On the calm b-differentiability of projector onto circular cone and its applications

Science & Technology Development Journal, 23(4):727-736 Open Access Full Text Article Research Article DongThap University Correspondence Vo Duc Thinh, Dong Thap University Email: vdthinh@dthu.edu.vn History  Received: 2020-07-30  Accepted: 2020-09-06  Published: 2020-10-10 DOI : 10.32508/stdj.v23i4.2426 Copyright © VNU-HCM Press. This is an open- access article distributed under the terms of the Creative Commons Attribution 4.0 International license. On the calm b-differentiability of projector onto circular cone and its applications Vo Duc Thinh* Use your smartphone to scan this QR code and download this article ABSTRACT In this paper, we study a concept on the calm B-differentiability, a new kind of generalized differen- tiabilities for a given vector function introduced by Ye and Zhou in 2017, of the projector onto the circular cone. Then, we discuss its applications inmathematical programming problemswith circu- lar cone complementarity constraints. Here, this problem can be considered to be a generalization of mathematical programming problems with second-order cone complementarity constraints, and thus it includes a large class ofmathematical models in optimization theory. Consequently, the obtained results for this problem are generalized, and then corresponding results for some special mathematical problems can be implied from them directly. For more detailed information, we will first prove the calmly B-differentiable property of the projector onto the circular cone. This result is not easy to be shownby simply resorting to those of the projection operator onto the second-order cone. By virtue of exploiting variational techniques, we next establish the exact formula for the reg- ular (Fréchet) normal cone (this concept was proposed by Kruger and Mordukhovich in 1980) to the circular cone complementarity set. Note that this set can be considered to be a generalization of the second-order cone complementarity set. In finally, the exact formula for the regular (Fréchet) normal cone to the circular cone complementarity set would be useful for us to study first-order necessary optimality conditions for mathematical programming problems with circular cone com- plementarity constraints. Our obtained results in the paper are new, and they are generalized to some existing ones in the literature. Key words: calmly B-differentiable, circular cone, complementarity set, Fréchet normal cone, optimality condition INTRODUCTION The second-order cone programming (SOCP) prob- lem plays an important role in the optimization the- ory and has attracted much attention from mathe- maticians, see, e.g., 1–7. We refer the reader to1,2,4–7 and the references therein for some remarkable re- sults on optimality conditions and stability analysis of (SOCP). Inspired by the second-order cone, many researchers have investigated optimization and complementar- ity problems where their constraints are involved in second-order cones. It is called the second-order cone complementarity problem (SOCCP), which in- cludes a large class of optimization problems