In this paper we consider the parameter space of continuous linear optimization problems with a given decision space and a given index set. We consider different partitions of this space, on the basis of the primal, the dual, and the primal-dual
status of each parameter. We define ill-posedness and relative ill-posedness w.r.t. a given set and absolute ill-posedness w.r.t. a given family of sets. These concepts are characterized for the elements of the partitions considered in this paper

Goberna, Miguel A.

Todorov, Maxim I.

Proceeding of the Bulgarian Academy of Sciences

Repositorio Institucional de la Universidad de Alicante

Äîêëàäè íà Áúëãàðñêàòà àêàäåìèß íà íàóêèòåComptes rendus de l’Acade´mie bulgare des SciencesTome 60, No 4, 2007MATHEMATIQUESThe´orie de l’optimisationILL-POSEDNESS IN CONTINUOUS LINEAR OPTIMIZATIONVIA PARTITIONS OF THE SPACE OF PARAMETERSM. A. Goberna, M. I. Todorov∗(Submitted by Academician P. Kenderov on March 13, 2007)AbstractIn this paper we consider the parameter space of continuous linear optimizationproblems with a given decision space and a given index set. We consider differentpartitions of this space, on the basis of the primal, the dual, and the primal-dualstatus of each parameter. We define ill-posedness and relative ill-posedness w.r.t. agiven set and absolute ill-posedness w.r.t. a given family of sets. These concepts arecharacterized for the elements of the partitions considered in this paper.1. Introduction. Let n ∈ N, n ≥ 2, and a non-empty compact Hausdorff topo-logical space T be given. We associate with each triplepi := (a, b, c) ∈ Π := C(T )n × C(T )× Rna primal optimization problem called primal,P : Min c′xs.t. a′tx ≥ bt, t ∈ T,with space of variables Rn, and its (Haar’s) dual problemD : Max∑t∈Tλtbts.t.∑tεTλtat = c,λt ≥ 0, t ∈ T,whose space of variables is the linear space of all functions λ : T 7→ R such that λt = 0for all t ∈ T except maybe for a finite number of indices. If |T | <∞ and we consider TSupported by MCYT of Spain and FEDER of EU, Grant BMF2002–04114–C02–01.∗Research partially supported by CONACyT of Mexico under Grant 44003.357equipped with the discrete topology, P and D are ordinary linear programming (LP)problems. Otherwise, P and D are continuous linear semi-infinite programming (LSIP)problems as far as either the number of variables or the number of constraints (butnot both) is infinite. Interesting applications of continuous LSIP can be found in ([7],Chapters 1–2), [6], and the references therein. Π is called the parameters space, and itcan be the result of all possible perturbations performed on a given continuous problemprovided the structure of the problem is preserved. In particular, Π := Rn|T |+|T |+n whenT is finite.We denote by ΠPC (ΠPI ) the set of parameters providing a consistent (inconsistent,respectively) primal problem. {ΠPC ,ΠPI } is the binary primal partition of Π. Replacing“primal” by “dual” we get the binary dual partition of Π, {ΠDC ,ΠDI }. The binary primal-dual partition of Π is formed by the four crossed intersections ΠCC := ΠPC ∩ ΠDC ,ΠCI := ΠPC ∩ΠDI , ΠIC := ΠPI ∩ΠDC , and ΠII := ΠPI ∩ΠDI .We denote by vP (pi) (vD(pi)) the optimal value of P (D), defining as usual vP (pi) =+∞ (vD(pi) = −∞, respectively) when the corresponding problem is inconsistent. Aproblem is bounded when its optimal value is a real number. Given pi := (a, b, c) ∈ Π,since P can be either inconsistent (IC) or bounded (B) or unbounded (UB), we canclassify pi in one of the elements of the ternary primal partition {ΠPI ,ΠPB,ΠPU}. Similarly,pi can be classified in one of the elements of the ternary dual partition {ΠDI ,ΠDB ,ΠDU }.The ternary primal-dual partition is formed by the non-empty pairwise intersectionsof the elements of the primal and the dual partitions. The elements of the ternaryprimal-dual partition are codified as shown in Table 1, where the set in each cell is theintersection of the entries of its column and its row.T a b l e 1Π ΠPI ΠPB ΠPUΠDI ΠII ΠBI ΠUIΠDB ΠIB ΠBBΠDU ΠIUThe null element of Π belongs to ΠBB := ΠPB ∩ ΠDB , the set of parameters withbounded associated problems. Each element of the primal-dual partition correspondsto a duality state ([1] and [10] have analysed the role played by the duality states in LPand LSIP).We consider Π as a metric space equipped with the following distance: given twoparameters pi1 =(a1, b1, c1)and pi2 =(a2, b2, c2),(1) d(pi1, pi2) := max{∥∥c1 − c2∥∥∞ ,maxt∈T∥∥∥∥( a1tb1t)−(a2tb2t)∥∥∥∥∞}.In fact, it can be easily seen that Π is also a Banach space with the usual sup norm.Throughout the paper the elements of Π will be distinguished by means of upperscripts,and the same (either as subscripts or as superscripts) applies for their correspondingobjects: pir = (ar, br, cr), Dr, Pr, and so on.In general LSIP, the functions a : T 7→ Rn and b : T 7→ R are not necessarilycontinuous, so that the space of parameters is (Rn)T ×RT ×Rn. Replacing “max” with358 M. A. Goberna, M. I. Todorov“sup” in (1), we get a pseudometric, which generates the topology of uniform conver-gence in this space and for which our parameters space Π is a topological subspace.Let us observe that the recent paper [4] provides characterizations of the interior,the boundary and the exterior of the sets ΠDC and ΠBB in general LSIP. Obviously,these characterizations become sufficient conditions in the context of continuous LSIP.In this paper we consider three different types of ill-posed parameters, two of themrelative to a given subset of Π and the third one relative to a family of subsets of Π. Theindividual sets will be the elements of the binary and the ternary partitions definedabove whereas the families will be the stable elements in each partition (in the sensethat they have non-empty interior). The way to do this consists of characterizing theelements of these partitions together with their respective interiors.2. Analysing the partitions. Given a non-empty set X ⊂ Rp, coneX denotesthe conical convex hull of X ∪ {0p} (0p denotes here the zero vector of Rp). Fromthe topological side, if X is a subset of any topological space, intX, clX and bdXrepresent the interior, the closure and the boundary of X, respectively.We associate with pi = (a, b, c) the first moment cone of pi, M := cone{at, t ∈ T},and the characteristic cone, K := cone{(at, bt), t ∈ T ; (0n,−1)}. If pi satisfies the Slatercondition (SC in brief), i.e., there exists x ∈ Rn such that a′tx > bt for all t ∈ T , thenK is closed.The next result summarizes well-known results on the primal and dual binarypartitions ({ΠPC ,ΠPI } and {ΠDC ,ΠDI }). The proof can be found in ([8], Theorem 3.1), [7]and ([9], Theorem 5).Lemma 1. The elements of the binary primal and dual partitions are neitheropen nor closed. Moreover, the following statements are true:(i) pi ∈ ΠPC if and only if (0n, 1) /∈ clK. In particular, pi ∈ intΠPC if and only ifP satisfies SC.