# ε-Properly Efficiency of Multiobjective Semidefinite Programming with Set-Valued Functions

Authors: Chun Hong Yuan; Wei Dong Rong
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

In this paper, we study theε-properly efficiency of multiobjective semidefinite programming with set-valued functions. Firstly, we obtain the scalarization theorems under the condition of the generalized cone-subconvexlikeness. Then, we establish the alternative theorem which contains matrixes and vectors, the ε-Lagrange multiplier theorems, and the ε-proper saddle point theorems of the primal programming under some suitable conditions.


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## Body

## 1. Introduction

Vector optimization with set-valued functions has been used widely in many fields, such as economics and engineering. In recent years, set-valued optimization problem has aroused extensive concerns among the researchers [1–6]. Many authors focus their attention on how to get the approximate solutions of optimization problems. Chicco et al. [7] proposed a new type of solution based on the upper comprehensive sets and discussed the existence of optimal points in multicriteria situations. Zhao and Yang [8] obtained some useful properties of E-Benson proper efficiency. Gutiérrez et al. [9] presented some properties of strict efficient solutions in vector optimization related to the ε-efficiency notion. In locally convex Hausdorff topological vector spaces, ε-strongly efficient solutions of vector optimization with set-valued maps were discussed by Wang [10]. Rong and Wu [11] proposed the concept of ε-weak efficient solutions and derived scalarization results, saddle point theorems, and Lagrangian multipliers theorems.Semidefinite programming involves optimization problems with a linear objective function over semidefinite constraints. It shares many interesting properties with linear programming. Semidefinite programming unifies several standard problems (linear and quadratic programming) and finds many applications in engineering [12].By combining approximate solutions of vector optimization problems with multiobjective semidefinite programming,ε-properly efficiency of multiobjective semidefinite programming with set-valued functions is discussed in this paper. The rest of the paper is organized as follows: In Section 2, we introduce some notations and definitions used throughout the text. In Section 3, we derive scalarization theorems expressing the necessary and sufficient optimality conditions for ε-properly efficient solutions of the primal programming. Under the condition of the generalized cone-subconvexlikeness, the alternative theorem which contains matrix and vector and the ε-Lagrange multiplier theorems are established in Section 4. In Section 5, the ε-proper saddle point results are presented.


## 2. Preliminaries

Letv={v1,…,vm}T∈Rm and (1)R+m=v∈Rm:vi≥0,i=1,…,m.Obviously, R+m⊂Rm is a pointed closed convex cone with nonempty interior.LetA be a nonempty subset of Rm; the generated cone of A is defined as (2)coneA=αa:α≥0,a∈A.cl(A) and int(A) represent the closure and the interior of A, respectively. The positive dual cone A+ of A is defined as (3)A+=y∗∈Rm:y·y∗≥0,∀y∈A,and the strict positive dual cone A+i of A is defined as (4)A+i=y∗∈Rm:y·y∗>0,∀y∈A∖0,where y·y∗=yTy∗. Obviously, we see that A+i⊂A+. Let A⊂R+m be a nonempty subset and let ε∈R+m; we define the following two important sets:(5)ε-PMinA,R+m=y¯∈A:clconeA+R+m+ε-y¯∩-R+m=0,(6)ε-PMaxA,R+m=y¯∈A:clconeA-R+m-ε-y¯∩R+m=0.LetSp be a set of real symmetric matrixes of p-order and S+p be a set of real symmetric positive semidefinite matrixes of p-order. For matrixes K1,K2∈Sp, (7)K1≥K2⟺K1-K2∈S+p.The dot product of K1 and K2 is defined as K1·K2=tr(K1K2), where tr(K1K2) is the trace of the matrix K1K2. For two vectors a,b∈Rm, their dot product denotes a·b=aTb.It is easy to prove the following results.Proposition 1.
(i) LetK1∈S+p, K2∈S+p; then K1·K2≥0.
(ii) LetK1∈Sp; if K1·K2≥0 for any K2∈S+p, then K1∈S+p.
We now state the following multiobjective semidefinite programming with set-valued functionsMSDP to be studied in the present paper: MSDPminFx,s.t.Gx∩-S+p≠∅,0∈Hx,x∈X0,where F:Rn→2Rm, G:Rn→2Sp, and H:Rn→2Rq are set-valued functions with nonempty value, and X0⊂Rn, X0≠∅. We use (8)X=x∈X0:Gx∩-S+p≠∅,0∈Hxto represent the feasible set of MSDP and assume that X is nonempty. Let (9)FX=⋃x∈XFx.Definition 2.
A pointx¯∈X is said to be an ε-properly efficient solution of MSDP, if (10)Fx¯∩ε-PMinFX,R+m≠∅.An ordered pair (x¯,y¯) is said to be an ε-properly efficient pair of MSDP, if (11)y¯∈Fx¯∩ε-PMinFX,R+m.
The following assumptions, (H1) and (H2), forMSDP will be imposed in the rest of this paper:(H1)
((H1.1)–(H1.3)):
(H1.1)
clcone(F(X0)+R+m) is a convex set(H1.2)
∀λ∈(0,1), x1,x2∈X0: (12)λGx1+1-λGx2⊂GX0+S+P(H1.3)
∀λ∈(0,1), x1,x2∈X0: (13)λHx1+1-λHx2⊂HX0(H2)
(Slater constraint qualification): for anyΩ∈S+p and ξ∈Rq, with (Ω,ξ)≠(0,0), there exists x0∈X0, such that (14)Ω·Gx0+ξ·Hx0∩R-≠∅,whereR-={r∈R:r<0}Remark 3.
F is said to be generalized R+m-subconvexlike on X0, if assumption (H1.1) is true [1]. G is said to be S+p-convexlike on X0, if assumption (H1.2) is true [2].


