Solving the Quadratic Eigenvalue Complementarity Problem by DC Programming

Transcription

1 Solvng the Quadratc Egenvalue Complementarty Problem by DC Programmng Y-Shua Nu 1, Joaqum Júdce, Le Th Hoa An 3 and Pham Dnh Tao 4 1 Shangha JaoTong Unversty, Maths Departement and SJTU-Parstech, Chna nuyshua@sjtu.edu.cn Insttuto de Telecomuncações, Portugal judce@co.t.pt 3 Unversty of Lorrane, France hoa-an.le-th@unv-lorrane.fr 4 Natonal Insttute of Appled Scences, France pham@nsa-rouen.fr Abstract. We present n ths paper some results for solvng the Quadratc Egenvalue Complementarty Problem (QECP) by usng DC(Dfference of Convex functons) programmng approaches. Two equvalent Nonconvex Polynomal Programmng (NLP) formulatons of QECP are ntroduced. We focus on the constructon of the DC programmng formulatons of the QECP from these NLPs. The correspondng numercal soluton algorthms based on the classcal DC Algorthm (DCA) are also dscussed. Keywords: Egenvalue Problem, Complementarty Problem, Nonconvex Polynomal Programmng, DC Programmng, DCA. 1 Introducton Gven three matrces A, B, C R n n, the Quadratc Egenvalue Complementarty Problem (QECP) conssts of fndng a λ R and an assocated nonzero vector x R n such that w = λ Ax + λbx + Cx x T w = 0 x 0, w 0 (1) Ths problem and some applcatons have been frstly ntroduced n [19] and s usually denoted by QECP(A, B, C). In any soluton (λ, x) of QECP(A, B, C), the λ-component s called a quadratc complementary egenvalue, and the vector x-component s a quadratc complementary egenvector assocated to λ. QECP s an extenson of the well-known Egenvalue Complementarty Problem (ECP) [18], whch conssts of fndng a complementary egenvalue λ R

2 YS Nu et al. and an assocated complementary egenvector x R n \ {0} such that w = λbx Cx x T w = 0 x 0, w 0 where B, C R n n are two gven matrces. Clearly, ECP s a specal case of QECP where the matrx A s null. Durng the past several years, many applcatons of ECP have been dscussed and a number of algorthms have been proposed for the soluton of ths problem and some extensons [1 3, 6 11, 15, 16]. ECP has at least one soluton f the matrx B of the leadng λ-term s postve defnte (PD) [9, 18]. Contrary to the ECP, the QECP may have no soluton even when the matrx A of leadng λ-term s PD. For nstance, f B = 0, A, C are PD matrces, there s no soluton for QECP snce x T w = λ x T Ax+x T Cx > 0, λ R, x R n \ {0}. The exstence of a soluton to QECP depends on the gven (A, B, C). If the matrx A s PD, QECP has at least a soluton f one of the two followng condtons holds: () C / S 0 [4], where S 0 s the class of matrces defned by C S 0 x 0, x 0, Cx 0. () co-hyperbolcty[19] : (x T Bx) 4(x T Ax)(x T Cx) for all x 0, x 0. In practce, nvestgatng whether C S 0 reduces to solvng a specal lnear program [4]. On the other hand, t s relatvely hard to prove that cohyperbolcty holds. However, there are some suffcent condtons whch mply the co-hyperbolcty. For nstance, ths occurs f A and C are both PD matrces. A number of algorthms have been proposed for the soluton of QECP when A PD and one of the condtons C / S 0 or co-hyperbolcty holds [1, 4 6, 19]. As dscussed n [4 6], some of these methods are based on nonlnear programmng (NLP) formulatons of QECP such that (λ, x) s a soluton of QECP f and only f (λ, x) s a global mnmum of NLP wth an optmal value equal to zero. In ths paper, we ntroduce two nonlnear programmng formulatons and ther correspondng DC programmng formulatons when co-hyperbolcty holds, and we brefly dscuss the DC Algorthm for the soluton of these DC programs. The paper s organzed as follows. Secton contans the nonlnear programmng formulatons of QECP, and the correspondng dc formulatons mentoned before. A new result on lower and upper bounds estmaton of the quadratc complementary egenvalue s gven n secton 3. The numercal soluton algorthms for solvng these DC programmng formulatons are dscussed n secton 4. Some conclusons are presented n the last secton. DC Programmng Formulatons for QECP In ths secton, we ntroduce two DC programmng formulatons of QECP when A PD and the co-hyperbolc property holds. These DC programs are based ()

