standard form (P ) min z = cx s.t. Ax = b x 0 where A R m n, c R 1 n, x R n 1, b R m 1,and its dual (D) max yb s.t. ya c.

More information

SMO vs PDCO for SVM: Sequential Minimal Optimization vs Primal-Dual interior method for Convex Objectives for Support Vector Machines

SMO vs PDCO for SVM: Sequential Minimal Optimization vs Primal-Dual interior method for Convex Objectives for Support Vector Machines vs for SVM: Sequential Minimal Optimization vs Primal-Dual interior method for Convex Objectives for Support Vector Machines Ding Ma Michael Saunders Working paper, January 5 Introduction In machine learning,

More information

Sum-Power Iterative Watefilling Algorithm

Sum-Power Iterative Watefilling Algorithm Sum-Power Iterative Watefilling Algorithm Daniel P. Palomar Hong Kong University of Science and Technolgy (HKUST) ELEC547 - Convex Optimization Fall 2009-10, HKUST, Hong Kong November 11, 2009 Outline

More information

Totally Unimodular Stochastic Programs

Totally Unimodular Stochastic Programs Totally Unimodular Stochastic Programs Nan Kong Weldon School of Biomedical Engineering, Purdue University 206 S. Martin Jischke Dr., West Lafayette, IN 47907 nkong@purdue.edu Andrew J. Schaefer Department

More information

MVE165/MMG631 Linear and integer optimization with applications Lecture 5 Linear programming duality and sensitivity analysis

MVE165/MMG631 Linear and integer optimization with applications Lecture 5 Linear programming duality and sensitivity analysis MVE165/MMG631 Linear and integer optimization with applications Lecture 5 Linear programming duality and sensitivity analysis Ann-Brith Strömberg 2017 03 29 Lecture 4 Linear and integer optimization with

More information

Multicommodity Flows and Column Generation

Multicommodity Flows and Column Generation Lecture Notes Multicommodity Flows and Column Generation Marc Pfetsch Zuse Institute Berlin pfetsch@zib.de last change: 2/8/2006 Technische Universität Berlin Fakultät II, Institut für Mathematik WS 2006/07

More information

maxz = 3x 1 +4x 2 2x 1 +x 2 6 2x 1 +3x 2 9 x 1,x 2

maxz = 3x 1 +4x 2 2x 1 +x 2 6 2x 1 +3x 2 9 x 1,x 2 ex-5.-5. Foundations of Operations Research Prof. E. Amaldi 5. Branch-and-Bound Given the integer linear program maxz = x +x x +x 6 x +x 9 x,x integer solve it via the Branch-and-Bound method (solving

More information

Lecture 10: Linear programming. duality. and. The dual of the LP in standard form. maximize w = b T y (D) subject to A T y c, minimize z = c T x (P)

Lecture 10: Linear programming. duality. and. The dual of the LP in standard form. maximize w = b T y (D) subject to A T y c, minimize z = c T x (P) Lecture 10: Linear programming duality Michael Patriksson 19 February 2004 0-0 The dual of the LP in standard form minimize z = c T x (P) subject to Ax = b, x 0 n, and maximize w = b T y (D) subject to

More information

Acceleration and Stabilization Techniques for Column Generation

Acceleration and Stabilization Techniques for Column Generation Acceleration and Stabilization Techniques for Column Generation Zhouchun Huang Qipeng Phil Zheng Department of Industrial Engineering & Management Systems University of Central Florida Sep 26, 2014 Outline

More information

The L-Shaped Method. Operations Research. Anthony Papavasiliou 1 / 38

The L-Shaped Method. Operations Research. Anthony Papavasiliou 1 / 38 1 / 38 The L-Shaped Method Operations Research Anthony Papavasiliou Contents 2 / 38 1 The L-Shaped Method 2 Example: Capacity Expansion Planning 3 Examples with Optimality Cuts [ 5.1a of BL] 4 Examples

More information

Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 2

Pedro Munari - COA 2017, February 10th, University of Edinburgh, Scotland, UK 2 Pedro Munari [munari@dep.ufscar.br] - COA 2017, February 10th, University of Edinburgh, Scotland, UK 2 Outline Vehicle routing problem; How interior point methods can help; Interior point branch-price-and-cut:

More information

(includes both Phases I & II)