such as quadratically constrained problems (see 8), the second-order cone programming, and nonlinear complementarity problem (see 9). In particular, re- cent attention is paid to the second-order cone com- plementarity set. Let us now mention some existing results concern- ing this set. In10, Liang et al. provided formula- tions for Fréchet normal cone to the second-order cone complementarity set. Unfortunately, the ob- tained results were shown to be inexact in 11. In that paper, Ye and Zhou gave exact formulas for the proximal/regular (Fréchet)/limiting normal cone to the second-order cone complementarity set by us- ing the projection operator onto second-order cones and the generalized differentiability called the calm B-differentiability. Some first-order optimality con- ditions formathematical programswith second-order cone complementarity constraints were obtained in 12 and sufficient conditions for error bound property of second-order cone complementarity problems were established in13. To obtain these results, the au- thors used the symmetric and self-dual property of the second-order cone. Recently, generalizations of second-order cones and second-order cone complementarity sets have been examined by many authors5,14–22. For example, au- thors in 14,19–22 considered circular cones, which are generalizations of second-order cones and are, in gen- eral, nonsymmetric and non-self-dual cones. The generalized differentiability of the projection operator Cite this article : Thinh V D. On the calm b-differentiability of projector onto circular cone and its applications. Sci. Tech. Dev. J.; 23(4):727-736. 727 Science & Technology Development Journal, 23(4):727-736 onto the circular cone was provided in 14,22. More- over, the differentiability and calmness of vector- valued functions associated with the circular cone were also studied in19,23. In particular, authors in21 showed that the results of the projection operator onto a circular cone could not be shown by simply resort- ing to the results of the projection operator onto the second-order cone, and hence, it is necessary to study the results of circular cone directly. To the best of our knowledge, there is no result on the calmly B-differentiable property concerning the circular cone and its extension. In this paper, in- spired by11,13,22, we first study in Section 3, the calm B-differentiability of the circular cone. We then pro- vide in Section 4 the formula for the Fréchet normal cone to a circular cone complementarity set, which can be considered as a generalization of the second- order cone complementarity set. This formula would be useful for us to study optimality conditions for mathematical programming problems with circular cone complementarity constraints. PRELIMINARIES Throughout the paper, if not otherwise specified, f(t)=o(t) (f(t)=O(t)) means f (t)jtj ! 0 (resp., f (t) jtj is uniformly bounded) as t ! 