The Value function of a Mixed-Integer Linear Program with a Single Constraint

Size: px
Start display at page:

Download "The Value function of a Mixed-Integer Linear Program with a Single Constraint"

Transcription

1 The Value Function of a Mixed Integer Linear Programs with a Single Constraint MENAL GUZELSOY TED RALPHS ISE Department COR@L Lab Lehigh University tkralphs@lehigh.edu University of Wisconsin February 8, 008 Thanks: Work supported in part by the National Science Foundation

2 Outline The Value Function MILP Duality Linear Approximations Structure of The Value Function Introduction Extending the Value Function Evaluating The Value Function 4

3 Motivation The Value Function MILP Duality The goal of this work is to study the structure of the value function of a general MILP. Eventually, we hope this will lead to methods for approximation useful for sensitivity analysis warm starting other methods that requires dual information Computing the value function (or even an approximation) is difficult even in a small neighborhood Our approach is to begin by considering the value functions of various single-row relaxations.

4 The Value Function MILP Duality Consider a general mixed-integer linear program (MILP) z P = min x S cx, (P) c R n, S = {x Z r + R n r + Ax = b} with A Q m n, b R m. The value function of the primal problem (P) is z(d) = min x S(d) cx, where for a given d R m, S(d) = {x Z r + R n r + Ax = d}. We let z(d) = if S(d) =.

5 Previous Work The Value Function MILP Duality Johnson [97,974,979], Jeroslow [978] : the theory of subadditive duality for integer linear programs. Pure Integer Programs: Jeroslow [98] : Gomory functions - maximum of finitely many subadditive functions. Lasserre [004] : generating functions, two-sided Z transformation. Loera et al. [004] : generating functions, global test set. Mixed Integer Programs: Jeroslow [98] : minimum of finitely many Gomory functions. Blair [995] : Jeroslow formula - consisting of a Gomory function and a correction term.

6 Subadditive Duality The Value Function MILP Duality A function F is subadditive over a domain Θ if F(λ ) + F(λ ) F(λ + λ ) λ,λ,λ + λ Θ. For b R, the subadditive dual of the primal problem: z D = max F(b) F(a j ) c j j I, F(a j ) c j j C, and F is subadditive, F(0) = 0, where a j is the j th column of A, I = {,...,r}, C = {r +,...,n} and F is defined by F(d) = lim sup δ 0 + F(δd) δ d R m. F (the upper d-directional derivative of F at zero) is positively homogeneous, subadditive, and bounds F from above.

7 Primal-Dual Relations The Value Function MILP Duality Theorem (Weak Duality) Let x be a feasible solution to the primal problem and let F be a feasible solution to the subadditive dual problem. Then F(b) cx. Theorem (Strong Duality) If the primal problem (resp., the dual) has a finite optimum, then so does the dual problem (resp., the primal) and they are equal. In other words, there exists a dual feasible F such that F (b) = z D = z P = z(b) and F (d) z(d) d R m. Extensions: complementarity, optimality conditions, etc.

8 The Value Function MILP Duality Chvátal and Gomory Functions Let L m = {f f : R m R, f is linear}. Chvátal functions are the smallest set of functions C m such that If f L m, then f C m. If f, f C m and α, β Q +, then αf + βf C m. If f C m, then f C m. Gomory functions are the smallest set of functions G m C m with the additional property that If f, f G m, then max{f, f } G m. Theorem For PILPs (r = n), if z(0) = 0, then there is a g G m such that g(d) = z(d) for all d R m with S(d).

9 Jeroslow Formula - MILP The Value Function MILP Duality Let the set E consist of the index sets of dual feasible bases of the linear program min{ M c Cx C : M A Cx C = b, x 0} where M Z + such that for any E E, MA E aj Z m for all j I. Theorem (Jeroslow Formular) There is a g G m such that z(d) = min E E g( d E ) + v E(d d E ) d R m with S(d), where for E E, d E = A E A E d and v E is the corresponding basic feasible solution.

10 Linear Approximations Structure of The Value Function We will consider min cx, x S (P) c R n, S = {x Z r + R n r + a x = b} with a Q n, b R. The value function of the primal problem (P) is now z(d) = min x S(d) cx, where for a given d R, S(d) = {x Z r + R n r + a x = d}. Assumptions: Let N = I C, z(0) = 0 = z : R R {+ }, N + = {i N a i > 0} and N = {i N a i < 0}, r < n, that is, C = z : R R.

11 Example Linear Approximations Structure of The Value Function min x + x + x 4 + x x 6 s.t x x + x + x 4 x 5 + x 6 = b and x, x, x Z +, x 4, x 5, x 6 R +. z F d

12 Lower Bound (LP Relaxation) Linear Approximations Structure of The Value Function The value function of the LP relaxation yields a lower bound. In this case, it has a convenient closed form: ηd if d > 0, F L (d) = max{ud ζ u η, u R} = 0 if d = 0, ζd if d < 0. where F L z. η = min{ c i a i i N + } and ζ = max{ c i a i i N }.

13 Linear Approximations Structure of The Value Function Upper Bound (Continuous Relaxation) To get an upper bound, we consider only the continuous variables: F U (d) = min{ c i x i η C d if d > 0 a i x i = d, x i 0 i C} = 0 if d = 0 i C i C ζ C d if d < 0 where η C = min{ c i a i i C + = {i C a i > 0}} and ζ C = max{ c i a i i C = {i C a i < 0}} By convention: C + η C =. C ζ C =. F U z

14 Example (cont d) Linear Approximations Structure of The Value Function We have η =, ζ = 0, ηc = and ζ C =. Consequently, { F L (d) = d if d 0 { d if d 0 and F 0 if d < 0 U (d) = d if d < 0 z(d) F U(d) F L(d) ζ C 5 η C ζ η d

15 Observations Linear Approximations Structure of The Value Function {η = η C } {z(d) = F U (d) = F L (d) d R + } {ζ = ζ C } {z(d) = F U (d) = F L (d) d R } z(d) F U(d) F L(d) ζ C 5 η C ζ η d

16 Observations Linear Approximations Structure of The Value Function Let d + U = sup{d 0 z(d) = F U (d)} d U = inf{d 0 z(d) = F U (d)}, d + L = inf{d > 0 z(d) = F L (d)} d L = sup{d < 0 z(d) = F L (d)}. z(d) F U(d) F L(d) 5 d d L U d + d + U L d

