Duality Theory for the Matrix Linear Programming Problem H. W. ...

Duality Theory for theMatrix Linear Programming ProblemH. W. CORLEYDepartmentof Industrial Engineering,The University of Texas, Arlington, Texas 76019Submittedby v. LakshmikanthamReceived March 25, 19831. INTRODUCTIONMultiple objective linear programming, also known as the linear vectormaximizationproblem, has been studied by a number of authors. Inparticular, its duality theory has been considered by Philip [9], Isermann[7], Gray and Sutherland [5], Brumelle [1], and Ponstein [10]. Neither thesymmetry nor all the relationships of the duality theory of conventionallinear programming, however, have been obtained in previous work. Suchresults are developed in this paper, which is most directly related to [7], byregarding the variables to be matrices. Previous work on matrixlinearprogramming [ 3] has been restricted to a scalar objective function anddiffers substantially from that presented here.The notation R m X n will denote the real inner product space of all realm X n matrices A = [au], and as usual Rn will be written for RnX1. R;xnwill be the set of matrices in R m X n with all nonnegative elements ande E R m X n the matrix consisting entirely of zeros with its order apparentfrom context. For A, B E Rmxn we write A ~B (or B :;:?;A) ifB -AE R;xnand A < B (or B > A) if B -A E R;xn\{e}. The relation ~ is obviously apartial order on Rmxn. Let S be a subset ofRmxn. The point Z* E S is saidto be a maximal element (or upper efficient point) of S if there does not existZ E S for which Z* < Z. Similarly Z* E S is a minimal element (or lowerefficient point) of S if there does not exist Z E S for which Z < Z*. The setof all maximal elements of S is denoted by max S and minimal elements bymin S.2. THE PRIMAL, DuAL, AND SADDLEPOINT PROBLEMSLet A E Rmxn, B E Rmxr, C E Rpxn' XE Rnxr, and YE Rpxmthroughout the paper. The primal matrix linear programming problem,written470022-247X/84 $3.00Copyright @ 1984 by Academic Press. Inc.All rights of reproduction in any form reserved.

(P) max CXs.t.AX~E,(I)X~ e, (2),is to find all optimal X* satisfying ( I) and (2) for which CX* Emax{CX: AX~ E, X~ e}. Any X satisfying (I) and (2) is termed feasible toP, and p is said to be unbounded if some component of CX is unboundedabove for feasible X. The dual matrix linear programming problem to P,written(D) min YEs.t.YA~C, (3)Y~ e, (4)is to determine all optimal Y* satifying (3) and (4) such thatY*E E min{EY: YA ~ C, Y~ e}. Any Y satisfying (3) and (4) is similarlytermed feasible to D, and D is unbounded if some component of YE isunbounded below for feasible Y.A related problem is the saddlepoint matrix linear programming problem.With the above notation the point (X*, Y*) is said to be a saddlepoint of theLagrangian functionL(X, Y) = CX + Y(E -AX) = YE + (C- YA)X (5)ifX* ~ e, Y* ~ e, (6)there does not exist Y ~ e such thatcx* + Y(E -AX*) < cx* + Y*(E -AX*), (7)and there does not exist X ~ e such thatCX* + Y*(E -AX*) < cx+ Y*(E -AX). (8)3. DUALITY RELATIONSHIPSThe duality relationships between problems p and D are now established.These results parallel the duality theory of conventional linear programming.Results 1-4 follow easily from the previous definitions and are stated withoutproof.

Result 1. The dual of the dual is the primal.Result 2.Result 3.If X is feasible to P and Y to D, then CX ~ YE.If X* is feasible to P, Y* is feasible to D, and CX* = Y*E,then X*, Y* are optimal to P and D, respectively.Result 4.If P or D is unbounded, then the other problem has no feasiblepoints.The remaining duality relationships are proved using the followinglemmas.LEMMA 1. If X* = [xkj] E Rnx, is optimal to P, then there exist scalars}..I} > 0, i = l,...,p, j = 1,..., r, such that x~, k = 1,..., n,j = 1,..., r, is optimalto the scalar linear programming problemp , n(Q) max ~ ~ ~ ).I}ClkXkj1=11=1 k=ls.t.alx1~bl}, i= 1,...,m,j= 1,...,r, (9)xkj~O' k=I,...,n,j=I,...,r, (10)where al represents row i of A and Xj columnj of x= [xkj] ERnx,.Proof It is obvious that P is equivalent to a linear vector maximizationproblem of finding an efficient point in R n' for the objective functionc1 e ecP e ee c1 e lx]e cP e ~: ERp"x,e e c1e e cPwhere Ci is row i of C, subject to (9), (10). As a result of Theorem 2 ofIsermann [6], X* = [x~] determines a properly efficient point(Xtl ,..., X:l ,..., xt"..., X:,)I for this problem. Then by Theorem 2 of GeofTrion[ 4] there exist ).I} > 0 for which the x~ solve Q. I

