The traditional perturbution (or lexicographic) methods for resolving degeneracy in linear programming impose decision rules that eliminate ties in the simplex ratio rule and, therefore,restrict the choice of exiting basic variables. Bland's combinatorial pivoting rule also restricts the choice of exiting variables. Using ideas from parametric linear programming, we develop anti-cycling pivoting rules that do not limit the choice of exiting variables beyond the simplex ratio rule. That is, any variable that ties for the ratio rule can leave the basis. A similar approach gives pivoting rules for the dual simplex method that do not restrict the choice of entering variables

Magnanti, Thomas L.

Orlin, James B., 1953-

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,- ... -. , , ; :' -:.. b i ...., , .. ....... ......... ; .._ ' ,. - -'3 - ...¥{'.' ',,,.!.C. ,..' ":',- .. .,'. ' ' .>. . r : : a o er.; ,,~~~~~~~~~~~~~~~~~~~~~~~~~ .' . - . . .~ . .... . . ... ." .: ~WS~ : "_::.: "'.; :0:  ::: ::_ .0D I;. 0 :.!:: t;0i f::t l .,~: .:. : - : :. :::~ ,: .: -,:I : ,--:-':: : :.:- .:-;Y -i : "' .. : ,'- '::. " -; ; . -.--- : . . ' ......... -' ,.: .... .-.--:-....' -: ,1*0 : X X ff 0 t 0 : X S t00:h .OFTECHY ...... :::::~~~ii~~~~~~li~~~~'!i7:- ° ' " -:-·-·: .. ·:  - - ·-·-·: ·;?::. : --- :- . i ·--····--.I ;:-- --.--.'...·. . ··i ..· --· . 'iI,: 1 ;·-·:.... ··; .j; ·; ·;· I :;:· - ;·'(·:·.;:..-.. ..·'::·I · ·.':· · I-L.-. I ,.·:II..PARAMETRIC LINEAR PROGRAMMING ANDANTI-CYCLING PIVOTING RULESbyThomas L. Magnanti*James B. Orlin**OR 144-85 October 1985* Support in part by grant ECS-83/6224 from the Systems Theory andOperations Research Division of the National Science Foundation.** Support in part by Presidential Young Investigator grant8451517-ECS of the National Science Foundation.ABSTRACTThe traditional perturbution (or lexicographic) methods forresolving degeneracy in linear programming impose decision rulesthat eliminate ties in the simplex ratio rule and, therefore,restrict the choice of exiting basic variables. Bland'scombinatorial pivoting rule also restricts the choice of exitingvariables. Using ideas from parametric linear programming, wedevelop anti-cycling pivoting rules that do not limit the choice ofexiting variables beyond the simplex ratio rule. That is, anyvariable that ties for the ratio rule can leave the basis. Asimilar approach gives pivoting rules for the dual simplex methodthat do not restrict the choice of entering variables.The primal simplex method for minimization problems permits anentering variable at each iteration to be any variable with anegative reduced cost and permits the exiting variable to be anyvariable that satisfies the minimum ratio rule. As is well-known,any implementation of the procedure is guaranteed to converge if theproblem is nondegenerate. In addition, there are two well-knownmethods for resolving degeneracy. The first of these, theperturbation (or equivalently, the lexicographic) method, avoidscycling by refining the selection rule for the exiting variable(Charnes 1952], Dantzig [1951], Wolfe [1963]). The second method,the combinatorial rule, developed by Bland [1979], avoids cycling byrefining the selection rule for both the exiting and enteringvariables. The situation raises the following natural question: Isthere a simplex pivoting procedure for avoiding cycling that doesnot restrict the minimum ratio rule choice of exiting variables? Inthis note, we answer this question affirmatively by describing ananti-cycling rule based on a "homotopy principle" that avoidscycling by refining the selection rule for only the enteringvariable. We also describe an analogous dual pivoting procedurethat avoids cycling by refining only the choice of exitingvariables.Our procedures are based upon a few elementary observationsconcerning parametric simplex methods. These observations may be ofsome importance in their own right, since they may shed light onsome theoretical issues encountered in several recent analyses2of average case performance of parametric simplex methods(e.g., Adler [1983], Borgward [1982], Haimovich [1983], Smale[1983]). In particular, whenever the probability distribution of aparametric linear program is chosen, as is frequently the case, sothat the problem satisfies a property that we call dualnondegeneracy, then the parametric algorithm converges finitely evenif it is degenerate.Parametric Linear