Some Properties of Interval Quadratic Programming Problem

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1 International Journal of Sstes Siene Applied Matheatis 07; (5): doi: 0.648/j.ijssa ISSN: (Print); ISSN: (Online) Soe Properties of Interval Quadrati Prograing Proble Qianqian Xu, Shengnan Jia, Haohao Li, *, Jinhua Huang 3 Shool of Sienes, Hangzhou Dianzi Universit, Hangzhou, China Shool of Data Sienes, Zhejiang Universit of Finane Eonois, Hangzhou, China 3 Shool of Autoation Siene Engineering, South China Universit of ehnolog, Guangzhou, China Eail address: hlzufe@6.o (Haohao Li) * Corresponding author o ite this artile: Qianqian Xu, Shengnan Jia, Haohao Li, Jinhua Huang. Soe Properties of Interval Quadrati Prograing Proble. International Journal of Sstes Siene Applied Matheatis. Vol., No. 5, 07, pp doi: 0.648/j.ijssa Reeived: June 5, 07; Aepted: June 7, 07; Published: Otober 4, 07 Abstrat: For interval liner prograing probles, Rohn proposed four equivalene relations regarding to the upper lower bounds of the interval optial value. In this paper, siilar probles of interval quadrati prograing proble have been disussed. Soe interesting properties have been proved an illustrative exaple rearks are given to get an insight of the properties. Kewords: Interval Quadrati Prograing, Lower Upper Bounds, Optial Value. Introdution Interval sstes interval optial odels are often used for odeling inforation sstes engineering probles, e.g. []. Over the past deades, the interval sstes interval atheatial prograing (IvMP) have been studied b an authors, see e.g. [-6, 0-7, 3-35] the referenes therein. Soe papers studied the proble of oputing the range of optial values of interval linear prograing probles, see e.g., [-7] aong others. Soe authors studied the proble of oputing the range of optial values of interval quadrati progras (IvQP) [7, 8, 8-9]. he other frequent probles is stud the properties on upper lower bounds of IvMP. here have been developed diverse ethods for oputing the lower upper bounds of IvQP. Liu [8] Li [9] desribed soe ethods to opute the lower upper bounds of IvQP with inequalit nonnegative onstraints. Hladik [7] foused on onvex quadrati prograing probles with interval data, the proble of oputing the best ase the worst ase optial values was disussed for interval onvex quadrati prograing probles of ertain fors, then he studied the ethod of the upper lower bounds of interval-valued onvex quadrati prograing probles in a general for. For oputing the upper bound, these ethods desribed in [8, 8, 9] are based on the dual proble of IvQP (dual ethod for short), under the ondition that the zero dualit gap of a pair of prial dual IvQP is speified. Reentl, Li et al. [] proposed a new ethod to opute the upper bound of optial values of IvQP, in this new ethod, onl prial progra is taken into onsideration, the dual proble is not required thus the ondition that the dualit gap is zero is also reoved, then Li desribed the properties on the upper lower bounds of interval quadrati prograing [9-]. However few was done on the relations aong the upper lower bounds. he relations aong the upper lower bounds of IvLP (interval linear prograing) have been established in [4, 8]. In this paper, IvQP several equivalent onditions for interval quadrati prograing proble have been studied. First, soe properties of interval quadrati progra are forulated. Based on these results, soe interesting useful relations of interval quadrati progra will be given, whih give an insight into the orresponding probles.. Preliinaries Fro notations fro [4], an interval atrix is defined as