0, and ( f (x))+ := maxf f (x);0g; and ( f (x)) := minf f (x);0g:Br(x) stands for the closed ball centered at x 2 Rn with ra- dius r > 0. Given x;y 2 Rn; xTy stands for the scalar product of x and y. For x := (x0;xr ) 2 RRn1; we use the following notation x? := fy 2 RnjxTy= 0g and exr :={ xrjjxr jj if xr ̸= 0; if otherwise:any unit vector e 2 Rn1 Let CRn be a nonempty subset, clC denotes its clo- sure. The polar cone C◦ and the dual cone C⋆ of C are C◦ := fy 2 Rnjy⊤x 0;8x 2Cg and C⋆ := fy 2 Rnjy⊤x 0;8x 2Cg respectively. The Fréchet normal cone to C at x 2 clC are defined respectively by, see24, bNC(x) := fx 2 Rn⟨x;x′ x⟩  o(∥ x′ x ∥);8x′ 2Cg: Lemma 2.1 (24, Theorem 1.14) Let D={x | h(x)2 C} and let Ñh(x) be surjective. Then bND(x) = Ñh(x)T bNC(x): Let f :Rn ! (¥; ¥] and x 2 Rn such that f ( x ) is finite. The Fréchet subdifferential of f at x is defined by, see [24, pages 89 and 90], b¶ f (x) := fx 2 Rngj limsup x!x ⟨x;xx⟩ f (x)+ f (x)xx   0 The indicator function of a setC  Rn is denoted by dC(x) := { 0 if x ̸2C; ¥ otherwise: It is known from [ 25, Proposition 1.18] that b¶dC(x) =bNC(x) for any x 2C. Let F:Rn ) Rm be a set-valued mapping, the domain and the graph of F are domF := fx 2 RnjF(x) ̸=∅g; gphF := f(x;y) 2 RnRmjy 2 F(x)g: The Fréchet coderivative of F at (x, y) 2 gphF are re- spectively defined by, see [24, Definition 1.32], for each y2Rm, bDF(x;y)(y) := fx 2 Rnj(x;y) 2 bNgphF (x;y)g: When F(x) is single-valued, y can be omitted in the above notations. Moreover, if F is continuously dif- ferentiable, then for all y2Rm, we get bDF(x)(y) = fÑF(x) yg : The derivative in the directionh2Rn ofF at x is defined by F ′(x;y) := lim t!0+ F(x+ th)F(x) t : The circular cone is defined (cf.14,19–23) by Kq := fx= (x0;xr) 2 RRmjx0 tanq  jjxrjjg (2.1) with angle q 2 (0; p2 ). When q = p4 , it re- duces to the second-order cone defined by Kq := fx= (x0;xr) 2 RRmjx0  jjxrjjg. In this case, the set Ω := { (x;y)jx 2K ;y 2K ;xTy= 0} ; (2.2) is called the second-order cone complementarity set. If q ̸= p4 thenKq is a nonsymmetric and non-self-dual 728 Science & Technology Development Journal, 23(4):727-736 cone. The boundary and the interior ofKq are given respectively by bdKq := fx= (x0;xr) 2 RRmjx0 tanq  jjxrjjg ; intKq :=Kqn(bdKq ): The positive dual cone and the polar cone ofKq are defined respectively by, see [ 20, Theorem 2.1], Kq := fy= (y0;yr) 2 RRmjy0 cotq  jjyrjjg ; K q :=K q = fy= (y0;yr) 2 RRmjy0 cotq  jjyrjjg : A relation between the boundary of Kq and that of K q is established as follows. Proposition 2.2 Let x 2 bdKqnf0g and y 2 bdK q nf0g. Then, yT x= 0 if and only if x = k(y0cot2q ;yr ) with k = x0 y0 tan 2 q (equivalently, y= k(x0 tan2 q ;xr) with k= y0 x0 cot 2q :). Proof. Let x 2 bdKqnf0g and y 2 bdK q nf0g. “ If ”: Suppose that there exists k 2 R++ := (0;¥) with x= k(y0cot2q ;yr ), then yT x= 0. “ Only if ”: Let yT x = 0, then we get x0 tanq =∥ xr ∥ ;y0 tan( p2 q) = jjyrjj and x0y0+yTr xr = 0:Thus, one has yTr xr =x0y0 = (jjxrjjcotq)(jjyrjjcot( p2 q))=jjxrjj : jjyrjj ; which implies the existence ofk 2R++ such that xr = kyr . Consequently, we obtain x0 tanq = ky0 tan( p 2 q); i.e., x0 tan2 q = ky0. Hence, x = k(y0cot2q ;yr) with k = x0y0 tan 2 q , and the proof is completed. □ We recall that for any given x := (x0;xr) 2RRm, it can be decomposed by (see [ 20, Theorem 3.1]) x= l1(x) u1x +l2(x)u2x ; where the spectral valuesl1(x);l2(x) and the spectral