1 / 9

2. Generating All Valid Inequalities

2. Generating All Valid Inequalities. Let Valid inequalities for ( conv ( )) can be generated using procedure . Also as inequalities. All valid inequalities for can be generated using these procedures. 0-1 problems: + n : All valid inequalities are inequalities.

shel
Download Presentation

2. Generating All Valid Inequalities

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 2. Generating All Valid Inequalities • Let • Valid inequalities for (conv()) can be generated using procedure. Also as inequalities. All valid inequalities for can be generated using these procedures. • 0-1 problems: +n: All valid inequalities are inequalities. We say that any valid inequality for dominated by a inequality (inequality) is also a inequality (inequality).

  2. Let . , Since contains all of the integral points in P, . Show that all valid inequalities for S (conv(S)) are D-inequalities by showing that all valid inequalities for S are D-inequalities for . (which yields that , hence ) • Thm 2.3: Every valid inequality for with is a inequality • Thm 2.4: • To get conv(), we only need to integralize one variable at a time.

  3. For 0-1 problems, all valid inequalities are inequalities. • Thm 2.8: Let with be a valid inequality for with . Then is a inequality for . • Thm 2.15: Let with be a valid inequality for with . Then is a inequality for . • Thm 2.16: Let with be a valid inequality for . Then is a inequality for .

  4. Rank of C-G Inequality • NW p.225. • procedure is used recursively • Def: elementary closure of P (first Chvatal closure): }, contains all of the nondominated inequalities that can be obtained by one application of the procedure. • Prop 2.17: If , then . ( : integer vector) Pf) Since , there exists +m such that and . Such is a feasible solution to the dual of max. Thus and .  (If , is not in .)

  5. Def: is of rank with respect to if is not equivalent to or dominated by any nonnegative linear combination of inequalities, each of which can be determined by no more than applications of the procedure, but is equivalent to or dominated by a nonnegative linear combination of some inequalities that require no more than applications of the procedure. (The smallest number of the procedure to get a given valid inequality for ) rank of : rank of : = max{: is valid for • For matching problem, . But, for most IP problems, the rank of the polyhedron increases without bound as a function of the dimension of the polyhedron. Even when dimension is fixed, there are examples such that the rank increases without bound as a function of the magnitude of the coefficients in the linear inequality description of .

  6. Suppose a family of polyhedra F has bounded rank ( Validity for conv() can be in . (only need original constraints for and weight vectors to show that is valid for conv()) • Certificate of optimality for IP problem: If a family of polyhedra F have bounded rank, we have short proof of optimality of to max{}. Only need to show that is a valid inequality, where , Using original constraints and weight vectors (provided that the weight vectors are polynomial in the description of .), we have short proof that is valid. Hence validity is in . If lower bound feasibility (complement of validity) is complete, we have which implies . Therefore it is unlikely that if a class of IP problem is hard, the polyhedra over which it is defined has bounded rank.

  7. 3. Gomory’s Fractional Cuts and Rounding • Why called Chvatal-Gomory procedure? procedure was implicitly used in the earlier work of Gomory’s finite cutting plane algorithm. • +n:, is integral +n+m: Let +m, , (3.1) Assume that this represents a row of the optimal simplex tableau for LP relaxation. (Technically, is included in the constraints and max is solved.  is a row of ) Let , , From modular arithmetic, (Gomory cutting plane) (3.2)

  8. We can get Gomory cut using C-G with weights . • Thm 3.1: Let +n: The fractional cut (3.2) derived from (3.1) is a C-G inequality for S obtained with weights for . Pf) Let and . Then . or Round down ------------------------------------------------------

  9. Coefficient of a basic variable in the optimal tableau of LP and all other basic variables have coefficient 0.  nonbasic (fractional term) Note that the current optimal LP solution violates this valid inequality since the values of nonbasic variables are all 0. • Gomory showed that this cutting plane algorithm converges in a finite number of steps if the cuts are chosen with some rule.

More Related