2 06 Qianqian Xu et al.: Soe Properties of Interval Quadrati Prograing Proble n A = [ A, A ] = {A R A A A} n where A, A R, A A, '' " is understood oponentwise. he enter the radius of atrix A is denoted b A = ( A + A), A = ( A A) So A = [ A A, A + A ] [4]. An interval vetor b = [ b, b] = { b R b b b} is understood as one-olun interval atrix. Let { ± } be the set of all {,} -diensional vetors, i.e. where e = ( ) { ± } = { R = e},, is the -diensional vetor of all s. For a given { ± }, let ( ) = diag,,, denote the orresponding diagonal atrix. For eah its sign vetor sgn x is defined b if xi (sgn x) i = if xi < 0 n x R, where i =,,, n. then x = zx, where z = sgn x { ± } n. Given an interval atrix A = [ A A, A + A ], for eah { ± } { } z ±, the atries Az = A A z are defined. Siilarl for an interval vetor b = [ b b, b + b ] n for eah { ± }, the vetors b = b + b are defined. n k n n k Let A R, B R, b R, R, d R n n Q R, onsider the quadrati prograing proble in x Qx + x subjet to Ax b, Bx = d, x, where Q is positive seide Briefl, the proble is rewritten as in{ x Qx + x Ax b, Bx = d, x }. () he Dorn dual proble[0] of the quadrati () is ax{ u Qu b v d w Qu + A v + B w +, v } () Let f ( A, B, b,, d, Q) = inf{ x Qx + x Ax b, Bx = d, x }, g( A, B, b,, d, Q) = sup{ u Qu b v d w Qu + A v + B w+, v }. denote the optial values of () (), respetivel. he set of all -b-n interval atries will be denoted b n IR IR. Given the set of all -diensional interval vetors b n n Q IR n k n n k A IR, B IR, b IR, IR, d IR, the interval onvex quadrati progra in{ x Qx + x Ax bb, x = d, x 0} (3) is the fail of onvex quadrati progras() with data satisfing A A, B B, b b,, d d, Q Q. where Q is positive seidefinite for all Q Q. he lower upper bounds of the optial values are respetivel defined as f = inf{ f ( A, B, b,, d, Q) A A, B B, b b,, d d, Q Q }, f = sup{ f ( A, B, b,, d, Q) A A, B B, b b,, d d, Q Q }. he following leas will be used in the proof of the ain results. Lea.. [] f = in{ x Qx + x Ax b, Bx d, Bx d, x } f ( ABbd,,,,, Q ) = sup f ( A, B, b,, d, Q). k { ± } Lea.. [9] Let f ( ABbd,,,,, Q ) =, then there exists a B0 B, suh that 0 e f ( A, B, b, d, Q) {, } holds for eah d d. Lea.3. [0] It is hold that he upper bound of the optial values of () is defined as ϕ = sup{ g( A, B, b,, d, Q) A A, B B, b b,, d d, Q Q }.

3 International Journal of Sstes Siene Applied Matheatis 07; (5): , if f = ; ( a) ϕ = f, if f is finite; ( b) annot derine, if f =. ( ) Lea.4. [9] Let f be finite let x be an optial solution of the proble(6). then f = f ( A, B B, b,, d + d, Q), where, for arbitrar α [, ] ( Bx d) i if ( Bx + d) i > 0 i = ( Bx + d) i i = k α if ( Bx + d) i = 0 3. Soe Properties of IvQP (,,, ) For an interval linear prograing proble (IvLP) with data Ab,,, Rohn proved that the following assertions are equivalent [4]. a. For eah A A, B B, b b, the proble in{ x Ax = b, x } has an optial solution. b. Both f ( Ab,, ) f ( Ab,, ) are. Both f ( Ab,, ) ϕ ( Ab,, ) are d. he sste A A p p is feasible ϕ ( Ab,, ) is Siilarl, the relationship of the following assertions is studied. (a) For eah A A, B B, b b,, d d, Q Q the proble in{ x Qx + x Ax b, Bx = d, x } (4) has an optial solution. (b) Both f ( ABbd,,,,, Q ) f ( ABbd,,,,, Q ) are () Both f ( ABbd,,,,, Q ) ϕ ( ABbd,,,,, Q ) are (d) he sste Qu + A v + B v B v3 +, vi, i =,, 3 (5) is solvable, ϕ ( ABbd,,,,, Q ) is heore.. For an interval quadrati prograing proble with data A A, B B, b b,, d d, Q Q there holds that ( a) ( b) ( ) ( d) Proof. ( a) ( b) : Sine eah proble (4) has an optial solution, it ust be f ( ABbd,,,,, Q ) <. Fro Lea., if f ( ABbd,,,,, Q ) =, then there exists a B0 B, suh that f ( A, B0, b, d, Q) {, }, hold for eah d d, so the possibilit of f ( ABbd,,,,, Q ) = is preluded b Lea.. Hene f ( ABbd,,,,, Q ) is Fro Lea.4 Lea., whih an be got is that f = f ( A, B B, b,, d + d, Q) k { ± } e f = sup f ( A, B, b,, d, Q), in other words, f gets the axiu in a finite group generated b, hene f ( A, B, b,, d, Q) is ( b) ( ) : Fro Lea.3, if f ( ABbd,,,,, Q ) is finite, f ( ABbd,,,,, Q) = ϕ( ABbd,,,,, Q ). Hene f ( ABbd,,,,, Q ) is finite iplies that ϕ ( ABbd,,,,, Q ) is Aording to the proess of above proof, soething an be easil got that if f ( ABbd,,,,, Q ) is finite, then f ( ABbd,,,,, Q ) is hus, the result of ( ) ( b) is obviousl true, sine that if f ( ABbd,,,,, Q ) ϕ ( ABbd,,,,, Q ) are finite, f ( ABbd,,,,, Q ) f ( ABbd,,,,, Q ) are obviousl hus, ( b ) ( ) are neessar suffiient onditions to eah other. ( ) ( d) : Sine f = inf{ x Qx + x Ax b, Bx d, Bx d, x } (6) he Dorn dual proble of (6) is ax{ u Qu b v d v + d v3 Qu + A v + B v B v3 +, vi, i =,, 3} (7) If f ( A, B, b,, d, Q) is finite, then b the strong dualit theor [9], the following forula is established