vectors u1x ;u2x are defined respectively by l1(x) := x0jjxrjjcotq ; l2(x) := x0+ jjxrjj tanq ; u1x := 1 1+cot2 q [ 1 0 0 cotq ][ 1 exr ] ; u2x := 1 1+tan2 q [ 1 0 0 tanq ][ 1exr ] : The metric projection of x onto Kq , denoted by PKq (x), is defined as follows PKq (x) := argminz2Kq jjx zjj = fz 2Kq j ∥ x z ∥∥ xu ∥;8u 2Kqg: From22 and the convexity ofKq , we get thatPKq (x) is a single-valued set and PKq (x) = (l1(x)+)u 1 x+(l2(x)+)u2x: (2.3) Moreover, [ 26, Proposition 2] states that, for all x 2 Rm+1, x=PKq (x)+PK ◦q (x) and ⟨PKq (x);PK ◦q (x)⟩= 0: SinceP(Kq )(x);P(K ◦q )(x), one gets PKq (x) =PK ◦q (x): Let us define the circular cone complementarity set as G := f(x;y)jx 2Kq ;y 2K q ;xTy= 0g; (2.4) which is a generalized type of (2.2). Given (x,y)2G and an arbitrary u 2Kq , it holds that jj(x y)ujj2jj(x y) xjj2 = jj(xu) yjj2jjyjj2 = jjxujj22⟨xu;y⟩ = jjxujj22⟨x;y⟩+2⟨u;y⟩  0; which means that x = PKq (x y). Similarly, we get that y 2PK q (y x). The above observation allows us to obtain a relation between the complementarity set G, and the projec- tion ontoKq as follows. Proposition 2.3 Let G be as in (2.4). Then, we get [(x;y) 2 G]() [x 2PKq (x y)] () [y 2PK q (y x)]() [y 2PK ◦q (x y)]: By Proposition 2.3, G can be expressed by G= f(x;y)j(x y;x) 2 gphPKq g: Let f :Rm+1Rm+1 ! Rm+1Rm+1 be defined by f (x;y) := (x y;x) for all (x;y) 2 Rm+1Rm+1, we can check that f is continuously differentiable and Ñ f (x;y) = [ Im+1 Im+1 Im+1 0 ] ; where Im+1 is the unit matrix of the degree m+1, has full rank. It follows from [ 27, Exercise 6.7] thatbNG(x;y) = fÑ f (x;y)(x;y)j (x;y) 2 bNgphPKq ( f (x;y))g = fx+ y;x)j (x;y) 2 bNgphPKq )(x y;x)g = f(u;v)j v 2 bDPKq (x y)(u v)g: 729 Science & Technology Development Journal, 23(4):727-736 From the above discussion, one obtains the following result, which plays an important role in computing the Fréchet normal cone to complementarity set. Proposition 2.4 Let G be as in (2.4) and (x,y)2G. Then, we getbNG(x;y) = f(u;v)j v 2 bDPKq (x y)(u v)g: (2.5) CALMB-DIFFERENTIABILITY OF THE PROJECTMAPPINGONTO A CIRCULAR CONE In this section, we first show that the projection oper- ator PKq is calmly B-differentiable at any x 2 Rm+1. Then, we provide a characterization for a proximal normal vector of G. Definition 3.1 (11) The function F : Rn ! Rm is called calmly B-differentiable at x if, for all h suffi- ciently close to 0, we get F(x+h)F(x)F ′(x;h) = O(∥ h ∥2 ): Theorem 3.2The projection mapping PKq is calmly B-differentiable for any x 2 Rm+1. Proof.Given an arbitrary x 2 Rm+1, it is enough to show that, for h sufficiently close to 0, PKq (x+h)PKq (x)P ′ Kq (x;h) = 0= O(∥ h ∥2): (3.1) We consider the following cases. Case 1: x 2 intKq . Then, we have PKq (x) = x, PKq (x + h) = x + h. Moreover, it follows from the definition of the directional derivative that P′Kq (x;h) = h. So, (3.1) is fulfilled. Case 2: x 2 intK . We get, in this case that PKq (x) = 0, PKq (x+h) = 0. On the other hand, by the definition ofP′Kq (x;h), one hasP ′ Kq (x;h) = 0, so (3.1) holds. Case 3: x 2 bdKqnf0g. It implies that l1(x) = 0 and l2(x)> 0. By (2.3) and Lemma 3.2(b) in22, we have PKq (x) = x;P ′ Kq (x;h) = h (1+ cot2 