## 3. Scalarization Theorems

The following scalar minimization problem ofMSDP is discussed in this section. Puminu·Fx=u·y:y∈Fx,u∈Rm∖0,s.t.x∈X.Definition 4.
An ordered pair(x¯,y¯) is said to be an u·ε-optimal pair of Pu, if there exist x¯∈X and y¯∈F(x¯), such that (15)u·y¯≤u·y+u·ε,∀y∈FX.Lemma 5 (see [13]).
LetU be a closed convex pointed cone in Rn and V be a closed convex cone in Rn with V≠∅; then (-U)∩V={0} if and only if U+i∩V+≠∅.Theorem 6.
Letu¯∈intR+m; if (x¯,y¯) is an u·ε-optimal pair of (Pu¯), then (x¯,y¯) is an ε-properly efficient pair of MSDP.Proof.
Taking arbitrarilyy∈clcone(F(X)+R+m+ε-y¯)∩(-R+m), we have that y∈(-R+m) and there exist sequences yn⊂F(X), rn⊂R+m, and λn⊂R+, such that (16)y=limn→∞λnyn+rn+ε-y¯.Therefore, we obtain(17)u¯·y=limn→∞λnu¯·yn+u¯·rn+u¯·ε-u¯·y¯.Since (x¯,y¯) is an u·ε-optimal pair of (Pu¯), by Definition 4, we have (18)u¯·y¯≤u¯·yn+u¯·ε.From rn⊂R+m and u¯∈intR+m, we obtain u¯·rn≥0. Therefore, the right side of (17) is nonnegative; sequentially u¯·y≥0. On the other hand, from y∈(-R+m), we have also u¯·y≤0. Thus u¯·y=0. Notice that u¯∈intR+m, and we obtain y=0. Then (19)clconeFX+R+m+ε-y¯∩-R+m=0.The proof is complete by Definition 2 and (5).Theorem 7.
If(x¯,y¯) is an ε-properly efficient pair of MSDP, and F+ε-y¯ is generalized R+m-subconvexlike on X, then there exists u¯∈intR+m, such that (x¯,y¯) is an u¯·ε-optimal pair of (Pu¯).Proof.
If(x¯,y¯) is an ε-properly efficient pair of MSDP, by Definition 2 and (5), we have (20)clconeFX+R+m+ε-y¯∩-R+m=0.Since F+ε-y¯ is generalized R+m-subconvexlike on X, from Remark 3, we have that clcone(F(X)+R+m+ε-y¯) is convex. By Lemma 5, we obtain (21)intR+m∩clconeFX+R+m+ε-y¯+≠∅;hence there exists u¯∈intR+m∩(clcone(F(X)+R+m+ε-y¯))+. Since F(X)+ε-y¯⊂clcone(F(X)+ε-y¯+R+m), we obtain (22)u¯·y+ε-y¯≥0,∀y∈FX.By Definition 4, (x¯,y¯) is an u¯·ε-optimal pair of (Pu¯).