(ii) pi ∈ ΠPI if and only if (0n, 1) ∈ clK. In particular, pi ∈ intΠPI if and only if(0n, 1) ∈ intK.(iii) pi ∈ ΠDC if and only if c ∈M . In particular, pi ∈ intΠDC if and only if c ∈ intM .(iv) pi ∈ ΠDI if and only if c /∈ M . In particular, pi ∈ intΠDI if and only if thereexists y ∈ Rn such that c′y < 0 and a′ty > 0 for all t ∈ T .From Lemma 1 it is possible to characterize the elements of the binary primal-dualpartition and their corresponding topological interiors.Theorem 2. The elements of the binary primal-dual partition are neither opennor closed and satisfy the following statements:(i) pi ∈ ΠCC if and only if (0n, 1) /∈ clK and c ∈ M . In particular, pi ∈ intΠCCif and only if SC holds and c ∈ intM . Moreover, intΠCC is dense in ΠCC ifand only if |T | ≥ n.(ii) pi ∈ ΠCI if and only if (0n, 1) /∈ clK and c /∈M . In particular, pi ∈ intΠCI ifand only if SC holds and there exists y ∈ Rn such that c′y < 0 and a′ty > 0for all t ∈ T . Moreover, intΠCI is dense in ΠCI .(iii) pi ∈ ΠIC if and only if (0n, 1) ∈ clK and c ∈M . In particular, pi ∈ intΠIC ifand only if (0n, 1) ∈ intK and c ∈ intM . Moreover, intΠIC is dense in ΠICif and only if |T | ≥ n+ 1.2 Compt. rend. Acad. bulg. Sci., 60, No 4, 2007 359(iv) pi ∈ ΠII if and only if (0n, 1) ∈ clK and c /∈M . Moreover, intΠII = ∅.The next theorem constitutes the counterpart of Theorem 2 for the ternary primal-dual partition (statements (i), because ΠBB = ΠCC , and (iv) are obviously redundant).Its proof is based on the following lemma.Lemma 3. The following statements are true:(i) pi ∈ ΠPB if and only if (0n, 1) /∈ clK and ({c} × R) ∩ clK 6= ∅. Moreover,intΠPB = intΠBB.(ii) pi ∈ ΠPU if and only if (0n, 1) /∈ clK and ({c} × R) ∩ clK = ∅. Moreover,intΠPU = intΠUI .(iii) pi ∈ ΠDB if and only if c ∈M and {c} ×R * K. Moreover, intΠDB = intΠBB.(iv) pi ∈ ΠDU if and only if c ∈M and {c} × R ⊂ K. Moreover, intΠDU = intΠIU .Theorem 4. The elements of the ternary primal-dual partition are neither opennor closed and satisfy the following statements:(i) pi ∈ ΠBB if and only if (0n, 1) /∈ clK and c ∈ M . In particular, pi ∈ intΠBBif and only if SC holds and c ∈ intM . Moreover, intΠBB is dense in ΠBB ifand only if |T | ≥ n.(ii) pi ∈ ΠUI if and only if (0n, 1) /∈ clK, c /∈ M and ({c} × R) ∩ clK = ∅. Inparticular, pi ∈ intΠUI if and only if there exists y ∈ Rn such that c′y < 0and a′ty > 0 for all t ∈ T . Moreover, intΠUI is dense in ΠUI .(iii) pi ∈ ΠIU if and only if (0n, 1) ∈ clK, c ∈M and {c} ×R ⊂ K. In particular,pi ∈ intΠIU if and only if (0n, 1) ∈ intK. Moreover, intΠIU is dense in ΠIUif and only if |T | ≥ n+ 1.(iv) pi ∈ ΠII if and only if (0n, 1) ∈ clK and c /∈M . Moreover, intΠII = ∅.(v) pi ∈ ΠBI if and only if (0n, 1) /∈ clK, c /∈ M and ({c} × R) ∩ clK 6= ∅. Inparticular, ΠBI = ∅ if |T | <∞. Moreover, intΠBI = ∅.(vi) pi ∈ ΠIB if and only if (0n, 1) ∈ clK, c ∈M and {c} ×R * K. In particular,ΠIB = ∅ if |T | <∞. Moreover, intΠIB = ∅.(vii)⋃i∈{BB,UI,IU}intΠi is a dense subset of Π.From Theorem 4 it can be shown that(2)⋃i∈{I,B,U}intΠPi =⋃i∈{I,B,U}intΠDi =⋃i∈{BB,UI,IU}intΠi.360 M. A. Goberna, M. I. TodorovExample 5. Let n = 2, T = {1, 2, 3} and pi := (a, b, c) such that (a1, b1) =(0, 0,−1), (a2, b2) = (1, 1,−1), (a3, b3) = (−1, 1,−1), and c = (0,−1). By Theo-rem 4(ii), pi ∈ ΠUI . Now consider the sequence {pir}∞r=1 ⊂ Π, where pir is obtainedfrom pi by replacing a1 = (0, 0) with ar1 = (0,−1r ). It is easy to see, applying The-orem 4(i), that pir ∈ ΠBB ∀r ∈ N and limr→∞ pir = pi. Thus, pi ∈ ΠUI ∩ bdΠBB.Hence,(intΠBB) ∪ (intΠUI) & int(ΠBB ∪ΠUI).3. Ill-posedness in continuous linear optimization. A mathematical pro-gramming problem is called ill-posed in the feasibility sense if arbitrarily small pertur-bations provide both consistent and inconsistent problems ([12], [5] and [2] give formulaefor the distance to ill-posedness in ordinary LP, in conic LP, and in general LSIP, re-spectively). In continuous linear optimization, the set of well-posed problems in primal(dual) feasibility sense is then the union of topological interiors (intΠPC) ∪ (intΠPI )((intΠDC ) ∪ (intΠDI ), respectively), whereas the the set of ill-posed problems in primal(dual) feasibility sense is bdΠPC (bdΠDC , respectively). The open sets intΠPC , intΠPI ,intΠDC , and intΠDI are characterized in Lemma 1. In [3] a general LSIP problem iscalled ill-posed in the solvability sense if arbitrarily small perturbations provide bothsolvable and non-solvable problems. In fact, it is shown that this set is the boundary ofthe set of parameters which have a finite primal optimal value. In our context of con-tinuous linear optimization this set is bdΠBB. In the same vein, the set of well-posedproblems w.r.t. the primal (dual) ternary partition is⋃i∈{I,B,U}intΠPi (⋃i∈{I,B,U}intΠDi ,respectively).On the other hand, [11] defines a conic programming problem to be ill-posed (inprimal-dual feasibility sense) when it lays on the boundary of the set of consistentproblems whose corresponding dual is also consistent. This class of primal-dual ill-posed parameters is, in our setting, bdΠBB. The interior of ΠBB was characterizedin [13]. Now the set of well-posed problems w.r.t. the primal-dual ternary partition is⋃i∈{BB,UI,IU}intΠi.First we recall the usual definition of ill-posedness in an arbitrary topological space.Definition 6. Let X be a topological space and ∅ 6= A ⊆ X. We say that thepoint y ∈ X is ill-posed w.r.t. to set A if each neighbourhood of y contains points of Aand its complement.We denote the set of ill-posed points w.r.t. the set A with AIP . Obviously, AIP =bdA = (clA) \ (intA). Moreover, (X \A)IP = AIP .Definition 7. Let X be a topological space and ∅ 6= A ⊆ X. We say that thepoint y ∈ A is relatively ill-posed w.r.t. to set A if each neighbourhood of y containspoints of the complement of A.We denote the set of relatively ill-posed points w.r.t. the set A with ARIP . Obvi-ously, ARIP = A\ intA = A∩AIP . This definition makes sense, because f.i. sometimeswe need only the ill-posed points which belong to the set A. Observe that A is open(closed) if and only if ARIP = ∅ (ARIP = AIP , respectively).Compt. rend. Acad. bulg. Sci., 60, No 4, 2007 361Definition 8. Let X be a topological space and let A = {Ai, i ∈ I} be a family ofpairwise disjoint subsets of X such that intAi 6= ∅ ∀i ∈ I and⋃i∈IAi is a dense subset ofX. We say that the point y ∈ X is absolutely ill-posed w.r.t. toA if y ∈ X\(⋃i∈IintAi).We denote the set of absolutely ill-posed points w.r.t. A as AAIP . In particular, ifA is a dense subset of X, {A,X \A}AIP = AIP .Let us go back to the linear continuous optimization. The proofs of the next char-acterizations of the relatively ill-posed problems w.r.t. the elements of the partitionsof Π considered in this paper follow from Lemma 3 and Theorems 2 and 4:pi ∈ (ΠPC)RIP if and only if (0n, 1) /∈ clK and SC fails.pi ∈ (ΠPI )RIP if and only if (0n, 1) ∈ bdK.pi ∈ (ΠDC )RIP if and only if c ∈M \ intM .pi ∈ (ΠDI )RIP