LEMMA 2. Let X* be optimal to P and E = {CX: AX ~ B, X ~ 8}. Thenthere exist a real number a and a linear functional I on Rpxrfor whichI(Z) > 0 forallZER:xr\{8}, (11)I(Z»a forallZE[R:xr\{8}]+CX*, (12)I(Z)~a forallZEE-R:xr. (13)Proof For Z = [ZI}] E Rpxr define I(Z) = Lf=1 LJ=I ).I}ZI} for ).I} > 0 asin Lemma 1, and set a = I(CX*). Then (11) holds immediately. Moreover,from (11) for Z E R:xr\{8},I(Z + CX*) = I(Z) + I(CX*) > I(CX*) = a,yielding (12). Finally, for CXE E and WE R:xr it follows from Lemma I,(II), and the fact that 1(8)=0 that I(CX- W)=I(CX)-I(W)~I(CX)~I(CX*) = a, so (13) is established. .Theorem I next relates the primal, dual, and saddlepoint problems.THEOREM I. If X* is optimal to P, then there exists Y* ~ 8 such that(X*, Y*) is a saddlepoint of L(X, Y) andY*(B-AX*)=8. (14)Similarly, if Y* is optimal to D, there exists X* ~ 8 such that (X*, Y*) is asaddlepoint of L(X, Y) and (C- Y*A)X* = 8.Proof Only the first statement will be proved since the second thenfollows readily from the symmetry of (5). From Lemmas I and 2 there existsa linear functional Ion R p x r such that ( 11) is satisfied and X* maximizesthe scalar objective function I(CX) subject to (I), (2). By a standardLagrange multiplier theorem in [8] there exists a linear functional u* onRmxr such that u*(W) ~ 0 for all WE R:.xr'andu*(B-AX*)=O, (15)I(CX*)+u*(B-AX*)~I(CX)+u*(B-AX) foraIIX~8. (16)Choose Z* E R:xr\ {8} for which I(Z*) = I and define Y* E Rpxm suchthat Y*W = u*(W) Z*. Then (14) follows from (15), and Y*WE R:xr forall WE R~xr. Hence Y* ~ e and (6) is proved since X* ~ e. The definitionof Y* and (16) next yield thatI[CX- CX* + Y*(B -AX) -Y*(B -AX*)] ~ 0 for all X~ e. (17)

It now followsfrom (11) and (17) thatX- CX* + Y*(B -AX) -Y*(B -AX*) e R:xr\ {8} for all X;;:!: 8to prove (8). Finally, suppose that there exists YE Rpxm' Y;;:!: 8, satisfyingCX* + Y*(B -AX*) -CX* -Y(B -AX*) E R:xr\{8}. (18)Then (14) and (18) give that Y(B-AX*)ER:xr\{8}, contradicting theassumption that Y;;:!: 8 and establishing (7) to complete the proof. ITHEOREM 2. If (X*, Y*) is a saddle point of L(X, Y), then X* is optimalto P, Y* is optimal to D, Y*(B -AX*) = 8, and (Y*A -C)X* = 8.Proof The result follows directly from Theorem I of Corley [2] and itsproof, with obvious changes in terminology, and from the symmetry of(5). ITheorem 3 next uses Theorems I and 2 to establish an analog to thestrong duality theorem of standard linear programming.THEOREM 3. If there exists an optimal X* to P, then there exists anoptimal Y* to D for which CX* = Y* B and Y*(B -AX*) = e. Similarly, ifthere exists an optimal Y* to D, there exists an optimal X* to P for whichCX* = Y*B and (Y*A -C)X* = e.Proof Only the first statement need be proved. If X* is optimal to P, byTheorem 1 there exists Y* such that (X*, Y*) is a saddlepoint of L(X, Y)and Y* (B -AX* ) = e. Then by Theorem 2 Y* is optimal to D and(Y* A -C) X* = e. It follows that CX* = Y* AX = Y* B to establish thetheorem.IAs opposed to scalar linear programming Theorem 3 does not say thatCX* = Y*B for any X* optimal to P and any Y* optimal to D. However,the following corollary to Theorem I does hold.COROLLARY. max{CX:AX~B, X;;:!: e} = min{YB: YA ;;:!: C, Y;;:!: e}.REFERENCESI. s. BRUMELLE, Duality for multiple objective convex programs, Math. Oper. Res. 6(1981),159-172.2. H. W. CORLEY, Duality theory for maximizations with respect to cones, I. Math. Anal.Appl. 84 (1981),560-568.3. B. D. CRAVEN AND B. MOND, Linear programming with matrix variables, Linear AlgebraAppl. 38 (1981), 73-80.

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