ProgrammingConsider the following parametric linear programming problem:Minimize (c. + ed)xsubject to Ax = b P(e)x > 0where A is an mxn constraint matrix with (for notationalconvenience) full row rank. For a given value e, we say that P(e)is nearly dual nondegenerate if for each primal feasible basis Bthere is at most one nonbasic variable xi whose reduced costci + edi is 0. We say that the parametric problem P is dualnondegenerate if P(e) is nearly dual nondegenerate for all eR.Consider the usual parametric simplex algorithm for solvingP(e) for all values of , starting with a basis that is optimal forall sufficiently large values of e. In the case that P is dualnondegenerate, we show that the procedure will not cycle (withoutany perturbations). We then apply this result to give new primalsimplex pivot rules that (1) are guaranteed to avoid cycling, and3(2) rely only on the selection of the entering variable; i.e., anybasic variable satisfying the minimum ratio rule may leave thebasis.The following procedure is a version of the usual parametricsimplex method as applied to a minimization problem.Beginlet B0 be an optimal basis for P(e) for all e > 00;let i=1;while d K 0 dobeginlet B = Bi-1;let c, d denote the reduced costs for the vectors c and dwith respect to basis B;select index s so that -cs/ds = max (-cj/dj:dj > 0);let ei = -/ds;if B-lA s < O then quit (since P(e) is unbounded from belowfor all e < e');else let Bi be obtained from B by pivoting in x andpivoting out any variable chosen by the usual simplexpivot rule (i.e., ties in the ratio rule can be brokenarbitrarily);let i = i+1;end.end.4We note that if d < 0 at any point in this algorithm, thenc + ed > c + ei-ld for all e < e i - 1 and hence B is an optimal basisfor all < ei- 1. The iterative modification of e in the parametricprogramming procedure can be conceptualized differently as expressedin the following observation.REMARK 1. Suppose that B=Bi- is an optimal basis for problem P(e*)for some * and let e i be selected as in the "while loop" of theparametric algorithm when applied to this basis B (consequently, cand d are defined by B). Then e i = min (: B is optimal for P(e)).PROOF. If e < e i , then cs + e d < 0 and B is non-optimal. Also,by our choice of ei,Cs + eids if dj > 0cj + ei dj >C + *dJ if dj < .But then each cj + idj > O, since c + ids = 0 and cj + *dj 0because B is an optimal basis for P(e*). Since cj + eidj = 0 forevery variable j corresponding to a column from B, B is optimal forp(ei) ).Let B be an optimal basis for P(e ° ) and let i and Bi for i > 1be defined recursively as in the parametric procedure. Moreover, letCi = reduced cost with respect to the basis Bi- 1 of the variablespivoted into B i- I to obtain Bi.5PROPOSITION 1. For all i > O,(i) Bi is optimal in [ei+l, ei];(ii) If P(e) is dual nondegenerate, then i+ 1 < ei;(iii) c1 < 0 for all i such that ei > 0.PROOF: Part (i) is a consequence of our previous remark, the factthat both Bi and B i - 1 are optimal at i, and the fact that (: Bi isoptimal for P(e)} is an interval. We prove (ii) via a contradiction.Let i be selected so that ei+ 1 = i and either i = 0 or else i < ei - 1Let xt be the variable pivoted into basis Bi to obtain Bi+1 andlet xp be the variable pivoted out of basis Bi -1 to obtain Bi. Alsolet d be the reduced cost for d with respec't to Bi. Then xt xpbecause dt > 0 and dp < 0 (dp < 0 since cp + 8 i dp = 0 andcp + dp > 0 for < i). Moreover, the assumption ei+l = eiimplies that ct + i+l dt = ct + ei dt = p + e i dp = 0,contradicting the near dual degeneracy of P(ei).To see (iii), note that e i = -ci/di and di > O.s S SThis proposition implies that the sequence of pivots generatedby the parametric algorithm defines a simplex algorithm for thenonparametric problem minimize {cx: Ax = b, x > 0). We record thisresult as follows:COROLLARY. Let BO, ... , Bt be a sequence of bases obtained bythe parametric algorithm, and let t be first index for which t +l isless than O. Then6(i) Bt is optimal for min {cx: Ax = b, x > O};(ii) the bases BO, ... , Bt are distinct;(iii) the pivot sequence satisfies the usual pivot selectionrule, viz., the entering variable has a negative reducedcost for cost vector c. 3REMARK 2. Note that Proposition remains valid if we replace dualnondegeneracy with the slightly weaker assumption that the argmax(-cjldj:dj > O) is unique and hence the index s is unique at eachiteration of the parametric