## 4.ε-Lagrange Multiplier Theorems

The mapT:Sp→Rm is defined as TK=(T1·K,…,Tm·K)T, where K∈Sp, Ti∈Sp(i=1,…,m). It is easy to prove that T, denoted by T=(T1,…,Tm)T, is a linear operator from Sp to Rm. A linear operator T is said to be nonnegative, denoted by T≥0, if Ti∈S+p(i=1,…,m). Let L denote the set of all linear operators T, and let L+ denote the set of all nonnegative linear operators; obviously, L+⊂L.Remark 8.
Let a linear operatorT∈L+, and let a matrix A≤0; by Proposition 1(i), we know that the vector TA≤0.The scalar valued Lagrangian function ofMSDP is defined as (23)lx,ϕ,Ω,ξ=ϕ·Fx+Ω·Gx+ξ·Hx,where x∈X0, ϕ∈R+m, Ω∈S+p, and ξ∈Rq. The vector valued Lagrangian function of MSDP is defined as (24)Lx,T,M=Fx+TGx+MHx,where x∈X0, T∈L+, and M∈Rm×q.Now we consider the following two unconstrained optimization problems:SPϕ,Ω,ξminx∈X0lx,ϕ,Ω,ξ,SMPT,Mminx∈X0Lx,T,M.Lemma 9 (see [1]).
LetA⊂X0; then clcone(F(A)+intR+m)=clcone(F(A)+R+m).Lemma 10.
LetA⊂X0; then F(A)∩(-intR+m)=∅ if and only if clcone(F(A)+R+m)∩(-intR+m)=∅.Proof.
LetF(A)∩(-intR+m)=∅. If clcone(F(A)+R+m)∩(-intR+m)≠∅, from Lemma 9, we have (25)clconeFA+intR+m∩-intR+m≠∅.It follows that there exist y¯∈clcone(F(A)+intR+m)∩(-intR+m) and sequences yn∈cone(F(A)+intR+m), such that limn→∞yn=y¯. Thus there exist αn∈R+, yn′∈F(A), and rn∈intR+m, such that(26)yn=αnyn′+rn.In view of y¯∈-intR+m and limn→∞yn=y¯, there exists a natural number N, such that n>N, yn∈-intR+m. It is easy to check that if n>N, then αn≠0. From (26) we have (27)yn′=ynαn-rn∈-intR+m-intR+m⊂-intR+m,n>N.This implies (28)yn′∈FA∩-intR+m,n>N.This is a contradiction to the assumption.
Conversely, the sufficient condition can be proved by consideringF(A)⊂clcone(F(A)+R+m).Lemma 11 (alternative theorem).
Suppose that the set-valued functionsF, G, and H satisfy assumptions (H1) and (H2) on X0; then exactly one of the following statements is true:(i)
There existsx∈X0, such that F(x)∩(-intR+m)≠∅, G(x)∩(-S+p)≠∅, and 0∈Hx.(ii)
There exists(ϕ,Ω,ξ)∈(R+m×S+p×Rq), ϕ≠0, such that (29)ϕ·Fx+Ω·Gx+ξ·Hx≥0,∀x∈X0.Proof.