(includes both Phases I & II) Minimize z=3x 5x 4x 7x 5x 4x subject to 2x x2 x4 3x6 0 x 3x3 x4 3x5 2x6 2 4x2 2x3 3x4 x5 5 and x 0 j, 6 2 3 4 5 6 j ecause of the lack of a slack variable in each constraint, we must use Phase I to find

More information

Operations Research Lecture 6: Integer Programming

Operations Research Lecture 6: Integer Programming Operations Research Lecture 6: Integer Programming Notes taken by Kaiquan Xu@Business School, Nanjing University May 12th 2016 1 Integer programming (IP) formulations The integer programming (IP) is the

More information

Distributed model predictive control based on Benders decomposition applied to multisource multizone building temperature regulation

Distributed model predictive control based on Benders decomposition applied to multisource multizone building temperature regulation Distributed model predictive control based on Benders decomposition applied to multisource multizone building temperature regulation Petru-Daniel Moroşan, Romain Bourdais, Didier Dumur, Jean Buisson Abstract

More information

Introduction to Nonlinear Stochastic Programming

Introduction to Nonlinear Stochastic Programming School of Mathematics T H E U N I V E R S I T Y O H F R G E D I N B U Introduction to Nonlinear Stochastic Programming Jacek Gondzio Email: J.Gondzio@ed.ac.uk URL: http://www.maths.ed.ac.uk/~gondzio SPS

More information

Decomposition Algorithms with Parametric Gomory Cuts for Two-Stage Stochastic Integer Programs

Decomposition Algorithms with Parametric Gomory Cuts for Two-Stage Stochastic Integer Programs Decomposition Algorithms with Parametric Gomory Cuts for Two-Stage Stochastic Integer Programs Dinakar Gade, Simge Küçükyavuz, Suvrajeet Sen Integrated Systems Engineering 210 Baker Systems, 1971 Neil

More information

Benders Decomposition for the Uncapacitated Multicommodity Network Design Problem

Benders Decomposition for the Uncapacitated Multicommodity Network Design Problem Benders Decomposition for the Uncapacitated Multicommodity Network Design Problem 1 Carlos Armando Zetina, 1 Ivan Contreras, 2 Jean-François Cordeau 1 Concordia University and CIRRELT, Montréal, Canada

More information

Large-scale optimization and decomposition methods: outline. Column Generation and Cutting Plane methods: a unified view

Large-scale optimization and decomposition methods: outline. Column Generation and Cutting Plane methods: a unified view Large-scale optimization and decomposition methods: outline I Solution approaches for large-scaled problems: I Delayed column generation I Cutting plane methods (delayed constraint generation) 7 I Problems

More information

Chap6 Duality Theory and Sensitivity Analysis

Chap6 Duality Theory and Sensitivity Analysis Chap6 Duality Theory and Sensitivity Analysis The rationale of duality theory Max 4x 1 + x 2 + 5x 3 + 3x 4 S.T. x 1 x 2 x 3 + 3x 4 1 5x 1 + x 2 + 3x 3 + 8x 4 55 x 1 + 2x 2 + 3x 3 5x 4 3 x 1 ~x 4 0 If we

More information

A Capacity Scaling Procedure for the Multi-Commodity Capacitated Network Design Problem. Ryutsu Keizai University Naoto KATAYAMA

A Capacity Scaling Procedure for the Multi-Commodity Capacitated Network Design Problem. Ryutsu Keizai University Naoto KATAYAMA A Capacity Scaling Procedure for the Multi-Commodity Capacitated Network Design Problem Ryutsu Keizai University Naoto KATAYAMA Problems 2006 1 Multi-Commodity Network Design Problem The basic model for

More information

An improved L-shaped method for two-stage convex 0-1 mixed integer nonlinear stochastic programs

An improved L-shaped method for two-stage convex 0-1 mixed integer nonlinear stochastic programs An improved L-shaped method for two-stage convex 0-1 mixed integer nonlinear stochastic programs Can Li a, Ignacio E. Grossmann a, a Center for Advanced Process Decision-Making, Department of Chemical

More information

Integer Programming Reformulations: Dantzig-Wolfe & Benders Decomposition the Coluna Software Platform

Integer Programming Reformulations: Dantzig-Wolfe & Benders Decomposition the Coluna Software Platform Integer Programming Reformulations: Dantzig-Wolfe & Benders Decomposition the Coluna Software Platform François Vanderbeck B. Detienne, F. Clautiaux, R. Griset, T. Leite, G. Marques, V. Nesello, A. Pessoa,