algorithm.An Application to a Primal Pivot RuleWe next show how to apply the previous proposition to defineprimal pivoting rules that, without any special provision forchoosing the exiting variable, avoid cycling.Our previous results demonstrate that the parametric simplexmethod defines an anti-cycling pivot sequence forminimize cxsubject to Ax = b (P)x > Owhenever we can choose the objective function coefficients c and d sothat(i) some basis B of A is optimal for all sufficiently largevalues of , and(ii) P is dual nondegenerate.To establish these two criteria, let B be any feasible basis for (P)7corresponding to the columns of B and with positive componentscorresponding to the columns of N will satisfy property (i). Toensure that P is dual nondegenerate, however, requires that we avoidties in computing the entering variables from the parametric ratiotest -Cs/d s = max (-cj/dj: dj > 0)}. One natural approach is to useperturbations of c or d. For example, following the lead from theusual (nonparametric) perturbation theory of linear programming, wemight perturb the nonbasic columns of c or d by distinct powers of for some small > O. Alternatively, we might use a variant of thefamiliar big M method: choose the nonbasic cost coefficients of d asdistinct powers of M for some large constant M. These procedureslead to the following parametric objective functions:(1) c + N + 8 IN,(2) c + 8e(N + IN), and(3) c + (l1/)N-In these expressions, N denotes a vector with zero componentscorresponding to columns in B and with components , 2 ... n - mfor all columns in N in, say, their natural order from A. AlsoI N = N with = 1, and (1/£)N = 6 N with = 1/c. The first two ofthese objective functions perturb c and d. The third objectivefunction is the polynomial big M method alluded to prior to the lastdisplay with the choice M = 1/c.For each of these objective functions, for all 0 < < 1, B is aunique optimal basis for sufficiently large values of . Therefore,each one defines an anti-cycling pivot rule if P is dualnondegenerate. To demonstrate this property, we first establish apreliminary result by a modification of the usual perturbationargument.8PROPOSITION 2Let A 1, ... , A be the columns of some basis B. Suppose thatD (D1, D 2, ... , D) is any other basis, possibly containingsome columns of B. Suppose that A = [B,N], that h = N andthat h = h - hDD-1A (hD is the subvector of h corresponding tothe columns in D.) Then hi and h are distinct nonzeropolynomials in with zero constant terms whenever i and j areindices corresponding to distinct columns of A other thanD1, D2 , .. , D.PROOF:We use a linear independence argument. First, let us duplicatethe columns of D and consider the following partitioned matrix:0 I CDB N Dcontaining an (n-m) x (n-m) identity matrix I, and a submatrix CD of[0,I] corresponding to the columns D from [B,N]. Therefore, eachcolumn of CD is either a copy of a column of the submatrix 0 or acopy of a column of the identity matrix I. We observe that thematrix M has linear rank of n. Let denote the vector (1, 2, ....£n-m). Let Q = [O,I]. Then h = Q. Next consider the followingmatrix obtained by pivoting on the basis D in M:CB CN 0M' =D-lB D-1N I .9Let Q' = [CB, CN]. Then Q' has full row rank n-m because M has fullrow rank n, and M' is obtained from M by pivoting. Moreover, the lastm columns of M' are copies of columns of the first n columns. Thecolumn of Q' corresponding to each of these "original columns" must bea 0 vector. Deleting these n zero vectors leaves an (n-m) x (n-m)nonsingular submatrix Q''.Finally, observe that h = h - hDD-1A = Q - CDD-1A = Q'.Consequently, the two polynomials hi and hj refered to in thestatement of the proposition may be obtained from two distinctelements of the vector Q'' and are thus distinct polynomials in cwith zero constant terms. eNow consider the selection rule for the incoming variable at anypoint in the parametric simplex method. Assume that the currentbasis is D and that c and d are the reduced costs of c and d = 1N withrespect to this basis. As in Proposition 2, let h = N. Then theratio rule for choosing the incoming variable for the three objectivefunctions (1), (2) and (3) becomes:(1') max {-(cj + hj)/dj: dj > 0)(2') max {-cj/(dj + hj): dj > O}), and(3') max {-cj/hj(1/E): h(1/c) > 0).In (3'), hj(l/c) denotes the polynomial in 1/c obtained by replacingE by 1/c in h. By the usual purturbation argument, if c is