if and only if c /∈M and c′y ≥ 0 is a consequence of {a′ty > 0, t ∈ T}.pi ∈ (ΠPB)RIP if and only if (0n, 1) /∈ clK, ({c} × R) ∩ clK 6= ∅ and SC fails ifc ∈ intM .pi ∈ (ΠPU )RIP if and only if (0n, 1) /∈ clK, ({c} × R) ∩ clK = ∅ and c′y ≥ 0 is aconsequence of {a′ty > 0, t ∈ T}.pi ∈ (ΠDB )RIP if and only if c ∈M , {c} × R * K and SC fails if c ∈ intM .pi ∈ (ΠDU )RIP if and only if c ∈M , {c} × R ⊂ K and (0n, 1) /∈ intK.pi ∈ ΠRIPCC = ΠRIPBB if and only if (0n, 1) /∈ clK, c ∈M and SC fails if c ∈ intM .pi ∈ ΠRIPCI if and only if (0n, 1) /∈ clK, c /∈M , and SC holds whenever c′y ≥ 0 is aconsequence of {a′ty > 0, t ∈ T}.pi ∈ ΠRIPIC if and only if (0n, 1) ∈ clK, c ∈ M , and (0n, 1) /∈ intK wheneverc ∈ intM .pi ∈ ΠRIPII = ΠII if and only if (0n, 1) ∈ clK and c /∈M .pi ∈ ΠRIPBB if and only if (0n, 1) /∈ clK, c ∈M and SC fails if c ∈ intM .pi ∈ ΠRIPUI if and only if (0n, 1) /∈ clK, c /∈M , ({c} ×R) ∩ clK = ∅ and c′y ≥ 0 isa consequence of {a′ty > 0, t ∈ T}.pi ∈ ΠRIPIU if and only if (0n, 1) ∈ bdK, c ∈M , and {c} × R ⊂ K.Moreover, we have{ΠPC ,ΠPI }AIP = (ΠPC)IP = (ΠPI )IP = (ΠPC)RIP ∪ (ΠPI )RIPand{ΠDC ,ΠDI }AIP = (ΠDC )IP = (ΠDI )IP = (ΠDC )RIP ∪ (ΠDI )RIP .Finally, taking into account (2) and Theorem 4(vii), we conclude that the set ofabsolutely ill-posed problems w.r.t. the primal-dual, the primal, and the dual ternary362 M. A. Goberna, M. I. Todorovpartitions coincide, and it can be expressed in different ways:{ΠBB,ΠUI ,ΠIU}AIP = ⋃i∈{BB,UI,IU}intΠiIP(3)= ⋃i∈{II,BI,IB}Πi ∪ ⋃i∈{BB,UI,IU}ΠRIPi ,{ΠPI ,ΠPB,ΠPU}AIP = ⋃i∈{I,B,U}intΠPiIP = ⋃i∈{I,B,U}(ΠPi )RIP ,(4)and(5) {ΠDI ,ΠDB ,ΠDU }AIP = ⋃i∈{I,B,U}intΠDiIP = ⋃i∈{I,B,U}(ΠDi )RIP .Moreover, pi is absolutely ill-posed w.r.t. any of the three ternary partitions (i.e.,pi ∈ {ΠBB,ΠUI ,ΠIU}AIP ) if and only if pi satisfies:1) SC fails if c ∈ intM ;2) c′y ≥ 0 is a consequence of {a′ty > 0, t ∈ T}; and3) (0n, 1) /∈ intK.The parameter pi in Example 5 satisfies these conditions, so that it is ill-posed inthe sense of (3), (4), and (5). This example shows that{ΠBB,ΠUI ,ΠIU}AIP ' (ΠBB ∪ΠUI ∪ΠIU )IP .REFERENCES[1] Ben-Israel A., A. Charnes, K. O. Kortanek. Bull. Amer. Math. Soc., 75, 1969,318–324.[2] Ca´novas M. J., M. A. Lo´pez, J. Parra, F. J. Toledo. Math. Programming (Ser. A),103, 2005, 95–126.[3] Id. Linear Algebra Appl., 416, 2006, 520–540.[4] Id. SIAM J. Optim., 16, 2006, 629–649.[5] Epelman M., R. M. Freund. Ibid., 12, 2002, 627–655.[6] Goberna M. A. In: Continuous Optimization: Current Trends and Applications, A. Ru-binov and V. Jeyakumar (eds), New York, Springer-Verlag, 2005.[7] Goberna M. A., M. A. Lo´pez. Linear Semi-Infinite Optimization, Chichester, JohnWiley & Sons, 1998.Compt. rend. Acad. bulg. Sci., 60, No 4, 2007 363[8] Goberna M. A., M. A. Lo´pez, M. I. Todorov. SIAM J. Matrix Anal. Appl., 17,1996, 730–743.[9] Id. Set-Valued Analysis, 9, 2001, 75–99.[10] Kortanek K. O., Q. Zhang. Math. Programming (Ser. A), 91, 2001, 127–144.[11] Nunez M. A. Ibid., 91, 2002, 375–390.[12] Renegar J. Ibid., 70, 1995, 279–351.[13] Todorov M. I. Numer. Funct. Anal. Optim., 8, 1985–86, 541–556.University of AlicanteAlicante, Spaine-mail: mgoberna@ua.es∗UDLAPuebla, MexicoOn leave fromInstitute of Mathematics and InformaticsBulgarian Academy of Sciences1113 Sofia, Bulgariae-mail: maxim.todorov@udlap.mx364 M. A. Goberna, M. I. Todorov

Ill-posedness in continuous linear optimization via partitions of the space of parameters

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