3 DCA for QECP 3 on two NLP formulatons of QECP. The constructon of the DC programmng problem requres lower and upper bounds on the λ-varable whch can be computed by the procedures dscussed n [6]. We wll also present a new procedure for such a goal n the secton 3..1 Nonlnear Programmng Formulatons As dscussed n [6], QECP s equvalent to the followng NLP: (P ) 0 = mn f(x, y, z, w, λ) := y λx + z λy + x T w s.t. w = Az + By + Cx e T x = 1 e T y = λ x 0, w 0, z 0. (3) As (x, y, z, w, λ) s an optmal soluton of the problem (P ) f and only f (λ, x) s a soluton of QECP. In fact, for any soluton of QECP (λ, x) that does not satsfy e T x x = 1, we can always construct a soluton (λ, ) of QECP satsfyng e T x such a constrant. The problem (P ) s a polynomal programmng problem where a nonconvex polynomal functon f(x, y, z, w, λ) s mnmzed subject to lnear constrants. Due to the fact that any polynomal functon s a dc functon, we can reformulate the problem (P ) as a dc program. On the other hand, observng that the complementarty constrant w T x = 0, x 0, w 0 holds f and only f w T x = n =1 mn(x, w ) = 0, we have the followng equvalent nonlnear programmng formulaton of (P ): (P ) 0 = mn f (x, y, z, w, λ) = y λx + z λy + n =1 mn(x, w ) s.t w = Az + By + Cx e T x = 1 e T y = λ x 0, w 0, z 0. The problems (P ) and (P ) have the same set of lnear constrants. The dffculty for solvng (P ) and (P ) reles on the non-convexty on ther objectve functons.. DC programmng formulatons The polynomal functon f n (P ) can be decomposed nto four parts: f(x, y, z, w, λ) = y + z λy T (x + z) + λ ( x + y ) + x T w = f 0 (y, z) + f 1 (x, y, z, λ) + f (x, y, λ) + f 3 (x, w) wth f 0 (y, z) = y + z f 1 (x, y, z, λ) = λy T (x + z) f (x, y, λ) = λ ( x + y ) f 3 (x, w) = x T w

4 4 YS Nu et al. The functon f 0 s convex quadratc functon, whle f 1, f, f 3 are nonconvex polynomal functons. Smlarly, the objectve functon f n (P ) s also decomposed nto the followng four terms as: f (x, y, z, w, λ) = f 0 (y, z) + f 1 (x, y, z, λ) + f (x, y, λ) + f 3 (x, w) where f 3 (x, w) defned by n =1 mn(x, w ) s a polyhedral concave functon. Both the blnear functon f 3 and the polyhedral concave functon f 3 are classcal dc functons whose dc decompostons are as follows: 1. DC decomposton of blnear functon f 3 : f 3 (x, w) = x + w 4 x w 4 n whch x+w 4 and x w 4 are both convex quadratc functons.. DC decomposton of polyhedral functon f 3 : (4) f 3 (x, w) = n n mn(x, w ) = (0) ( mn(x, w )) (5) =1 =1 where n =1 mn(x, w ) s a convex polyhedral functon. To obtan a dc decompostons of the nonconvex polynomal functons f 1 and f, we frst obtan the expressons of ther gradents and hessans: 1. Gradent and Hessan of f 1 : x f 1 (x, y, z, λ) f 1 (x, y, z, λ) = y f 1 (x, y, z, λ) z f 1 (x, y, z, λ) = λ f 1 (x, y, z, λ) λy λ(x + z) λy y T (x + z). 0 λi 0 y f 1 (x, y, z, λ) = λi 0 λi (x + z) 0 λi 0 y. y T (x + z) T y T 0. Gradent and Hessan of f : x f (x, y, λ) λ x f (x, y, λ) = y f (x, y, λ) = λ y. λ f (x, y, λ) λ( x + y ) λ I 0 4λx f (x, y, z, λ) = 0 λ I 4λy 4λx T 4λy T ( x + y )