17 Observations Linear Approximations Structure of The Value Function η C = d + U = 0 and ζc = d U = 0 η < η C {d z(d) = F U(d) = F L(d), d R +} {0} d + U ζ > ζ C {d z(d) = F U(d) = F L(d), d R } {0} d U < > z(d) F U(d) F L(d) 5 d d L U d + d + U L d

18 Observations Linear Approximations Structure of The Value Function z(d) = F U (d) d (d U, d+ U ) d + L d+ U if d+ L > 0 and d L d U if d L < 0 if b {d R z(d) = F L (d)}, then, z(kb) = kf L (b), k Z + z(d) F U(d) F L(d) 5 d d d L L U d + d + d + U L L d

19 Observations Linear Approximations Structure of The Value Function Notice the relation between F U and the linear segments of z: {η C,ζ C } z(d) F U(d) F L(d) 5 d d d L L U d + d + d + U L L d

20 Redundant Variables Linear Approximations Structure of The Value Function Let T C be such that t + T if and only if η C < and η C = c t + a t + t T if and only if ζ C > and ζ C = c t a t. and define and similarly, ν(d) = min s.t. c I x I + c T x T a I x I + a T x T = d x I Z I +, x T R T + Then ν(d) = z(d) for all d R. The variables in C\T are redundant. z can be represented with at most continuous variables.

21 Back to the Jeroslow Formula Linear Approximations Structure of The Value Function Let M Z + be such that for any t T, Maj a t Z for all j I. Then there is a Gomory function g such that z(d) = min {g( d t T t ) + c t (d d a t )} d R t where d t = at Md M a t. It is possible to show that such a Gomory function can be obtained from the value function of a related PILP: g(q) = min s.t for all q R, where ϕ = M c I x I + M c Tx T + z(ϕ)v a I x I + M a Tx T + ϕv = q x I Z I +, x T Z+, T v Z + t T a t.

22 Linear Approximations Structure of The Value Function Piecewise-Linearity of the Value Function For t T, setting we can write ω t (d) = g( d t ) + c t a t (d d t ) d R, z(d) = min t T ω t(d) d R For t T, ω t is piecewise linear with finitely many linear segments on any closed interval and each of those linear segments has a slope of η C if t = t + or ζ C if t = t. Thus, z is also piecewise-linear with finitely many linear segments on any closed interval. Furthermore, each of those linear segments has a slope of η C or ζ C.

23 Structure of Linear Pieces Linear Approximations Structure of The Value Function Theorem If the value function z is linear on an interval U R, then there exists a ȳ Z I + such that ȳ is the integral part of an optimal solution for any d U. Consequently, for some t T, z can be written as z(d) = c I ȳ + c t a t (d a I ȳ) d U. Furthermore, for any d U, we have d a I ȳ 0 if t = t + and d a I ȳ 0 if t = t.

24 Example (cont d) Linear Approximations Structure of The Value Function T = {4, 5} and hence, x 6 is redundant. η C = and ζ C =, z d U = [0, /], U = [/, ], U = [, 7/6], U 4 = [7/6, /],... y = (0 0 0), y = ( 0 0), y = ( 0 0), y 4 = (0 0 ),... d if d U d + / if d U z(d) = d / if d U d + if d U 4...

25 Continuity Linear Approximations Structure of The Value Function ω t + is continuous from the right and ω t is continuous from the left. ω t + and ω t are both lower-semicontinuous. Theorem If z is discontinuous at a right-hand-side b R, then there exists a ȳ Z I + such that b a I ȳ = 0. z is lower-semicontinuous. η C < if and only if z is continuous from the right. ζ C > if and only if z is continuous from the left. Both η C and ζ C are finite if and only if z is continuous everywhere.

26 Example Linear Approximations Structure of The Value Function η c = /, ζ C =. min x /4x + /4x s.t 5/4x x + /x = b, x, x Z +, x R For each discontinuous point d i, we have d i (5/4y i y i ) = 0 and each linear segment has the slope of η C = /.

27 Introduction Extending the Value Function Evaluating The Value Function Let f : [0, h] R, h > 0 be subadditive and f(0) = 0. The maximal subadditive extension of f from [0, h] to R + is f(d) if d [0, h] f S (d) = inf f(ρ) if d > h, C C(d) ρ C C(d) is the set of all finite collections {ρ,..., ρ R} such that ρ i [0, h], i =,..., R and P R i= ρi = d. Each collection {ρ,..., ρ R} is called an h-partition of d. We can also extend a subadditive function f : [h, 0] R, h < 0 to R similarly. f S is subadditive and if g is any other subadditive extension of f from [0, h] to R +, then g f S (maximality).

28 Introduction Extending the Value Function Evaluating The Value Function Lemma What if we use z as the seed function? We can change the inf to min : Let the function f : [0, h] R be defined by f(d) = z(d) d [0, h]. Then, z(d) if d [0, h] f S (d) = min z(ρ) if d > h. C C(d) ρ C For any h > 0, z(d) f S (d) d R +. Observe that for d R +, f S (d) z(d) while h. Is there an h < such that f S (d) = z(d) d R +?

29 Introduction Extending the Value Function Evaluating The Value Function We can get the value function by extending it from a specific neighborhood. Theorem Let d r = max{a i i N} and d l = min{a i i N} and let the functions f r and f l be the maximal subadditive extensions of z from the intervals [0, d r ] and [d l, 0] to R + and R, respectively. Let { fr (d) d R F(d) = + f l (d) d R then, z = F. Outline of the Proof. z F : By construction. z F : Using MILP duality, F is dual feasible. In other words, the value function is completely encoded by the breakpoints in [d l, d r ] and slopes.

30 Introduction Extending the Value Function Evaluating The Value Function The next question is whether we can obtain z(d) for some d [d l, d r ] from this encoding. Consider extending z from [0, d r ] to R + : write z(d) = min z(ρ) d > d r, C C(d) ρ C Can we limit C, C C(d)? Yes! Using subadditivity! Can we limit C(d)? Yes! Using the break points of z!