More information

Extended Formulations, Lagrangian Relaxation, & Column Generation: tackling large scale applications

Extended Formulations, Lagrangian Relaxation, & Column Generation: tackling large scale applications Extended Formulations, Lagrangian Relaxation, & Column Generation: tackling large scale applications François Vanderbeck University of Bordeaux INRIA Bordeaux-Sud-Ouest part : Defining Extended Formulations

More information

Algorithms and Theory of Computation. Lecture 13: Linear Programming (2)

Algorithms and Theory of Computation. Lecture 13: Linear Programming (2) Algorithms and Theory of Computation Lecture 13: Linear Programming (2) Xiaohui Bei MAS 714 September 25, 2018 Nanyang Technological University MAS 714 September 25, 2018 1 / 15 LP Duality Primal problem

More information

Stochastic Uncapacitated Hub Location

Stochastic Uncapacitated Hub Location Stochastic Uncapacitated Hub Location Ivan Contreras, Jean-François Cordeau, Gilbert Laporte HEC Montréal and Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT),

More information

Logic-Based Benders Decomposition

Logic-Based Benders Decomposition Logic-Based Benders Decomposition J. N. Hooker Graduate School of Industrial Administration Carnegie Mellon University, Pittsburgh, PA 15213 USA G. Ottosson Department of Information Technology Computing

More information

Relation of Pure Minimum Cost Flow Model to Linear Programming

Relation of Pure Minimum Cost Flow Model to Linear Programming Appendix A Page 1 Relation of Pure Minimum Cost Flow Model to Linear Programming The Network Model The network pure minimum cost flow model has m nodes. The external flows given by the vector b with m

More information

Part 1. The Review of Linear Programming

Part 1. The Review of Linear Programming In the name of God Part 1. The Review of Linear Programming 1.5. Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Formulation of the Dual Problem Primal-Dual Relationship Economic Interpretation

More information

An Integrated Column Generation and Lagrangian Relaxation for Flowshop Scheduling Problems

An Integrated Column Generation and Lagrangian Relaxation for Flowshop Scheduling Problems Proceedings of the 2009 IEEE International Conference on Systems, Man, and Cybernetics San Antonio, TX, USA - October 2009 An Integrated Column Generation and Lagrangian Relaxation for Flowshop Scheduling

More information

Distributed Estimation via Dual Decomposition

Distributed Estimation via Dual Decomposition Proceedings of the European Control Conference 27 Kos, Greece, July 2 5, 27 TuC9.1 Distributed Estimation via Dual Decomposition Sikandar Samar Integrated Data Systems Department Siemens Corporate Research

More information

COMP3121/9101/3821/9801 Lecture Notes. Linear Programming

COMP3121/9101/3821/9801 Lecture Notes. Linear Programming COMP3121/9101/3821/9801 Lecture Notes Linear Programming LiC: Aleks Ignjatovic THE UNIVERSITY OF NEW SOUTH WALES School of Computer Science and Engineering The University of New South Wales Sydney 2052,

More information

Primal-Dual Interior-Point Methods for Linear Programming based on Newton s Method

Primal-Dual Interior-Point Methods for Linear Programming based on Newton s Method Primal-Dual Interior-Point Methods for Linear Programming based on Newton s Method Robert M. Freund March, 2004 2004 Massachusetts Institute of Technology. The Problem The logarithmic barrier approach

More information

Stochastic Integer Programming An Algorithmic Perspective

Stochastic Integer Programming An Algorithmic Perspective Stochastic Integer Programming An Algorithmic Perspective sahmed@isye.gatech.edu www.isye.gatech.edu/~sahmed School of Industrial & Systems Engineering 2 Outline Two-stage SIP Formulation Challenges Simple

More information

ECE Optimization for wireless networks Final. minimize f o (x) s.t. Ax = b,

ECE Optimization for wireless networks Final. minimize f o (x) s.t. Ax = b, ECE 788 - Optimization for wireless networks Final Please provide clear and complete answers. PART I: Questions - Q.. Discuss an iterative algorithm that converges to the solution of the problem minimize

More information

min3x 1 + 4x 2 + 5x 3 2x 1 + 2x 2 + x 3 6 x 1 + 2x 2 + 3x 3 5 x 1, x 2, x 3 0.