5 DCA for QECP 5 The spectral radus of the hessan matrces f 1 and f (denoted by ρ( f 1 ) and ρ( f )) can be bounded above by the nduced 1-norm as follows: ρ( f 1 ) f 1 1 = max{ λ + y, x + z + λ, ( y + x + z )} ρ( f ) f 1 = max{λ + λ x, λ + λ y, x + y + λ ( x + y )} Thus ρ( f 1 ) and ρ( f )) are bounded when the varables (x, y, z, w, λ) of (P ) and (P ) are bounded. The next proposton shows that f the quadratc complementary egenvalue λ of QECP s bounded, then the varables x, y, z, w are bounded wth respect to the bounds of λ. Proposton 1. If the quadratc complementary egenvalue λ of QECP s bounded n an nterval [l, u], then any optmal soluton of (P ) and (P ) satsfes: x [0, 1] n ; y [mn{0, l}, max{0, u}] n ; z [0, max{u, l }] n ; max{u, l } j A 1j + max{ l, u } j B 1j + j C 1j 0 w. max{u, l } j A nj + max{ l, u } j B nj +. j C nj Proof. Suppose that we could determne some values l and u such that λ- component of QECP s located n the nterval [l, u]. 1. e T x = 1, x 0 mples x [0, 1] n.. y = λx, x [0, 1] n and λ [l, u] mply y [mn{0, l}, max{0, u}] n. 3. z = λy, y = λx z = λ x, wth x [0, 1] n, λ [l, u], leads to z [0, max{u, l }] n. 4. Snce w 0, the upper bound of w s obtaned from the defnton of w as Az+By+Cx. As x [0, 1] n, y [mn{0, l}, max{0, u}] n, z [0, max{u, l }] n, then w s also bounded: w max{u, l } j A 1j + max{ l, u } j B 1j + j C 1j. max{u, l } j A nj + max{ l, u } j B nj + j C nj Let us defne the convex polyhedral set: C := {(x, y, z, w, λ) : w = Az + By + Cx, e T x = 1, e T y = λ, x [0, 1] n,. y [mn{0, l}, max{0, u}] n, z [0, max{u, l }] n, w 0, l λ u}. The problems (P ) and (P ) defned on C have the same set of optmal solutons, and ρ( f 1 ) and ρ( f ) are bounded. In fact, the followng proposton holds:

6 6 YS Nu et al. Proposton. For (x, y, z, w, λ) C, ρ( f 1 ) + n(p + p) = ρ 1 ρ( f ) (3np + p + 1) = ρ where p = max{ l, u }. Proof. Snce λ [l, u], then λ max{ l, u } = p. Hence, ρ( f 1 ) max{ λ + y, x + z + λ, ( y + x + z )}. But, Hence, x + z y np. x + z 1 + np. ρ( f 1 ) max{p, 1 + p + p, 1 + n(p + p)} = + n(p + p) = ρ 1 Smlarly, ρ( f ) max{λ + λ x, λ + λ y, x + y + λ ( x + y )} max{p + p, 3p, 3np + p + 1} = (3np + p + 1) = ρ. Thus, we get a dc decomposton for f 1 and f as follows: f 1 (x, y, z, λ) = ρ 1 (x, y, z, λ) ( ρ 1 (x, y, z, λ) f 1 (x, y, z, λ)) f (x, y, λ) = ρ (x, y, λ) ( ρ (x, y, λ) f (x, y, λ)) where ρ1 (x, y, z, λ) and ρ (x, y, λ) are quadratc convex functons. Whle ρ 1 (x, y, z, λ) f 1 (x, y, z, λ) and ρ (x, y, λ) f (x, y, λ) are locally convex restrcted on C. Usng the dc decompostons of f 1,f,f 3 and f 3 derved n ths secton, we get the followng dc decomposton for the objectve functons f and f. 1. A dc decomposton for f s gven by: where g(x, y, z, w, λ) = f(x, y, z, w, λ) = g(x, y, z, w, λ) h(x, y, z, w, λ) x + w 4 + ρ 1 + ρ x +( ρ 1 + ρ +1) y +( ρ 1 h(x, y, z, w, λ) = g(x, y, z, w, λ) f(x, y, z, w, λ). +1) z + ρ 1 + ρ λ,