4 08 Qianqian Xu et al.: Soe Properties of Interval Quadrati Prograing Proble f = sup{ u Qu b v d v + d v3 Qu + A v + B v B v3 +, vi, i =,, 3} (8) so that(7) is finite, thus the sste(5)is solvable. 4. An Illustrative Exaple In this setion, an illustrative exaple is given for heore., whih helps us to underst heore.. Exaple Consider the interval quadrati progra in[, ] x + x [, ] x x = [0, ] [,] x x = [, ] x, x he orresponding interval atries vetors of (9) are = [, ] B = [0, ] d = [,] [,] he lower bound of the optial values an be deterined b onvex quadrati progra in x + x x x x x x x x x x, x (9) (0) It an be shown that f = 0 hen the upper bound is oputed b Lea.. his proble an be deoposed into four onvex quadrati progras x x = x x = x, x 4 x x = x x = x, x (3) (4) It is eas to see that the onvex quadrati progras (), () (3) are infeasible the onvex quadrati progra (4) is feasible, hene the optial solution is exist in the onvex quadrati progra (4), it an be shown that optial values of four onvex quadrati progras are f =, f =, f =, f =. hus f =. 3 4 herefore f, f are finite while the subproble () has no optial solution. his exaple shows that ( b) ( a) in heore.. 5. Conlusion In appliations we are ostl interested in interval quadrati prograing probles having finite optial solutions. here for probles of interval optiization when all subprobles have optial solutions is of partiular interest. his paper disuss soe interesting finite solution properties of interval quadrati prograing proble with stard onstraints. A topi is worth of further stud is the properties of interval quadrati prograing proble with ixed onstraints. Aknowledgeents he authors were partiall supported b the NNSF of China (Grant Nos , , U5097). x x = 0 x x = x, x x x = 0 x x = x, x 3 () () Referenes [] Worrawate Leela-apiradee, WeldonA. Lodwik, Phantipa hipwiwatpotjana, An algorith for solving two-sided interval sste of ax-plus linear equations, Inforation Sienes 399 (07) [] R. E. Moore, R. B. Kearfott, M. J. Cloud, Introdution to interval analsis, SIAM, Philadelphia, 009. [3] A. Neuaier, Interval Methods for Sstes of Equations, Cabridge Universit Press, Cabridge, 990. [4] Fiedler M, Rohn J, Nedoa J. Linear optiization probles with inexat date [M]. New York: Springer, 006:35-66.

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