31 Special Case Introduction Extending the Value Function Evaluating The Value Function Theorem Let d l, d r be defined as in Theorem 8. If z is concave in [0, d r ], then for any d R + z(d) = kz(d r ) + z(d kd r ), kd r d < (k + )d r k Z + Similarly, if z is concave in [d l, 0], then for any d R See Figure. z(d) = kz(d l ) + z(d kd l ), (k + )d l < d kd l k Z +

32 Introduction Extending the Value Function Evaluating The Value Function Theorem Let d > d r and let k d be the integer such that d ( k d d r, k d+ d r ]. Then z(d) = min{ k d i= z(ρ i ) k d i= ρ i = d,ρ i [0, d r ], i =,..., k d }. Therefore, C k d for any C C(d). How about C(d)?

33 Lower Break Points Introduction Extending the Value Function Evaluating The Value Function Theorem Set Ψ be the lower break points of z in [0, d r ]. For any d R + \[0, d r ] there is an optimal d r -partition C C(d) such that C\Ψ. In particular, we only need to consider the collection Λ(d) {H {µ} H C(d µ), H = k d, H Ψ, ρ H ρ + µ = d,µ In other words, z(d) = min C Λ(d) ρ C z(ρ) d R + \[0, d r ] Observe that the set Λ(d) is finite, since Ψ is finite due to the fact that z has finitely many linear segments on [0, d r ] and since for each H Λ(d), µ is uniquely determined.

34 Example (cont d) Introduction Extending the Value Function Evaluating The Value Function For the interval [ 0, ], we have Ψ = {0,, }. For b = 5 8, C = { 8,, } is an optimal d r -partition with C\Ψ =. z(d) U-bp L-bp d

35 Introduction Extending the Value Function Evaluating The Value Function Evaluating the breakpoints in [d l, d r ] Note that z overlaps with F U = η C d in a right-neighborhood of origin and with F U = ζ C d in a left-neighborhood of origin. The slope of each linear segment is either η C or ζ C. Furthermore, if both η C and ζ C are finite, then the slopes of linear segments alternate between η C and ζ C (continuity). For d, d [0, d r] (or [d l, 0]), if z(d ) and z(d ) are on the line with the slope of η C (or ζ C ), then z is linear over [d, d ] with the same slope (subadditivity). With these observations, we can formulate a finite algorithm to evaluate z in [d l, d r ].

36 Example (cont d) Introduction Extending the Value Function Evaluating The Value Function η C ζ C 0 Figure: Evaluating z in [0, ]

37 Example (cont d) Introduction Extending the Value Function Evaluating The Value Function η C ζ C 0 Figure: Evaluating z in [0, ]

38 Example (cont d) Introduction Extending the Value Function Evaluating The Value Function η C ζ C 0 Figure: Evaluating z in [0, ]

39 Example (cont d) Introduction Extending the Value Function Evaluating The Value Function η C ζ C 0 Figure: Evaluating z in [0, ]

40 Combinatorial Approach Introduction Extending the Value Function Evaluating The Value Function We can formulate the problem of evaluating z(d) for d R + \[0, d r ] as a Constrained Shortest Path Problem. Among the paths (feasible partitions) of size k b with exactly k b edges (members of each partition) of each path chosen from Ψ, we need to find the minimum-cost path with a total length of d.

41 Recursive Construction Introduction Extending the Value Function Evaluating The Value Function Set Ψ(p) to the set of the lower break points of z in the interval (0, p] p R +. Let p := d r. For any d `p, p + p, let z(d) = min{z(ρ ) + z(ρ ) ρ + ρ = d, ρ Ψ(p), ρ (0, p]} Let p := p + p and repeat this step.

42 Introduction Extending the Value Function Evaluating The Value Function We can also do the following: z(d) = min g j (d) d j ( p, p + p ] where, for each d j Ψ(p), the functions g j : [ 0, p + p ] R { } are defined as z(d) if d d j, g j (d) = z(d j ) + z(d d j ) if d j < d p + d j, otherwise. Because of subadditivity, we can then write z(d) = min g j (d) d j ( 0, p + p ].

43 Example (cont d) Introduction Extending the Value Function Evaluating The Value Function Extending the value function of (6) from [ [ ] 0, ] to 0, 9 4 z(d) F U(d) F L(d) d

44 Example (cont d) Introduction Extending the Value Function Evaluating The Value Function Extending the value function of (6) from [ [ ] 0, ] to 0, 9 4 z(d) F U(d) F L(d) 5 g g d

45 Example (cont d) Introduction Extending the Value Function Evaluating The Value Function Extending the value function of (6) from [ [ ] 0, ] to 0, 9 4 z(d) F U(d) F L(d) d

46 Example (cont d) Introduction Extending the Value Function Evaluating The Value Function Extending the value function of (6) from [ [ ] 0, 4] 9 to 0, 7 8 z(d) F U(d) F L(d) d

47 Example (cont d) Introduction Extending the Value Function Evaluating The Value Function Extending the value function of (6) from [ [ ] 0, 4] 9 to 0, 7 8 z(d) F U(d) F L(d) 5 g g g g d

48 Example (cont d) Introduction Extending the Value Function Evaluating The Value Function Extending the value function of (6) from [ [ ] 0, 4] 9 to 0, 7 8 z(d) F U(d) F L(d) d

49 Computational Experiments Approximating the value function of a MILP with a single constraint. Extending our results to bounded case. Approximating the value function of a general MILP using the value functions of single constraint relaxations.

The Value function of a Mixed-Integer Linear Program with a Single Constraint

The Value function of a Mixed-Integer Linear Program with a Single Constraint The Value Function of a Mixed Integer Linear Programs with a Single Constraint MENAL GUZELSOY TED RALPHS ISE Department COR@L Lab Lehigh University tkralphs@lehigh.edu OPT 008, Atlanta, March 4, 008 Thanks:

More information

On the Value Function of a Mixed Integer Linear Program

On the Value Function of a Mixed Integer Linear Program On the Value Function of a Mixed Integer Linear Program MENAL GUZELSOY TED RALPHS ISE Department COR@L Lab Lehigh University ted@lehigh.edu AIRO, Ischia, Italy, 11 September 008 Thanks: Work supported

More information

Integer Programming Duality

Integer Programming Duality Integer Programming Duality M. Guzelsoy T. K. Ralphs July, 2010 1 Introduction This article describes what is known about duality for integer programs. It is perhaps surprising that many of the results

More information

Bilevel Integer Linear Programming

Bilevel Integer Linear Programming Bilevel Integer Linear Programming TED RALPHS SCOTT DENEGRE ISE Department COR@L Lab Lehigh University ted@lehigh.edu MOPTA 2009, Lehigh University, 19 August 2009 Thanks: Work supported in part by the

More information

Duality for Mixed-Integer Linear Programs

Duality for Mixed-Integer Linear Programs Duality for Mixed-Integer Linear Programs M. Guzelsoy T. K. Ralphs Original May, 006 Revised April, 007 Abstract The theory of duality for linear programs is well-developed and has been successful in advancing

More information

Bilevel Integer Programming

Bilevel Integer Programming Bilevel Integer Programming Ted Ralphs 1 Joint work with: Scott DeNegre 1, Menal Guzelsoy 1 COR@L Lab, Department of Industrial and Systems Engineering, Lehigh University Norfolk Southern Ralphs, et al.