7 DCA for QECP 7. A dc decomposton for f s gven by: where f (x, y, z, w, λ) = g (x, y, z, w, λ) h (x, y, z, w, λ) g (x, y, z, λ) = ρ 1 + ρ x + ( ρ 1 + ρ + 1) y + ( ρ 1 + 1) z + ρ 1 + ρ λ, h (x, y, z, w, λ) = g (x, y, z, λ) f (x, y, z, w, λ). The functons g and g are both convex quadratc functons, whle h and h are locally convex functons restrcted on the convex polyhedral set C. Fnally, we get the followng equvalent DC programs of (P ) and (P ) as below: (P DC ) 0 = mn g(x, y, z, w, λ) h(x, y, z, w, λ) (6) s.t. (x, y, z, w, λ) C. (P DC ) 0 = mn g (x, y, z, λ) h (x, y, z, w, λ) s.t. (x, y, z, w, λ) C. (7) 3 Lower and upper bounds for the quadratc complementary egenvalue λ Snce the bounds of the varables x, y, z, w n C, as well as the dc decompostons gven n the prevous secton depend on the bounds of λ, we need to estmate ts upper and lower bounds. The followng theorem gves these values. Proposton 3. If A PD and the co-hyperbolc condton holds, the λ-component of any soluton of QECP satsfes l = β α λ γ + α = u wth s = mn{x T Ax : e T x = 1, x 0}, α = max{γ, β } + max,j{ Cj} β = γ = { mn{ Bj} max{a, f mn{ B j} j} > 0; mn{ B j} s, f mn{ B j } 0. { max{ Bj} s, f max{ B j } > 0; max{ B j} max{a j}, f max{ B j} 0. s, Proof. Snce A PD and the co-hyperbolc condton holds, the λ-component of any soluton of QECP satsfes λ = xt Bx ± (x T Bx) 4(x T Ax)(x T Cx) x T. Ax

8 8 YS Nu et al. Let U = {e T x = 1, x 0}. For a gven matrx M R n n and for any x U, we next prove that: mn,j M j x T Mx max M j, x U. (8),j If fact, let (Mx) denote the -th element of the vector Mx. Then But, (Mx) s bounded by j=1 x T Mx = n x (Mx) =1 n n mn{ M j x j : x U} (Mx) max{ M j x j : x U}, x U. Snce the lnear programs mn{ n j=1 M jx j : x U} and max{ n j=1 M jx j : x U} have optmal solutons on vertces, the optmal values of the above lnear programs are exactly mn j {M j } and max j {M j }. Hence, we can compute bounds for x T Mx on U as follows: j=1 mn,j {M j} = mn{ x mn{m j } : x U} j x mn{m j } j x (Mx) = x T Mx x max{m j } max{ j x max j {M j } : x U} = max,j {M,j}. Hence, (8) s true. Usng the bounds (8) for the matrces B and C, we have: Snce A PD, we have Accordngly, xt Bx x T Ax mn{ B j } x T Ax mn { B j} x T Bx max { B j},,j,j mn { C j} x T Cx max,j,j { C j} 0 < s = mn{x T Ax : x U} x T Ax max,j {A j}, x U. s bounded by: xt Bx x T Ax max{ B j} x T γ = Ax { max{ Bj} s, f max{ B j } > 0; max{ B j} max{a j}, f max{ B j} 0. and mn{ B j } x T Ax β = { mn{ Bj} max{a j}, f mn{ B j} > 0; mn{ B j} s, f mn{ B j } 0.

9 DCA for QECP 9 Then ( xt Bx x T Ax ) + xt Cx x T Ax max{γ, β } + max{ C j} = α. s Fnally, we can compute bounds for λ as follows: β α xt Bx x T Ax ( xt Bx x T Ax ) + xt Cx x T Ax λ xt Bx x T Ax + ( xt Bx x T Ax ) + xt Cx x T Ax γ + α. In practce, t s nterestng to compare n the future the bound proposed here wth the one gven n [6]. The bounds gven n ths paper have been desgned such that they can be computed n a small amount of effort, even for large-scale problems. 4 DC Algorthms for solvng P DC and P DC In ths secton, we nvestgate how to solve the DC programmng formulatons (P DC ) and (P DC ). Gven a general DC program: mn{g(x) h(x) : x C}, where C s a non-empty convex set, the general DC algorthm (DCA) conssts of constructng two sequences {x k } and {y k } va the followng scheme[1 14]: x k y k h(x k ) x k+1 g (y k ) = argmn{g(x) x, y k : x C}. The symbol h stands for the sub-dfferental of the convex functon h, and g s the conjugate functon of g. These defntons are fundamental and can be found n any textbook of the convex analyss (see for example [17]). The sequence {x k } and {y k } are respectvely canddates for optmal solutons of the prmal and dual DC programs. In DCA, two major computatons should be consdered: 1. Computng h(x k ) to get y k.. Solvng the convex program argmn{g(x) x, y k : x C} to obtan x k+1. Now, we nvestgate the use of DCA to solve the DC programs (P DC ) and (P DC ). Concernng to (P DC ), snce the functon h s dfferentable, h(x, y, z, w, λ) s