min3x 1 + 4x 2 + 5x 3 2x 1 + 2x 2 + x 3 6 x 1 + 2x 2 + 3x 3 5 x 1, x 2, x 3 0. ex-.-. Foundations of Operations Research Prof. E. Amaldi. Dual simplex algorithm Given the linear program minx + x + x x + x + x 6 x + x + x x, x, x. solve it via the dual simplex algorithm. Describe

More information

Nonconvex Generalized Benders Decomposition Paul I. Barton

Nonconvex Generalized Benders Decomposition Paul I. Barton Nonconvex Generalized Benders Decomposition Paul I. Barton Process Systems Engineering Laboratory Massachusetts Institute of Technology Motivation Two-stage Stochastic MINLPs & MILPs Natural Gas Production

More information

Section Notes 9. Midterm 2 Review. Applied Math / Engineering Sciences 121. Week of December 3, 2018

Section Notes 9. Midterm 2 Review. Applied Math / Engineering Sciences 121. Week of December 3, 2018 Section Notes 9 Midterm 2 Review Applied Math / Engineering Sciences 121 Week of December 3, 2018 The following list of topics is an overview of the material that was covered in the lectures and sections

More information

Weighted Acyclic Di-Graph Partitioning by Balanced Disjoint Paths

Weighted Acyclic Di-Graph Partitioning by Balanced Disjoint Paths Weighted Acyclic Di-Graph Partitioning by Balanced Disjoint Paths H. Murat AFSAR Olivier BRIANT Murat.Afsar@g-scop.inpg.fr Olivier.Briant@g-scop.inpg.fr G-SCOP Laboratory Grenoble Institute of Technology

More information

A DECOMPOSITION PROCEDURE BASED ON APPROXIMATE NEWTON DIRECTIONS

A DECOMPOSITION PROCEDURE BASED ON APPROXIMATE NEWTON DIRECTIONS Working Paper 01 09 Departamento de Estadística y Econometría Statistics and Econometrics Series 06 Universidad Carlos III de Madrid January 2001 Calle Madrid, 126 28903 Getafe (Spain) Fax (34) 91 624

More information

to work with) can be solved by solving their LP relaxations with the Simplex method I Cutting plane algorithms, e.g., Gomory s fractional cutting

to work with) can be solved by solving their LP relaxations with the Simplex method I Cutting plane algorithms, e.g., Gomory s fractional cutting Summary so far z =max{c T x : Ax apple b, x 2 Z n +} I Modeling with IP (and MIP, and BIP) problems I Formulation for a discrete set that is a feasible region of an IP I Alternative formulations for the

More information

F 1 F 2 Daily Requirement Cost N N N

F 1 F 2 Daily Requirement Cost N N N Chapter 5 DUALITY 5. The Dual Problems Every linear programming problem has associated with it another linear programming problem and that the two problems have such a close relationship that whenever

More information

Linear Programming Inverse Projection Theory Chapter 3

Linear Programming Inverse Projection Theory Chapter 3 1 Linear Programming Inverse Projection Theory Chapter 3 University of Chicago Booth School of Business Kipp Martin September 26, 2017 2 Where We Are Headed We want to solve problems with special structure!

More information

CS711008Z Algorithm Design and Analysis

CS711008Z Algorithm Design and Analysis CS711008Z Algorithm Design and Analysis Lecture 8 Linear programming: interior point method Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China 1 / 31 Outline Brief

More information

Review Solutions, Exam 2, Operations Research

Review Solutions, Exam 2, Operations Research Review Solutions, Exam 2, Operations Research 1. Prove the weak duality theorem: For any x feasible for the primal and y feasible for the dual, then... HINT: Consider the quantity y T Ax. SOLUTION: To

More information

Lecture 9 Tuesday, 4/20/10. Linear Programming

Lecture 9 Tuesday, 4/20/10. Linear Programming UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Spring, 2010 Lecture 9 Tuesday, 4/20/10 Linear Programming 1 Overview Motivation & Basics Standard & Slack Forms Formulating

More information

Scenario grouping and decomposition algorithms for chance-constrained programs

Scenario grouping and decomposition algorithms for chance-constrained programs Scenario grouping and decomposition algorithms for chance-constrained programs Yan Deng Shabbir Ahmed Jon Lee Siqian Shen Abstract A lower bound for a finite-scenario chance-constrained problem is given