More information

Bilevel Integer Linear Programming

Bilevel Integer Linear Programming Bilevel Integer Linear Programming SCOTT DENEGRE TED RALPHS ISE Department COR@L Lab Lehigh University ted@lehigh.edu Université Bordeaux, 16 December 2008 Thanks: Work supported in part by the National

More information

On the Value Function of a Mixed Integer Linear Optimization Problem and an Algorithm for its Construction

On the Value Function of a Mixed Integer Linear Optimization Problem and an Algorithm for its Construction On the Value Function of a Mixed Integer Linear Optimization Problem and an Algorithm for its Construction Ted K. Ralphs and Anahita Hassanzadeh Department of Industrial and Systems Engineering, Lehigh

More information

Duality, Warm Starting, and Sensitivity Analysis for MILP

Duality, Warm Starting, and Sensitivity Analysis for MILP Duality, Warm Starting, and Sensitivity Analysis for MILP Ted Ralphs and Menal Guzelsoy Industrial and Systems Engineering Lehigh University SAS Institute, Cary, NC, Friday, August 19, 2005 SAS Institute

More information

A Benders Algorithm for Two-Stage Stochastic Optimization Problems With Mixed Integer Recourse

A Benders Algorithm for Two-Stage Stochastic Optimization Problems With Mixed Integer Recourse A Benders Algorithm for Two-Stage Stochastic Optimization Problems With Mixed Integer Recourse Ted Ralphs 1 Joint work with Menal Güzelsoy 2 and Anahita Hassanzadeh 1 1 COR@L Lab, Department of Industrial

More information

Bilevel Integer Optimization: Theory and Algorithms

Bilevel Integer Optimization: Theory and Algorithms : Theory and Algorithms Ted Ralphs 1 Joint work with Sahar Tahernajad 1, Scott DeNegre 3, Menal Güzelsoy 2, Anahita Hassanzadeh 4 1 COR@L Lab, Department of Industrial and Systems Engineering, Lehigh University

More information

Asteroide Santana, Santanu S. Dey. December 4, School of Industrial and Systems Engineering, Georgia Institute of Technology

Asteroide Santana, Santanu S. Dey. December 4, School of Industrial and Systems Engineering, Georgia Institute of Technology for Some for Asteroide Santana, Santanu S. Dey School of Industrial Systems Engineering, Georgia Institute of Technology December 4, 2016 1 / 38 1 1.1 Conic integer programs for Conic integer programs

More information

Generation and Representation of Piecewise Polyhedral Value Functions

Generation and Representation of Piecewise Polyhedral Value Functions Generation and Representation of Piecewise Polyhedral Value Functions Ted Ralphs 1 Joint work with Menal Güzelsoy 2 and Anahita Hassanzadeh 1 1 COR@L Lab, Department of Industrial and Systems Engineering,

More information

IP Duality. Menal Guzelsoy. Seminar Series, /21-07/28-08/04-08/11. Department of Industrial and Systems Engineering Lehigh University

IP Duality. Menal Guzelsoy. Seminar Series, /21-07/28-08/04-08/11. Department of Industrial and Systems Engineering Lehigh University IP Duality Department of Industrial and Systems Engineering Lehigh University COR@L Seminar Series, 2005 07/21-07/28-08/04-08/11 Outline Duality Theorem 1 Duality Theorem Introduction Optimality Conditions

More information

Integer Programming ISE 418. Lecture 8. Dr. Ted Ralphs

Integer Programming ISE 418. Lecture 8. Dr. Ted Ralphs Integer Programming ISE 418 Lecture 8 Dr. Ted Ralphs ISE 418 Lecture 8 1 Reading for This Lecture Wolsey Chapter 2 Nemhauser and Wolsey Sections II.3.1, II.3.6, II.4.1, II.4.2, II.5.4 Duality for Mixed-Integer

More information

- Well-characterized problems, min-max relations, approximate certificates. - LP problems in the standard form, primal and dual linear programs

- Well-characterized problems, min-max relations, approximate certificates. - LP problems in the standard form, primal and dual linear programs LP-Duality ( Approximation Algorithms by V. Vazirani, Chapter 12) - Well-characterized problems, min-max relations, approximate certificates - LP problems in the standard form, primal and dual linear programs

More information

Strong Dual for Conic Mixed-Integer Programs

Strong Dual for Conic Mixed-Integer Programs Strong Dual for Conic Mixed-Integer Programs Diego A. Morán R. Santanu S. Dey Juan Pablo Vielma July 14, 011 Abstract Mixed-integer conic programming is a generalization of mixed-integer linear programming.

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

Optimization for Communications and Networks. Poompat Saengudomlert. Session 4 Duality and Lagrange Multipliers

Optimization for Communications and Networks. Poompat Saengudomlert. Session 4 Duality and Lagrange Multipliers Optimization for Communications and Networks Poompat Saengudomlert Session 4 Duality and Lagrange Multipliers P Saengudomlert (2015) Optimization Session 4 1 / 14 24 Dual Problems Consider a primal convex

More information

Mixed-integer nonlinear programming, Conic programming, Duality, Cutting

Mixed-integer nonlinear programming, Conic programming, Duality, Cutting A STRONG DUAL FOR CONIC MIXED-INTEGER PROGRAMS DIEGO A. MORÁN R., SANTANU S. DEY, AND JUAN PABLO VIELMA Abstract. Mixed-integer conic programming is a generalization of mixed-integer linear programming.