Obviously, (i) and (ii) cannot hold simultaneously. Otherwise, there existx¯∈X0, y¯∈F(x¯), Z¯∈G(x¯), and w¯∈H(x¯), such that (30)y¯∈-intR+m,Z¯∈-S+p,w¯=0.By Proposition 1, there exists (ϕ,Ω,ξ)∈R+m×S+p×Rq, ϕ≠0, such that (31)ϕ·y¯+Ω·Z¯+ξ·w¯<0,and this contradicts (ii). Now we show that if (i) is not true, then (ii) is true. Let(32)S=u,V,w∈Rm×Sp×Rq∣∃x∈X0,s.t.u∈clconeFx+R+m+intR+m,V∈Gx+S+p,w∈Hx.It is easy to show that S is a convex set and intS≠∅. If (i) is not true, then (0,0,0)∉S. Otherwise, assume that (0,0,0)∈S; from (32) there exists x∈X0, such that (33)clconeFx+R+m∩-intR+m≠∅,Gx∩-S+p≠∅,0∈Hx.By Lemma 10, there exists x∈X0, such that (34)Fx∩-intR+m≠∅,Gx∩-S+p≠∅,0∈Hx,which is inconsistent with the assumed condition. Hence by the separation theorem of convex sets, there exists (ϕ,Ω,ξ)∈(Rm×Sp×Rq)∖{0,0,0}, such that (35)ϕ·u+Ω·V+ξ·w≥0,∀u,v,w∈clS.Thus for any x∈X0, y∈F(x), Z∈G(x), w∈H(x), r∈intR+m, s∈S+p, ε>0, and η>0, we have(36)ϕ·y+εr+Ω·Z+ηs+ξ·w≥0.Letting ε→+∞ in (36), we obtain (37)ϕ·r≥0,∀r∈intR+m;hence ϕ∈R+m. Letting η→+∞ in (36), we have also Ω∈S+p. Now letting ε→0 and η→0 in (36), we have (38)ϕ·y+Ω·Z+ξ·w≥0,∀x∈X0,y∈Fx,Z∈Gx,w∈Hx,which implies that there exists (ϕ,Ω,ξ)∈(R+m×S+p×Rq)∖{0,0,0}, such that(39)ϕ·Fx+Ω·Gx+ξ·Hx≥0,∀x∈X0.
At last we findϕ≠0. Contrarily, taking ϕ=0 in (39), we obtain (40)Ω·Gx+ξ·Hx≥0,∀x∈X0,and this contradicts (H2). Therefore, (ii) holds.Theorem 12.
Letx¯∈X, y¯∈F(x¯); suppose that F+ε-y¯, G, and H satisfy (H1) and (H2) on X0 and F+ε-y¯ is generalized R+m-subconvexlike on X. If (x¯,y¯) is an ε-properly efficient pair of MSDP, then there exists (ϕ,Ω,ξ)∈intR+m×S+p×Rq, such that (41)ϕ·y¯≤lx,ϕ,Ω,ξ+ϕ·ε,∀x∈X0.Proof.
If(x¯,y¯) is an ε-properly efficient pair of MSDP and F+ε-y¯ is generalized R+m-subconvexlike on X, by Theorem 7 and Definition 4, there exists u¯∈intR+m, such that (42)u¯·y¯≤u¯·Fx+u¯·ε,∀x∈X.Therefore the following conditions are not true: (43)u¯·Fx+ε-y¯∩-intR+≠∅,Gx∩-S+p≠∅,0∈Hx,∀x∈X0.Since (F+ε-y¯) is generalized R+m-subconvexlike on X0, we can easily prove that u¯·(F+ε-y¯) is generalized R+-subconvexlike on X0. Hence u¯·(F+ε-y¯), G, and H satisfy (H1) and (H2) on X0. From Lemma 11, there exists (α¯,Ω,ξ)∈(R+×S+p×Rq), α¯≠0, such that(44)α¯u¯·Fx+ε-y¯+Ω·Gx+ξ·Hx≥0,∀x∈X0.Let ϕ=α¯u¯; we have ϕ∈intR+m, and (44) can be written as (45)ϕ·y¯≤lx,ϕ,Ω,ξ+ϕ·ε,∀x∈X0.This completes the proof.Theorem 13.