q)((ux1)Th)_ux1; PKq (x+h) = ((x0+h0 ∥ xr+hr ∥ cotq)+)  11+cot2 q [ 1 (xrhr)cotq jjxr+hr jj ] +((x0+h0+ ∥ xr+hr ∥ tanq)+)  11+tan2 q [ 1 (xr+hr) tanq jjxr+hr jj ] : (3:2) Let ePh = (ePh0; ePhr ) 2 Rm+1 be defined byePh =PKq (x+h)PKq (x)P′Kq (x;h) (3.3) we now show computations for ePh0 and ePhr . By (3.2) and (3.3), ePh0 is given by Figure 1. By the expression of ∥ xr ∥, for all hr sufficiently close to 0, we get ∥ xr+hr ∥=∥ xr ∥+⟨exr;hr⟩+O(∥ hr ∥2), which implies that x0+h0 ∥ xr+hr ∥ cotq = h0⟨exr;hr⟩cotq +O(∥ hr ∥2): (3.5) It follows from (3.4), (3.5), and the Lipschitz property of the function (
)_ with modulus 1 that j (x0+h0 ∥ xr+hr ∥ cotq)_ +(h0⟨exr;hr⟩cotq)_j  O(∥ h_r ∥2): (3.6) On the other hand, we also get equation 3.7 in Figure 2 Using the Taylor expression of the function x∥x∥ , one has xr+hr jjxr+hrjj = exr+ 1jjxjj (exr;exTr )hr+O(jjhrjj2) : (3.8) Thus, (3.7) is given by Figure 3 Moreover, from (3.5) and (3.6), we obtain ePhr = (exr;exTr )hr+O(∥ hr ∥2)∥ xr ∥ (cotq + tanq)  ((h0⟨xr;hr⟩cotq +O(∥ hr ∥2))_) + exr cotq + tanq (∥ hr ∥2)): (3.10) It is necessary to show that ∥ ePhr ∥ O(∥ h ∥2). In- deed, it follows from (3.10) that ∥ ePhr ∥ ∥ (exr;exTr )hr+O(∥ hr ∥2) ∥∥xr∥(cotq+tanq) ( ∥ h ∥ p 1+ cot2 q +O(∥ h ∥2 ) ) +O(∥ h ∥2 ) = O(∥ h ∥2 ): Combining to (3.6), we get ∥ ePh ∥=O(∥ h ∥2 ), which means that (3.1) holds. Case 4: x2bdK q nf0g. It is obvious thatPKq (x)= 0. Then, for each h 2 Rm+1 sufficiently close to 0, we have l1(x) < 0;l2(x) = 0 and l1(x+ h) = (x0+ h0) ∥ xr+hr ∥ cotq < 0:. It follows from (2.3) and Lemma 3.2 in22 that PKq (x+h) = ((x0+h0+ tanq jjxr+hrjj)+)u2x+h P′Kq (x;h) = (1+ tan 2 q)(⟨u2x ;h⟩)+u2x ; where u2x = 1 1+ tan2 q [ 1exr tanq ] ; u2x+h = 1 1+tan2 q [ 1 xr+hr jjxr+hr jj tanq ] = u2x + 1 1+tan2 q [ 0 1 jjxr jj (exrexTr )hr+O(jjhrjj2) ] 730 Science & Technology Development Journal, 23(4):727-736 Figure 1: Equation 3.4 Figure 2: Equation 3.7 Figure 3: Equation 3.9 Thus, one gets ePh = (x0+h0+ tanq jjxr+hrjj)+u2x+h (1+ tan2 q)(⟨u2x ;h⟩)+u2x = (h0+ ⟨exr tanq ;hr⟩+O(jjhrjj2))+( u2x + 1 1+tan2 q [ 0 1 jjxr jj (exrexTr )hr+O(jjhrjj2) ]) (h0+ ⟨exr tanq ;hr⟩)+u2x (3:11) with ePh :=PKq (x+h)PKq (x)P′Kq (x;h). Since the function (
)+ is Lipschitz with modulus 1, from (3.11), we obtainePh O(jjhrjj2)+ u2x +jj(h0+ ⟨exr tanq⟩+O(jjhrjj2))+( 1 1+tan2 q )[ 0 1 jjxr jj (exrexTr )hr+O(jjhrjj2) ] jj  O(jjhrjj2)+(jjhjj p 1+ tan2 q +O(jjhrjj2)  ( 1 1+tan2 q √( 1 jjxr jj2 exr;hr)2+O(jjhrjj4) ) = O(jjhjj2): Note that the last inequality holds by ∥hr∥∥ h∥. Consequently, (3.1) is implied. Case 5: x=0. Then, for all h 2 Rm+1, we get l1(x) = l2(x) = 0 and PKq (x) = 0;PKq (x+h) =PKq (h); P′Kq (x) =PKq (h): 731 Science & Technology Development Journal, 23(4):727-736 Thus, one has jjPKq (x+h)PKq (x)P ′ Kq (x;h)jj= 0; which means that (3.1) holds. Case 6: x 2 Rm+1n(Kq [ (K q )). Since Rm+1n(Kq [ (K q )) is open, for h sufficiently close to 0, one has x+ h 2 Rm+1n(Kq [ (K q )). Moreover, we can check that l1(x)< 0; l1(x+h)< 0 and l2(x) > 0; l2(x+ h) > 0. Thus, it follows from (2.3) and [ 22, Lemma 3.2(a) and (3.6)] that PKq (x) = (x0+ tanq jjxrjj)u2x ; PKq (x+h) = (x0+h0+ tanq jjxr+hrjj)u2x+h P′Kq (x;h) = 1 tanq+cotq[ cotq