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

A non-standart approach to Duality

A non-standart approach to Duality Department of Industrial and Systems Engineering Lehigh University COR@L Seminar Series, 2005 10/27/2005 Outline 1 Duality Integration & Counting Duality for Integration Brion and Vergne s continuous formula

More information

4. The Dual Simplex Method

4. The Dual Simplex Method 4. The Dual Simplex Method Javier Larrosa Albert Oliveras Enric Rodríguez-Carbonell Problem Solving and Constraint Programming (RPAR) Session 4 p.1/34 Basic Idea (1) Algorithm as explained so far known

More information

Midterm Review. Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A.

Midterm Review. Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. Midterm Review Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye (LY, Chapter 1-4, Appendices) 1 Separating hyperplane

More information

Structure of Valid Inequalities for Mixed Integer Conic Programs

Structure of Valid Inequalities for Mixed Integer Conic Programs Structure of Valid Inequalities for Mixed Integer Conic Programs Fatma Kılınç-Karzan Tepper School of Business Carnegie Mellon University 18 th Combinatorial Optimization Workshop Aussois, France January

More information

Structure in Mixed Integer Conic Optimization: From Minimal Inequalities to Conic Disjunctive Cuts

Structure in Mixed Integer Conic Optimization: From Minimal Inequalities to Conic Disjunctive Cuts Structure in Mixed Integer Conic Optimization: From Minimal Inequalities to Conic Disjunctive Cuts Fatma Kılınç-Karzan Tepper School of Business Carnegie Mellon University Joint work with Sercan Yıldız

More information

Chapter 1 Linear Programming. Paragraph 5 Duality

Chapter 1 Linear Programming. Paragraph 5 Duality Chapter 1 Linear Programming Paragraph 5 Duality What we did so far We developed the 2-Phase Simplex Algorithm: Hop (reasonably) from basic solution (bs) to bs until you find a basic feasible solution

More information

1 Review of last lecture and introduction

1 Review of last lecture and introduction Semidefinite Programming Lecture 10 OR 637 Spring 2008 April 16, 2008 (Wednesday) Instructor: Michael Jeremy Todd Scribe: Yogeshwer (Yogi) Sharma 1 Review of last lecture and introduction Let us first

More information

Multiobjective Mixed-Integer Stackelberg Games

Multiobjective Mixed-Integer Stackelberg Games Solving the Multiobjective Mixed-Integer SCOTT DENEGRE TED RALPHS ISE Department COR@L Lab Lehigh University tkralphs@lehigh.edu EURO XXI, Reykjavic, Iceland July 3, 2006 Outline Solving the 1 General

More information

I.3. LMI DUALITY. Didier HENRION EECI Graduate School on Control Supélec - Spring 2010

I.3. LMI DUALITY. Didier HENRION EECI Graduate School on Control Supélec - Spring 2010 I.3. LMI DUALITY Didier HENRION henrion@laas.fr EECI Graduate School on Control Supélec - Spring 2010 Primal and dual For primal problem p = inf x g 0 (x) s.t. g i (x) 0 define Lagrangian L(x, z) = g 0

More information

Subgradients. subgradients and quasigradients. subgradient calculus. optimality conditions via subgradients. directional derivatives

Subgradients. subgradients and quasigradients. subgradient calculus. optimality conditions via subgradients. directional derivatives Subgradients subgradients and quasigradients subgradient calculus optimality conditions via subgradients directional derivatives Prof. S. Boyd, EE392o, Stanford University Basic inequality recall basic

More information

CO350 Linear Programming Chapter 6: The Simplex Method

CO350 Linear Programming Chapter 6: The Simplex Method CO350 Linear Programming Chapter 6: The Simplex Method 8th June 2005 Chapter 6: The Simplex Method 1 Minimization Problem ( 6.5) We can solve minimization problems by transforming it into a maximization

More information

Nonlinear Programming

Nonlinear Programming Nonlinear Programming Kees Roos e-mail: C.Roos@ewi.tudelft.nl URL: http://www.isa.ewi.tudelft.nl/ roos LNMB Course De Uithof, Utrecht February 6 - May 8, A.D. 2006 Optimization Group 1 Outline for week

More information

DUALITY AND INTEGER PROGRAMMING. Jean B. LASSERRE

DUALITY AND INTEGER PROGRAMMING. Jean B. LASSERRE LABORATOIRE d ANALYSE et d ARCHITECTURE des SYSTEMES DUALITY AND INTEGER PROGRAMMING Jean B. LASSERRE 1 Current solvers (CPLEX, XPRESS-MP) are rather efficient and can solve many large size problems with

More information

March 2002, December Introduction. We investigate the facial structure of the convex hull of the mixed integer knapsack set

March 2002, December Introduction. We investigate the facial structure of the convex hull of the mixed integer knapsack set ON THE FACETS OF THE MIXED INTEGER KNAPSACK POLYHEDRON ALPER ATAMTÜRK Abstract. We study the mixed integer knapsack polyhedron, that is, the convex hull of the mixed integer set defined by an arbitrary

More information

Lagrangian Duality Theory

Lagrangian Duality Theory Lagrangian Duality Theory Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapter 14.1-4 1 Recall Primal and Dual

More information

Subgradients. subgradients. strong and weak subgradient calculus. optimality conditions via subgradients. directional derivatives

Subgradients. subgradients. strong and weak subgradient calculus. optimality conditions via subgradients. directional derivatives Subgradients subgradients strong and weak subgradient calculus optimality conditions via subgradients directional derivatives Prof. S. Boyd, EE364b, Stanford University Basic inequality recall basic inequality

More information

Discrete Optimization 2010 Lecture 7 Introduction to Integer Programming

Discrete Optimization 2010 Lecture 7 Introduction to Integer Programming Discrete Optimization 2010 Lecture 7 Introduction to Integer Programming Marc Uetz University of Twente m.uetz@utwente.nl Lecture 8: sheet 1 / 32 Marc Uetz Discrete Optimization Outline 1 Intro: The Matching

More information

Linear and Combinatorial Optimization

Linear and Combinatorial Optimization Linear and Combinatorial Optimization The dual of an LP-problem. Connections between primal and dual. Duality theorems and complementary slack. Philipp Birken (Ctr. for the Math. Sc.) Lecture 3: Duality