Letx¯∈X and y¯∈F(x¯). If there exists (ϕ,Ω,ξ)∈intR+m×S+p×Rq, such that(46)ϕ·y¯≤lx,ϕ,Ω,ξ+ϕ·ε,∀x∈X0,then (x¯,y¯) is an ε-properly efficient pair of MSDP.Proof.
Ifx∈X, there exist Zx∈G(x) and wx∈H(x), such that Zx∈-S+p and wx=0; then we have Ω·Zx≤0, ξ·wx=0, and(47)ϕ·Fx+Ω·Zx+ξ·wx≤ϕ·Fx+ϕ·ε,∀x∈X.By (46) and (47), we obtain (48)ϕ·y¯≤ϕ·Fx+ϕ·ε,∀x∈X.From ϕ∈intR+m and Definition 4, we know that (x¯,y¯) is an ϕ·ε-optimal pair of (Pϕ). Therefore, (x¯,y¯) is an ε-properly efficient pair of MSDP by Theorem 6.Theorem 14.
Letx¯∈X, y¯∈F(x¯), and 0∈G(x¯), and let F+ε-y¯, G, and H satisfy (H1) and (H2) on X0; F+ε-y¯ is also generalized R+m-subconvexlike on X. If (x¯,y¯) is an ε-properly efficient pair of MSDP, then there exist T∈L+ and M∈Rm×q, such that (x¯,y¯) is an ε-properly efficient pair of SMPT,M.Proof.
If(x¯,y¯) is an ε-properly efficient pair of MSDP, by Theorem 12, there exists (ϕ,Ω,ξ)∈intR+m×S+p×Rq, such that(49)ϕ·y¯≤ϕ·Fx+Ω·Gx+ξ·Hx+ϕ·ε,∀x∈X0.Taking c∈intR+m satisfying ϕ·c=1, we can define a map T:Sp→Rm as (50)TA=Ω·Ac=c1Ω·A,…,cmΩ·AT.Then T≥0, TG(x)=(Ω·G(x))c. Let M=cξT∈Rm×q, MH(x)=(ξ·H(x))c. From (49) and ϕ·c=1, we have (51)ϕ·y¯≤ϕ·Fx+TGx+MHx+ε,∀x∈X0.Since x¯∈X, 0∈G(x¯), and 0∈H(x¯), y¯∈F(x¯)+TG(x¯)+MH(x¯). By Definition 4, (x¯,y¯) is a ϕ·ε-optimal pair of the scalar problem: (52)minx∈X0ϕ·Lx,T,M.From ϕ∈intR+m, we obtain that (x¯,y¯) is an ε-properly efficient pair of SMPT,M by Theorem 6.Theorem 15.
Letx¯∈X and y¯∈F(x¯); if there exist T∈L+ and M∈Rm×q, such that (x¯,y¯) is an ε-properly efficient pair of SMPT,M, then (x¯,y¯) is an ε-properly efficient pair of MSDP.Proof.
If there existT∈L+ and M∈Rm×q, such that (x¯,y¯) is an ε-properly efficient pair of SMPT,M, by Definition 2 and (5), we have(53)clconeLX0,T,M+ε-y¯+R+m∩-R+m=0.Let Z¯∈G(x¯)∩(-S+p); we have -TZ¯∈R+m and R+m-TZ¯⊂R+m+R+m⊂R+m. Hence, we obtain (54)R+m⊂TZ¯+R+m⊂TGX+R+m;then(55)FX+ε-y¯+R+m⊂FX+TGX+M0+R+m+ε-y¯⊂LX,T,M+ε-y¯+R+m.This implies (56)0∈clconeFX+ε-y¯+R+m⊂clconeLX0,T,M+ε-y¯+R+m.By the above expression and (53), we have (57)clconeFX+ε-y¯+R+m∩-R+m=0.Therefore, (x¯,y¯) is an ε-properly efficient pair of MSDP by Definition 2 and (5).