10 10 YS Nu et al. reduced to a sngleton { h(x, y, z, w, λ)}, where h(x, y, z, w, λ) = g(x, y, z, w, λ) f(x, y, z, w, λ) x+w + (ρ 1 + ρ )x + λy λ x w (ρ 1 + ρ λ )y + λ(x + z) = ρ 1 z + λy w x. (9) (ρ 1 + ρ ( x + y ))λ + y T (x + z) For (P DC ), snce the functon h s non-dfferentable, we compute the convex set h (x, y, z, w, λ) as follows: (ρ 1 + ρ )x + λy λ x u (ρ 1 + ρ λ )y + λ(x + z) h (x, y, z, w, λ) = ρ 1 z + λy (10) v (ρ 1 + ρ ( x + y ))λ + y T (x + z) where 1, x < w ; u = (u ) =1,...,n, u = {0, 1}, x = w ; 0, x > w. 0, x < w ; v = (v ) =1,...,n, v = {0, 1}, x = w ; 1, x > w. Fnally, DCA appled to (P DC ) and (P DC ) requres solvng respectvely one convex quadratc program over a polyhedral convex set n each teraton. The followng two fxed-pont schemes descrbe our dc algorthms: (x k+1, y k+1, z k+1, w k+1, λ k+1 ) = argmn{g(x, y, z, w, λ) (x, y, z, w, λ), h(x k, y k, z k, w k, λ k ) : (x, y, z, w, λ) C} (11) wth g(x, y, z, w, λ) = x+w ρ 1+ρ λ. 4 + ρ1+ρ x + ( ρ1+ρ + 1) y + ( ρ1 + 1) z + (x k+1, y k+1, z k+1, w k+1, λ k+1 ) = argmn{g (x, y, z, λ) (x, y, z, w, λ), Y k : (x, y, z, w, λ) C} (1) wth Y k h (x k, y k, z k, w k, λ k ) and g (x, y, z, λ) = ρ1+ρ x + ( ρ1+ρ 1) y + ( ρ1 + 1) z + ρ1+ρ λ. + These convex quadratc programs can be effcently solved va a quadratc programmng solver such as CPLEX, Gurob, XPress, etc. DCA should termnate f one of the followng stoppng crtera s satsfed for gven tolerances ɛ 1, ɛ and ɛ 3.

11 DCA for QECP 11 (1) The sequence {(x k, y k, z k, w k, λ k )} converges,.e., (x k+1, y k+1, z k+1, w k+1, λ k+1 ) (x k, y k, z k, w k, λ k ) ɛ 1 () The sequence {f(x k, y k, z k, w k, λ k )} (resp. {f (x k, y k, z k, w k, λ k )}) converges,.e., f(x k+1, y k+1, z k+1, w k+1, λ k+1 ) f(x k, y k, z k, w k, λ k ) ɛ (resp. f (x k+1, y k+1, z k+1, w k+1, λ k+1 ) f (x k, y k, z k, w k, λ k ) ɛ ). (3) The suffcent global ɛ-optmalty condton holds,.e., f(x k, y k, z k, w k, λ k ) ɛ 3 (resp. f (x k, y k, z k, w k, λ k ) ɛ 3 ). The followng theorem ndcates the convergence of DCA: Theorem 1 (Convergence theorem of DCA). DCA appled to QECP generates convergence sequences {(x k, y k, z k, w k, λ k )} and {f(x k, y k, z k, w k, λ k )} (resp. {f (x k, y k, z k, w k, λ k )}) such that: The sequence {f(x k, y k, z k, w k, λ k )} (resp. {f (x k, y k, z k, w k, λ k )}) s decreasng and bounded below. The sequence {(x k, y k, z k, w k, λ k )} converges ether to a soluton of QECP when the thrd stoppng condton s satsfed or to a general KKT pont of (P DC ) (resp. (P DC )). Proof. The proof of the theorem s an obvous consequence of the general convergence theorem of DCA [1 14]. The suffcent global optmalty condton s due to the fact that the optmal value of the dc program s equal to zero. 5 Conclusons In ths paper, we have presented two DC programmng formulatons of the Quadratc Egenvalue Complementarty Problem. The correspondng numercal soluton algorthms based on the classcal DCA for solvng these dc programs were brefly dscussed. The numercal results and the analyss of the performance of DCA for solvng QECP wll be gven n a future paper. We wll dscuss a new local dc decomposton algorthm that s desgned to speed up the convergence of DCA. Furthermore, that paper wll also be devoted to the soluton of QECP when the condton A PD and C / S 0 holds. A new DC formulaton of QECP based on the reformulaton of an equvalent extended ECP wll be ntroduced to deal wth ths case and the correspondng DC Algorthm wll be dscussed.