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

Computational Integer Programming. Lecture 2: Modeling and Formulation. Dr. Ted Ralphs

Computational Integer Programming. Lecture 2: Modeling and Formulation. Dr. Ted Ralphs Computational Integer Programming Lecture 2: Modeling and Formulation Dr. Ted Ralphs Computational MILP Lecture 2 1 Reading for This Lecture N&W Sections I.1.1-I.1.6 Wolsey Chapter 1 CCZ Chapter 2 Computational

More information

Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem

Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem Michael Patriksson 0-0 The Relaxation Theorem 1 Problem: find f := infimum f(x), x subject to x S, (1a) (1b) where f : R n R

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

Semidefinite Programming

Semidefinite Programming Chapter 2 Semidefinite Programming 2.0.1 Semi-definite programming (SDP) Given C M n, A i M n, i = 1, 2,..., m, and b R m, the semi-definite programming problem is to find a matrix X M n for the optimization

More information

Linear Programming. Operations Research. Anthony Papavasiliou 1 / 21

Linear Programming. Operations Research. Anthony Papavasiliou 1 / 21 1 / 21 Linear Programming Operations Research Anthony Papavasiliou Contents 2 / 21 1 Primal Linear Program 2 Dual Linear Program Table of Contents 3 / 21 1 Primal Linear Program 2 Dual Linear Program Linear

More information

Multi-Row Cuts in Integer Programming. Tepper School of Business Carnegie Mellon University, Pittsburgh

Multi-Row Cuts in Integer Programming. Tepper School of Business Carnegie Mellon University, Pittsburgh Multi-Row Cuts in Integer Programming Gérard Cornuéjols Tepper School o Business Carnegie Mellon University, Pittsburgh March 2011 Mixed Integer Linear Programming min s.t. cx Ax = b x j Z or j = 1,...,

More information

Subadditive Approaches to Mixed Integer Programming

Subadditive Approaches to Mixed Integer Programming Subadditive Approaches to Mixed Integer Programming by Babak Moazzez A thesis submitted to the Faculty of Graduate and Postdoctoral Affairs in partial fulfillment of the requirements for the degree of

More information

MVE165/MMG631 Linear and integer optimization with applications Lecture 8 Discrete optimization: theory and algorithms

MVE165/MMG631 Linear and integer optimization with applications Lecture 8 Discrete optimization: theory and algorithms MVE165/MMG631 Linear and integer optimization with applications Lecture 8 Discrete optimization: theory and algorithms Ann-Brith Strömberg 2017 04 07 Lecture 8 Linear and integer optimization with applications

More information

Minimal Valid Inequalities for Integer Constraints

Minimal Valid Inequalities for Integer Constraints Minimal Valid Inequalities for Integer Constraints Valentin Borozan LIF, Faculté des Sciences de Luminy, Université de Marseille, France borozan.valentin@gmail.com and Gérard Cornuéjols Tepper School of

More information

Some cut-generating functions for second-order conic sets

Some cut-generating functions for second-order conic sets Some cut-generating functions for second-order conic sets Asteroide Santana 1 and Santanu S. Dey 1 1 School of Industrial and Systems Engineering, Georgia Institute of Technology June 1, 2016 Abstract

More information

Lagrangean relaxation

Lagrangean relaxation Lagrangean relaxation Giovanni Righini Corso di Complementi di Ricerca Operativa Joseph Louis de la Grange (Torino 1736 - Paris 1813) Relaxations Given a problem P, such as: minimize z P (x) s.t. x X P

More information

EE364a Review Session 5

EE364a Review Session 5 EE364a Review Session 5 EE364a Review announcements: homeworks 1 and 2 graded homework 4 solutions (check solution to additional problem 1) scpd phone-in office hours: tuesdays 6-7pm (650-723-1156) 1 Complementary

More information

Lagrange duality. The Lagrangian. We consider an optimization program of the form

Lagrange duality. The Lagrangian. We consider an optimization program of the form Lagrange duality Another way to arrive at the KKT conditions, and one which gives us some insight on solving constrained optimization problems, is through the Lagrange dual. The dual is a maximization

More information

Lecture 8 Plus properties, merit functions and gap functions. September 28, 2008

Lecture 8 Plus properties, merit functions and gap functions. September 28, 2008 Lecture 8 Plus properties, merit functions and gap functions September 28, 2008 Outline Plus-properties and F-uniqueness Equation reformulations of VI/CPs Merit functions Gap merit functions FP-I book:

More information

On the Relative Strength of Split, Triangle and Quadrilateral Cuts

On the Relative Strength of Split, Triangle and Quadrilateral Cuts On the Relative Strength of Split, Triangle and Quadrilateral Cuts Amitabh Basu Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 53 abasu@andrew.cmu.edu Pierre Bonami LIF, Faculté

More information

Convexification of Mixed-Integer Quadratically Constrained Quadratic Programs

Convexification of Mixed-Integer Quadratically Constrained Quadratic Programs Convexification of Mixed-Integer Quadratically Constrained Quadratic Programs Laura Galli 1 Adam N. Letchford 2 Lancaster, April 2011 1 DEIS, University of Bologna, Italy 2 Department of Management Science,

More information

LINEAR PROGRAMMING. Relation to the Text (cont.) Relation to Material in Text. Relation to the Text. Relation to the Text (cont.

LINEAR PROGRAMMING. Relation to the Text (cont.) Relation to Material in Text. Relation to the Text. Relation to the Text (cont. LINEAR PROGRAMMING Relation to Material in Text After a brief introduction to linear programming on p. 3, Cornuejols and Tϋtϋncϋ give a theoretical discussion including duality, and the simplex solution

More information

3.10 Lagrangian relaxation

3.10 Lagrangian relaxation 3.10 Lagrangian relaxation Consider a generic ILP problem min {c t x : Ax b, Dx d, x Z n } with integer coefficients. Suppose Dx d are the complicating constraints. Often the linear relaxation and the

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

On Sublinear Inequalities for Mixed Integer Conic Programs

On Sublinear Inequalities for Mixed Integer Conic Programs Noname manuscript No. (will be inserted by the editor) On Sublinear Inequalities for Mixed Integer Conic Programs Fatma Kılınç-Karzan Daniel E. Steffy Submitted: December 2014; Revised: July 2015 Abstract

More information

Lagrangian Duality and Convex Optimization

Lagrangian Duality and Convex Optimization Lagrangian Duality and Convex Optimization David Rosenberg New York University February 11, 2015 David Rosenberg (New York University) DS-GA 1003 February 11, 2015 1 / 24 Introduction Why Convex Optimization?