asufficiently small constant (i.e., < c(D) for some constant (D) <1), then a single index j gives the unique minimum in each of theseratios. Consequently, for all c < min {c(D): D is a basis of A), the10entering variable is unique and Proposition 1 (see Remark 2 as well)and its Corollary apply. (Similarly, if we express the reduced costfor D as a function of and , then by Proposition 2, each reducedcost is a different nonconstant polynomial in and is linear in .Therefore, if is sufficiently small, then the value of e for whichthe reduced of xj is 0 is different for all nonbasic variables j.That is, P is dual nondegenerate.)Each of the perturbations (1), (2), and (3) lead to differentpivot rules that can be interpreted as certain lexicographic selectionprocedures. Since our purpose in this note has been merely toestablish the possibility of simplex point rules that do not restrictthe leaving variable, we will not specify the details of theseprocedures, nor do we discuss their computation requirements (or claimthat they are efficient).In concluding this section, we might note that the parametrics(or homotopies) seem essential for the results given in this paper.Simply perturbing the objective function to avoid ties in theselection of an entering variable will not suffice. For example, instandard examples of cycling in the simplex method, there is no dualdegeneracy and the choice of an entering variable is unique.Dual Pivot RulesArguments similar to those used previously apply to right-handside parametrics in a dual simplex algorithm. That is, consider theparametric problemminimize cxsubject to Ax = b + egx > 0.11We say that this problem is primal nondegenerate if for any dualfeasible basis B (i.e., c - cgB B-1A > 0) and parameter value ', thereis at most one basic variable x whose value is zero. Then, assumingthat B is an optimal basis for sufficiently large values of theparameter e and assuming primal nondegeneracy, we can mimic ourearlier arguments to show that the usual parametric dual simplexmethod will(i) compute optimal solutions to P'(e) for all e < 0°,(it) for values of 8 > 0, defines a dual simplex method(i.e., at any step (i), the leaving variable xrsatisfies br = [(Bi)-lb]r < 0).Consequently, the primal nondegeneracy assumption results in afinitely convergent dual simplex algorithm that permits any variablesatisfying the dual minimum ratio rule to leave the basis. In orderto ensure primal nondegeneracy, we can introduce right hand sideperturbations; that is consider right hand sides such as the following[here £ is a column vector defined by = (1, 2, .... m)]:(i) b + eg + , or(ii) b + (g + ).Choosing the m-vector g so that (Bo)-1 g > 0 will ensure that B is aunique optimal basis for sufficiently large values for e and that thisperturbation will ensure primal nondegeneracy.AcknowledgmentWe are grateful to Robert Freund for a suggestion that led to animprovement in the proof of Proposition 2.12ReferencesAdler, I. (1983). "The Expected number of pivots needed to solveparametric linear programs and the efficency of the self-dualsimplex method," Technical Report, Department of IndustrialEngineering and Operations Research, University of California,Berkeley, California.Bland, R.G. (1977). New finite pivoting rules for the simplex method.Math. of Oper. Research 2, 103-107.Borgwardt, K.H. (1982). "The average number of steps required by thesimplex method is polynomial," Zeitschrift fur OperationsResearch 26, 157-177.Charnes, A. (1952). Optimality and degeneracy in linear programming.Econometrica 20, 160-170.Dantzig, G.B. (1951). Maximization of a linear function of variablessubject to linear inequalities. In T.C. Koopmans (ed.), ActivityAnalysis of Production and Allocation, John Wiley and Sons, Inc.,New York.Haimovich, M. (1983). "The simplex method is very good! - On theexpected number of pivot steps and related properties of randomlinear programs," Technical Report, Columbia University, NewYork.Smale, S. (1983). "On the average number of steps of the simplexmethod of linear programming," Mathematical Programming 27, 241-262.Wolfe, P. (1963). A technique for resolving degeneracy in linearprogramming. J. SIAM 11, 205-211.13

Parametric Linear Programming and Anti-Cycling Pivoting Rules

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Parametric Linear Programming and Anti-Cycling Pivoting Rules

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