12 1 YS Nu et al. Acknowledgements The research of Y-Shua Nu n ths project s partally supported and fnanced by the Innovatve Research Fund of Shangha Jao Tong Unversty 985 Program. Joaqum Júdce was partally supported n the scope of R&D Unt UID/EEA/5008/013, fnanced by the applcable fnancal framework (FCT/MEC through natonal funds and the applcable co-funded by FEDER-PT00 partnershp agreement). References 1. Adly, S., Seeger, A. A non-smooth algorthm for cone constraned egenvalue problems. Computatonal Optmzaton and Applcatons. Vol. 49, (011).. Adly, S., Rammal, H. A new method for solvng second-order cone egenvalue complementarty problem. Journal of Optmzaton Theory and Applcatons. (014), do: /s Brás, C, Fukushma, M.. Júdce, J., Rosa, S. Varatonal nequalty formulaton for the asymmetrc egenvalue complementarty problem and ts soluton by means of a gap functon. Pacfc Journal of Optmzaton. Vol. 8, (01). 4. Brás, C, Iusem, A.N.. Júdce, J. On the quadratc egenvalue complementarty problem. To appear n Journal of Global Optmzaton. 5. Fernandes, L. M., Júdce, J., Fukushma, M., Iusem, A. On the symmetrc quadratc egenvalue complementarty problem. Optmzaton Methods and Software, Vol. 9, (014). 6. Fernandes, L.M., Júdce, J., Sheral, H.D., Forjaz M.A., On an enumeratve algorthm for solvng egenvalue complementarty problems. Computatonal Optmzaton and Applcatons. Vol. 59, (014). 7. Júdce, J, Sheral, H. D., Rbero, I. The egenvalue complementarty problem. Computatonal Optmzaton and Applcatons. Vol. 37, (007). 8. Júdce, J., Raydan, M., Rosa, S., Santos, S. On the soluton of the symmetrc complementarty problem by the spectral projected gradent method. Numercal Algorthms. Vol. 44, (008). 9. Júdce, J., Sheral, H. D., Rbero, I., Rosa, S. On the asymmetrc egenvalue complementarty problem. Optmzaton Methods and Software. Vol. 4, (009). 10. Le Th, H.A., Moen, M., Pham, D.T., Judce, J. A DC programmng approach for solvng the symmetrc Egenvalue Complementarty Problem, Computatonal Optmzaton and Applcatons, Vol. 51, (01). 11. Nu, Y.S., Le Th, H.A., Pham, D.T., Júdce, J., Effcent dc programmng approaches for the asymmetrc egenvalue complementarty problem. Optmzaton Methods and Software. Vol. 8, (013). 1. Pham Dnh, T., Le Th, H.A.: DC optmzaton algorthms for solvng the trust regon subproblem, SIAM Journal of Optmzaton, Vol. 8, (1998). 13. Pham Dnh, T., Le Th, H.A.: DC Programmng. Theory, Algorthms, Applcatons: The State of the Art. Frst Internatonal Whorkshop on Global Constraned Optmzaton and Constrant Satsfacton, Nce, October 4 (00). 14. Pham Dnh, T., Le Th, H.A.: The DC programmng and DCA Revsted wth DC Models of Real World Nonconvex Optmzaton Problems. Annals of Operatons Research, Vol. 133, 3 46 (005).

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