More information

Lecture 1: Entropy, convexity, and matrix scaling CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016

Lecture 1: Entropy, convexity, and matrix scaling CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 Lecture 1: Entropy, convexity, and matrix scaling CSE 599S: Entropy optimality, Winter 2016 Instructor: James R. Lee Last updated: January 24, 2016 1 Entropy Since this course is about entropy maximization,

More information

Sufficiency of Cut-Generating Functions

Sufficiency of Cut-Generating Functions Sufficiency of Cut-Generating Functions Gérard Cornuéjols 1, Laurence Wolsey 2, and Sercan Yıldız 1 1 Tepper School of Business, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, United

More information

On Sublinear Inequalities for Mixed Integer Conic Programs

On Sublinear Inequalities for Mixed Integer Conic Programs Noname manuscript No. (will be inserted by the editor) On Sublinear Inequalities for Mixed Integer Conic Programs Fatma Kılınç-Karzan Daniel E. Steffy Submitted: December 2014; Revised: July 7, 2015 Abstract

More information

On the Relative Strength of Split, Triangle and Quadrilateral Cuts

On the Relative Strength of Split, Triangle and Quadrilateral Cuts On the Relative Strength of Split, Triangle and Quadrilateral Cuts Amitabh Basu Pierre Bonami Gérard Cornuéjols François Margot Abstract Integer programs defined by two equations with two free integer

More information

Today: Linear Programming (con t.)

Today: Linear Programming (con t.) Today: Linear Programming (con t.) COSC 581, Algorithms April 10, 2014 Many of these slides are adapted from several online sources Reading Assignments Today s class: Chapter 29.4 Reading assignment for

More information

Separation, Inverse Optimization, and Decomposition. Some Observations. Ted Ralphs 1 Joint work with: Aykut Bulut 1

Separation, Inverse Optimization, and Decomposition. Some Observations. Ted Ralphs 1 Joint work with: Aykut Bulut 1 : Some Observations Ted Ralphs 1 Joint work with: Aykut Bulut 1 1 COR@L Lab, Department of Industrial and Systems Engineering, Lehigh University MOA 2016, Beijing, China, 27 June 2016 What Is This Talk

More information

CSCI5654 (Linear Programming, Fall 2013) Lecture-8. Lecture 8 Slide# 1

CSCI5654 (Linear Programming, Fall 2013) Lecture-8. Lecture 8 Slide# 1 CSCI5654 (Linear Programming, Fall 2013) Lecture-8 Lecture 8 Slide# 1 Today s Lecture 1. Recap of dual variables and strong duality. 2. Complementary Slackness Theorem. 3. Interpretation of dual variables.

More information

Understanding the Simplex algorithm. Standard Optimization Problems.

Understanding the Simplex algorithm. Standard Optimization Problems. Understanding the Simplex algorithm. Ma 162 Spring 2011 Ma 162 Spring 2011 February 28, 2011 Standard Optimization Problems. A standard maximization problem can be conveniently described in matrix form

More information

Optimality Conditions for Constrained Optimization

Optimality Conditions for Constrained Optimization 72 CHAPTER 7 Optimality Conditions for Constrained Optimization 1. First Order Conditions In this section we consider first order optimality conditions for the constrained problem P : minimize f 0 (x)

More information

Cutting Plane Methods I

Cutting Plane Methods I 6.859/15.083 Integer Programming and Combinatorial Optimization Fall 2009 Cutting Planes Consider max{wx : Ax b, x integer}. Cutting Plane Methods I Establishing the optimality of a solution is equivalent

More information

LP Duality: outline. Duality theory for Linear Programming. alternatives. optimization I Idea: polyhedra

LP Duality: outline. Duality theory for Linear Programming. alternatives. optimization I Idea: polyhedra LP Duality: outline I Motivation and definition of a dual LP I Weak duality I Separating hyperplane theorem and theorems of the alternatives I Strong duality and complementary slackness I Using duality

More information

ORIE 6300 Mathematical Programming I August 25, Lecture 2

ORIE 6300 Mathematical Programming I August 25, Lecture 2 ORIE 6300 Mathematical Programming I August 25, 2016 Lecturer: Damek Davis Lecture 2 Scribe: Johan Bjorck Last time, we considered the dual of linear programs in our basic form: max(c T x : Ax b). We also

More information

Conic Linear Optimization and its Dual. yyye

Conic Linear Optimization and its Dual.   yyye Conic Linear Optimization and Appl. MS&E314 Lecture Note #04 1 Conic Linear Optimization and its Dual Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A.

More information

Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm

Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm Chapter 13 Radon Measures Recall that if X is a compact metric space, C(X), the space of continuous (real-valued) functions on X, is a Banach space with the norm (13.1) f = sup x X f(x). We want to identify

More information

Example Problem. Linear Program (standard form) CSCI5654 (Linear Programming, Fall 2013) Lecture-7. Duality

Example Problem. Linear Program (standard form) CSCI5654 (Linear Programming, Fall 2013) Lecture-7. Duality CSCI5654 (Linear Programming, Fall 013) Lecture-7 Duality Lecture 7 Slide# 1 Lecture 7 Slide# Linear Program (standard form) Example Problem maximize c 1 x 1 + + c n x n s.t. a j1 x 1 + + a jn x n b j

More information

THEOREMS, ETC., FOR MATH 516

THEOREMS, ETC., FOR MATH 516 THEOREMS, ETC., FOR MATH 516 Results labeled Theorem Ea.b.c (or Proposition Ea.b.c, etc.) refer to Theorem c from section a.b of Evans book (Partial Differential Equations). Proposition 1 (=Proposition

More information

Cutting Plane Methods II

Cutting Plane Methods II 6.859/5.083 Integer Programming and Combinatorial Optimization Fall 2009 Cutting Plane Methods II Gomory-Chvátal cuts Reminder P = {x R n : Ax b} with A Z m n, b Z m. For λ [0, ) m such that λ A Z n, (λ

More information

14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness.