## 5.ε-Proper Saddle Point Theorem

In this section, first we give the concept ofε-proper saddle point of the set-valued Lagrangian function; then we establish an ε-proper saddle point theorem.Definition 16.
Letx¯∈X0, T¯∈L+, and M¯∈Rm×q, and an ordered group (x¯,T¯,M¯) is said to be an ε-proper saddle point of Lagrangian function L(x,T,M) if (58)Lx¯,T¯,M¯∩ε-PMinx∈X0Lx,T¯,M¯∩ε-PMaxT≥0,MLx¯,T,M≠∅.Theorem 17.
Letx¯∈X0, T¯∈L+, and M¯∈Rm×q. If (x¯,T¯,M¯) is an ε-proper saddle point of Lagrangian function L(x,T,M), then there exist y¯∈F(x¯) and Z¯∈G(x¯), such that the following conditions are satisfied:(i)
y¯+T¯Z¯∈ε-PMinx∈X0Lx,T¯,M¯.(ii)
G(x¯)⊂-S+p, ω¯=0.(iii)
-T¯Z¯∈R+m∖(ε+intR+m).Proof.
By Definition16, there exist y¯∈F(x¯), Z¯∈G(x¯), and w¯∈H(x¯), such that(59)y¯+T¯Z¯+M¯w¯∈ε-PMinx∈X0Lx,T¯,M¯,(60)y¯+T¯Z¯+M¯w¯∈ε-PMaxT≥0,MLx¯,T,M.From (60) and (6), we have (61)clconeFx¯+TGx¯+MHx¯-y¯-T¯Z¯-M¯w¯-ε-R+m∩intR+m=∅,∀T≥0,M∈Rm×q.That is,(62)Fx¯+TGx¯+MHx¯-y¯-T¯Z¯-M¯w¯-ε-R+m∩intR+m=∅,∀T≥0,M∈Rm×q.Taking M=M¯ in (62), since y¯∈F(x¯), Z¯∈G(x¯), and w¯∈H(x¯), we have (63)TZ¯-T¯Z¯-ε-R+m∩intR+m=∅,∀T≥0.From the above expression and 0∈R+m, we obtain(64)TZ¯-T¯Z¯-ε∉intR+m,∀T≥0.We assert Z¯∈-S+p. Contrarily, by Proposition 1(ii), there exists Ω∈S+p∖{0}, such that Ω·Z¯>0. Letting c∈intR+m, we define a map T:Sp→Rm as (65)TZ=Ω·ZΩ·Z¯c+ε+T¯Z.It is easy to check that T∈L+, and (66)TZ¯-T¯Z¯-ε=c∈intR+m.This contradicts (64), so Z¯∈-S+p and -T¯Z¯∈R+m. Taking T=0 in (64), we obtain -T¯Z¯∉ε+intR+m. Therefore, condition (iii) holds.
Now, we assertG(x¯)⊂-S+p. Contrarily, there exists Z~∈G(x¯), but Z~∉-S+p. By Proposition 1(ii), there exists Ω∈S+p∖{0}, such that Ω·Z~>0. Letting c¯∈intR+m, the map T~:Sp→Rm is defined as (67)T~Z=Ω·ZΩ·Z~c¯+ε.Obviously, by -T¯Z¯∈R+m, we obtain(68)T~Z~-T¯Z¯-ε∈c¯+R+m⊂intR+m+R+m⊂intR+m.On the other hand, taking T=T~ in (64), we have (69)T~Z~-T¯Z¯-ε∉intR+m,and this contradicts (68), so G(x¯)⊂-S+p.
TakingT=T¯ in (62), since y¯∈F(x¯), Z¯∈G(x¯), and 0∈R+m, we have(70)MHx¯-M¯w¯-ε∩intR+m=∅,∀M∈Rm×q.From w¯∈H(x¯), we obtain(71)Mw¯-M¯w¯-ε∉intR+m,∀M∈Rm×q.Suppose that w¯≠0 and ∃ξ∈Rq, such that ξ·w¯>0, letting c¯∈intR+m. The map M~∈Rm×q is defined as (72)M~w=ξ·wξ·w¯c¯+ε+M¯w.We obtain (73)M~w¯-M¯w¯-ε=c¯∈intR+m.This contradicts (71); hence w¯=0. Therefore, condition (ii) holds. From the fact that w¯=0 and (59), condition (i) holds.Theorem 18.
If an order group(x¯,T¯,M¯) is an ε-proper saddle point of Lagrangian function L(x,T,M) and 0∈G(x¯), then there exist y¯∈F(x¯) and Z¯∈G(x¯), such that (x¯,y¯) is an ε′-properly efficient pair of MSDP, where ε′=ε-T¯Z¯.Proof.
By Theorem17(i), there exist y¯∈F(x¯) and Z¯∈G(x¯), such that y¯+T¯Z¯∈ε-PMinx∈X0L(x,T¯,M¯); from (5), we have (74)clconeLX0,T,M+R+m-y¯+ε-T¯Z¯∩-R+m=0,∀T≥0,M∈Rm×q.By Theorem 17(iii), -T¯Z¯∈R+m; hence ε′=ε-T¯Z¯∈R+m. On the other hand, from the fact that 0∈G(x¯), we obtain y¯∈L(X0,T,M). Applying Theorem 15 yields the result.



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