More information

Economic MPC for large and distributed energy systems

Economic MPC for large and distributed energy systems Economic MPC for large and distributed energy systems WP4 Laura Standardi, Niels Kjølstad Poulsen, John Bagterp Jørgensen August 15, 2014 1 / 34 Outline Introduction and motivation Problem definition Economic

More information

Overview of course. Introduction to Optimization, DIKU Monday 12 November David Pisinger

Overview of course. Introduction to Optimization, DIKU Monday 12 November David Pisinger Introduction to Optimization, DIKU 007-08 Monday November David Pisinger Lecture What is OR, linear models, standard form, slack form, simplex repetition, graphical interpretation, extreme points, basic

More information

Lecture Note 18: Duality

Lecture Note 18: Duality MATH 5330: Computational Methods of Linear Algebra 1 The Dual Problems Lecture Note 18: Duality Xianyi Zeng Department of Mathematical Sciences, UTEP The concept duality, just like accuracy and stability,

More information

SELECT TWO PROBLEMS (OF A POSSIBLE FOUR) FROM PART ONE, AND FOUR PROBLEMS (OF A POSSIBLE FIVE) FROM PART TWO. PART ONE: TOTAL GRAND

SELECT TWO PROBLEMS (OF A POSSIBLE FOUR) FROM PART ONE, AND FOUR PROBLEMS (OF A POSSIBLE FIVE) FROM PART TWO. PART ONE: TOTAL GRAND 1 56:270 LINEAR PROGRAMMING FINAL EXAMINATION - MAY 17, 1985 SELECT TWO PROBLEMS (OF A POSSIBLE FOUR) FROM PART ONE, AND FOUR PROBLEMS (OF A POSSIBLE FIVE) FROM PART TWO. PART ONE: 1 2 3 4 TOTAL GRAND

More information

Notes on Dantzig-Wolfe decomposition and column generation

Notes on Dantzig-Wolfe decomposition and column generation Notes on Dantzig-Wolfe decomposition and column generation Mette Gamst November 11, 2010 1 Introduction This note introduces an exact solution method for mathematical programming problems. The method is

More information

Part 1. The Review of Linear Programming

Part 1. The Review of Linear Programming In the name of God Part 1. The Review of Linear Programming 1.2. Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Basic Feasible Solutions Key to the Algebra of the The Simplex Algorithm

More information

A BRANCH&BOUND ALGORITHM FOR SOLVING ONE-DIMENSIONAL CUTTING STOCK PROBLEMS EXACTLY

A BRANCH&BOUND ALGORITHM FOR SOLVING ONE-DIMENSIONAL CUTTING STOCK PROBLEMS EXACTLY APPLICATIONES MATHEMATICAE 23,2 (1995), pp. 151 167 G. SCHEITHAUER and J. TERNO (Dresden) A BRANCH&BOUND ALGORITHM FOR SOLVING ONE-DIMENSIONAL CUTTING STOCK PROBLEMS EXACTLY Abstract. Many numerical computations

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL Y -» $ 5 Y 7 Y Y -Y- Q x Q» 75»»/ q } # ]»\ - - $ { Q» / X x»»- 3 q $ 9 ) Y q - 5 5 3 3 3 7 Q q - - Q _»»/Q Y - 9 - - - )- [ X 7» -» - )»? / /? Q Y»» # X Q» - -?» Q ) Q \ Q - - - 3? 7» -? #»»» 7 - / Q

More information

Primal/Dual Decomposition Methods

Primal/Dual Decomposition Methods Primal/Dual Decomposition Methods Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2018-19, HKUST, Hong Kong Outline of Lecture Subgradients

More information

Lagrange Relaxation: Introduction and Applications

Lagrange Relaxation: Introduction and Applications 1 / 23 Lagrange Relaxation: Introduction and Applications Operations Research Anthony Papavasiliou 2 / 23 Contents 1 Context 2 Applications Application in Stochastic Programming Unit Commitment 3 / 23

More information

Scenario Grouping and Decomposition Algorithms for Chance-constrained Programs

Scenario Grouping and Decomposition Algorithms for Chance-constrained Programs Scenario Grouping and Decomposition Algorithms for Chance-constrained Programs Siqian Shen Dept. of Industrial and Operations Engineering University of Michigan Joint work with Yan Deng (UMich, Google)

More information