14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness. CS/ECE/ISyE 524 Introduction to Optimization Spring 2016 17 14. Duality ˆ Upper and lower bounds ˆ General duality ˆ Constraint qualifications ˆ Counterexample ˆ Complementary slackness ˆ Examples ˆ Sensitivity

More information

Discrete (and Continuous) Optimization WI4 131

Discrete (and Continuous) Optimization WI4 131 Discrete (and Continuous) Optimization WI4 131 Kees Roos Technische Universiteit Delft Faculteit Electrotechniek, Wiskunde en Informatica Afdeling Informatie, Systemen en Algoritmiek e-mail: C.Roos@ewi.tudelft.nl

More information

Chapter 2: Linear Programming Basics. (Bertsimas & Tsitsiklis, Chapter 1)

Chapter 2: Linear Programming Basics. (Bertsimas & Tsitsiklis, Chapter 1) Chapter 2: Linear Programming Basics (Bertsimas & Tsitsiklis, Chapter 1) 33 Example of a Linear Program Remarks. minimize 2x 1 x 2 + 4x 3 subject to x 1 + x 2 + x 4 2 3x 2 x 3 = 5 x 3 + x 4 3 x 1 0 x 3

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

Lagrangian Duality. Richard Lusby. Department of Management Engineering Technical University of Denmark

Lagrangian Duality. Richard Lusby. Department of Management Engineering Technical University of Denmark Lagrangian Duality Richard Lusby Department of Management Engineering Technical University of Denmark Today s Topics (jg Lagrange Multipliers Lagrangian Relaxation Lagrangian Duality R Lusby (42111) Lagrangian

More information

Introduction to Mathematical Programming IE406. Lecture 13. Dr. Ted Ralphs

Introduction to Mathematical Programming IE406. Lecture 13. Dr. Ted Ralphs Introduction to Mathematical Programming IE406 Lecture 13 Dr. Ted Ralphs IE406 Lecture 13 1 Reading for This Lecture Bertsimas Chapter 5 IE406 Lecture 13 2 Sensitivity Analysis In many real-world problems,

More information

On Subadditive Duality for Conic Mixed-Integer Programs

On Subadditive Duality for Conic Mixed-Integer Programs arxiv:1808.10419v1 [math.oc] 30 Aug 2018 On Subadditive Duality for Conic Mixed-Integer Programs Diego A. Morán R. Burak Kocuk Abstract It is known that the subadditive dual of a conic mixed-integer program

More information

Reformulation and Sampling to Solve a Stochastic Network Interdiction Problem

Reformulation and Sampling to Solve a Stochastic Network Interdiction Problem Network Interdiction Stochastic Network Interdiction and to Solve a Stochastic Network Interdiction Problem Udom Janjarassuk Jeff Linderoth ISE Department COR@L Lab Lehigh University jtl3@lehigh.edu informs

More information

Chapter 1. Optimality Conditions: Unconstrained Optimization. 1.1 Differentiable Problems

Chapter 1. Optimality Conditions: Unconstrained Optimization. 1.1 Differentiable Problems Chapter 1 Optimality Conditions: Unconstrained Optimization 1.1 Differentiable Problems Consider the problem of minimizing the function f : R n R where f is twice continuously differentiable on R n : P

More information

Lecture 10: Linear programming duality and sensitivity 0-0

Lecture 10: Linear programming duality and sensitivity 0-0 Lecture 10: Linear programming duality and sensitivity 0-0 The canonical primal dual pair 1 A R m n, b R m, and c R n maximize z = c T x (1) subject to Ax b, x 0 n and minimize w = b T y (2) subject to

More information

Numerical Optimization

Numerical Optimization Linear Programming - Interior Point Methods Computer Science and Automation Indian Institute of Science Bangalore 560 012, India. NPTEL Course on Example 1 Computational Complexity of Simplex Algorithm

More information

On the knapsack closure of 0-1 Integer Linear Programs

On the knapsack closure of 0-1 Integer Linear Programs On the knapsack closure of 0-1 Integer Linear Programs Matteo Fischetti 1 Dipartimento di Ingegneria dell Informazione University of Padova Padova, Italy Andrea Lodi 2 Dipartimento di Elettronica, Informatica

More information

Introduction to linear programming using LEGO.

Introduction to linear programming using LEGO. Introduction to linear programming using LEGO. 1 The manufacturing problem. A manufacturer produces two pieces of furniture, tables and chairs. The production of the furniture requires the use of two different

More information

On duality theory of conic linear problems

On duality theory of conic linear problems On duality theory of conic linear problems Alexander Shapiro School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 3332-25, USA e-mail: ashapiro@isye.gatech.edu

More information

Mixing Inequalities and Maximal Lattice-Free Triangles

Mixing Inequalities and Maximal Lattice-Free Triangles Mixing Inequalities and Maximal Lattice-Free Triangles Santanu S. Dey Laurence A. Wolsey Georgia Institute of Technology; Center for Operations Research and Econometrics, UCL, Belgium. 2 Outline Mixing

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

Convex Sets and Minimal Sublinear Functions

Convex Sets and Minimal Sublinear Functions Convex Sets and Minimal Sublinear Functions Amitabh Basu Gérard Cornuéjols Giacomo Zambelli April 2010 Abstract We show that, given a closed convex set K containing the origin in its interior, the support

More information

A new Hellinger-Kantorovich distance between positive measures and optimal Entropy-Transport problems

A new Hellinger-Kantorovich distance between positive measures and optimal Entropy-Transport problems A new Hellinger-Kantorovich distance between positive measures and optimal Entropy-Transport problems Giuseppe Savaré http://www.imati.cnr.it/ savare Dipartimento di Matematica, Università di Pavia Nonlocal

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

Lecture 7 Duality II

Lecture 7 Duality II L. Vandenberghe EE236A (Fall 2013-14) Lecture 7 Duality II sensitivity analysis two-person zero-sum games circuit interpretation 7 1 Sensitivity analysis purpose: extract from the solution of an LP information

More information

Lift-and-Project Inequalities

Lift-and-Project Inequalities Lift-and-Project Inequalities Q. Louveaux Abstract The lift-and-project technique is a systematic way to generate valid inequalities for a mixed binary program. The technique is interesting both on the

More information

5. Duality. Lagrangian

5. Duality. Lagrangian 5. Duality Convex Optimization Boyd & Vandenberghe Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized

More information