exTrexr x0+jjxr jj tanqjjxr jj I x0jjxr jjexrexTr ][ h0 hr ] : By directly computations, we get jjxr+hrjj= jjxrjj+ ⟨exr;hr⟩+O(jjhrjj2) ; xr+hr jjxr+hr jj = xr jjxr jj + 1jjxrjj ( IexrexTr )hr+O(jjhrjj2) ; and P′Kq (x;h) =[ 1 1+tan2 q (h0+ tanq⟨exr;hr⟩) 1 cotq+tanq ( h0exr+ x0+jjxr jj tanqjjxr jj hr)+B ] B= x0jjxr jj ⟨exr;hr⟩exr: By letting ePh :=( 1+ tan2 q )( PKq (x+h)PKq (x)P ′ Kq (x;h) ) , then one has ePh0 = O(jjhrjj2)= O(jjhjj2) ; (3.12) O ( jjhrjj2 ) (x0+ tanq jjxrjjexr tanq) h0 tanqexr x0 tanq + jjxrjj tan2 qjjxrjj hr + x0 tanq jjxrjj ⟨exr;hr⟩exr = h0⟨exr;hr⟩+O(jjhrjj2)= O(jjhjj2) ; (3.13) where the last equation holds by the fact that jjh0⟨exr ;hr⟩jj jjhjj2  jh0j:jjxrjj:jjhrjj jjhjj2  jjxr jj:jjhjj 2 2jjhjj2 = jjxr jj 2 : Thus, (3.1) is fulfilled. □ APPLICATION In this section, we first establish the formulation for the Fréchet normal cone to the circular cone comple- mentarity set G. Theorem 4.1 Let G be defined as in (2.4) and (x;y) 2 G. Then, we get Figure 4 Proof . We consider the following cases. Case 1: x=0 and y 2 int K q . Then, we get x-y=-y, which implies that l1(x y) =y0jjyrjjcotq < 0; l2(x y) =y0+ jjyrjj tanq : Since y 2 int K q , we get y0 tan ( p 2 q ) > jjyrjj ; i:e:; y0 > jjyrjj tanq . Consequently, one has l2(x y) =y0+ jjyrjj tanq < 0: It follows from [22, Lemma 3.1(a) and (3.6)] that ¶B(PKq )(x y) = { ÑPKq (x y) } = 0: By [22, Theorem 3.5(a)], we have bDPKq (x y)(y) = {ÑPKq (x y)y}= 0: Consequently, from (2.5), one obtains bNG(x;y) = {(u;v) ju 2 Rm+1;v= 0} : Case 2: x 2 intKq and y=0. Then we get x-y=x, so l1(x y) = x0jjxrjjcotq < 0; l2(x y) = x0+ jjxrjj tanq : It follows from [22, Lemma 3.1(a), (3.6) andTheorem 3.5(a)] that ¶B(PKq )(x y) = { ÑPKq (x y) } = I;bDPKq (x y)(y) = {ÑPKq (x y)y}= y: Therefore, by (2.5), we have bNG(x;y) = {(u;v) ju 2 Rm+1;v 2 Rm+1} : Case 3: x 2 bdKqnf0g,y 2 bdK q nf0g and y⊤x = 0. We get from Lemma 2.2 that y = k(x0 tan2 q ;xr) with k = y0x0 cot 2q > 0, which implies that x y= (x0;xr) k ( x0 tan2 q ;xr ) = ( (1 k tan2 q)x0;(1+ k)xr ) : Moreover, we have l1(x y) = (1 k tan2 q)x0 (1+ k) jjxrjjcotq =k(1 tan2 q)x0 < 0 (4.1) 732 Science & Technology Development Journal, 23(4):727-736 Figure 4: Theorem 4.1 and l2(x y) = (1 k tan2 q)x0+(1+ k) jjxrjjcotq = (1 tan2 q)x0 > 0: (4.2) By (3.6) in 22, we get ÑPKq (x y) = 1tanq+cotq[ cotq exTrexr 1+tan2 qtanq(1+k) I 1k tan2 q(1+k) tanq exrexTr ] : On the other hand, it follows from [22, Theorem 3.5] and (2.5) that bNG(x;y) = { (u;v) j v 2 bDPKq (x y)(u v)} = { (u;v) j v 2 ÑPKq (x y)(u v) } : (4.4) Let (u;v) 2 bNG(x;y) and x′ 2 bd Kqnf0g, x′ 2 bdK q nf0gwith y′= k(x ′ 0 tan 2 q ;x′r). Then y′Tx′= 0, which implies that (x’,y’ )2G. Consequently, one has ⟨(u;v);(x′;y′ )(x;y)⟩ jj(x′;y′ )(x;y)jj = ⟨u;x′x⟩+⟨v;y′y⟩ jj(x′x;y′y)jj = ⟨u;x′x⟩+⟨kv;(x′0 tan2 q ;x ′ r)(x0 tan2 q ;xr)⟩ jj(x′x;k(x′0 tan2 q ;xr)k(0tan2 q ;xr ))jj  ⟨u+k(v0 tan2 q ;vr );x′x⟩p 1+k2jjx′xjj : (4:5) Since (u;v)2 bNG(x;y), passing to the limit in (4.5), we get limsup x′ bdKq nf0g!x ⟨u+k(v0 tan2 q ;vr );x′x⟩ jjx′xjj  limsup (x′;y′) G!(x;y) p 1+k2(⟨(u;v);(x′;y′ )(x;y)⟩) jj(x′;y′ )(x;y)jj  0 which implies that u+ k(v0 tan 2 q ;vr ) 2 bNbdKq nf0g(x) (4.6.) By x2 bdKqnf