Stability of efficient solutions for semi-infinite vector optimization problems
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1 Stability of efficient solutions for semi-infinite vector optimization problems Z. Y. Peng, J. T. Zhou February 6, 2016 Abstract This paper is devoted to the study of the stability of efficient solutions for semi-infinite vector optimization problems (SIO). We first obtain the closeness, Berge-lower semicontinuity and Painlevé-Kuratowski convergence of constraint set mapping. Then, under the assumption of continuous convergence of the objective function, we establish some sufficient conditions of the upper Painlevé-Kuratowski stability of efficient solution mappings to the (SIO). Examples are also given to illustrate the results. Keywords: upper Painlevé-Kuratowski stability, semi-infinite vector optimization, perturbation, efficient solution, continuous convergence Mathematics Subject Classification (2000) 49K40 90C29 90C31 1 Introduction Let X be a Hausdorff topological space, Y and Z be real Banach spaces with norms denoted by. Let D (resp. K) be closed, convex and pointed cone in Y (resp. Z) with nonempty interior intd (resp. intk). Let A be a nonempty compact convex subset of X. We denote the space U[A, Y ] be the set of all vector-valued functions from A to Y. Let T be a nonempty compact subset of a Hausdorff topological space, denote USC[A T, Z] be the set of all K-upper semicontinous vector-valued functions with respect to the first variable, where the metric of the function h USC[A T, Z] is defined as ρ(h 1, h 2 ) := min{ sup h 1 (x, t) h 2 (x, t), 1 x A,t T 5 }. Consider parametric semi-infinite vector optimization problems (SIO for brevity), or generalized parametric vector optimization problems, under functional perturbations of both College of Mathematices and Statistices, Chongqing JiaoTong University, Chongqing , China. pengzaiyun@126.com. College of Civil Engineering, Chongqing JiaoTong University, Chongqing , P.R. China. jt-zhou@163.com. 1
2 objective function and constraint set on the parameter space G 0 := U[A, Y ] USC[A T, Z] formulated as follows: for every a double of parameter p := (f, h) G 0, we have the semi-infinite vector optimization problem { D min f(x) (SIO) s.t. x M(h) where M(h) := {x A : h(x, t) K 0, t T }. (1.2) (1.1) We know that the semi-infinite optimization problem plays a very important role in optimization theory and applications. The models of semi-infinite optimization problems cover, e.g., optimal control, approximation theory, popular semi-definite programming and numerous engineering problems, etc. The semi-infinite optimization problem and its wide range of applications have been an active research area in mathematical programming in recent years. Many paper are published on theory, methods and applications for semiinfinite optimization problem and its extensions; examples of fresh literatures include, the existence results in [4, 5, 15], the optimality and/or characterizations of the solution set in [14, 22, 23], the stability results of solution mappings in [6, 7, 11, 12], etc. Since the semi-infinite vector optimization problem has been acting more and more important role in optimization theory and applications, some new methods and skills will appear gradually. On the other hand, the stability of solution mappings under certain perturbations, either of the feasible set or the objective function, has been of great interest in the optimization theory and related field. There are some stability results for vector optimization problems and related issues with a sequence of sets converging in the sense of Painlevé-Kuratowski. Examples of fresh literatures include, for vector optimization problems, we can see Attouch and Riahi [2], Huang [13], Lucchetti and Miglierina [17], Lalitha and Chatterjee [18]; for vector equilibrium problem, we can refer to Durea [9], Fang and Li [10], Zhao et al. [27], Peng and Yang [25], etc. However, to the best of our knowledge, the Painlevé-Kuratowski stability of efficient solutions set for semi-infinite vector optimization problems has not been found. Thus, it is interesting to investigate the Painlevé-Kuratowski convergence of the efficient solution mapping for semi-infinite vector optimization problems! The rest of the paper is organized as follows. In Sect. 2, we recall some basic definitions and preliminaries from set-valued analysis and vector optimization, which will be used in next section. The main result is presented in Sect.3. In Sect. 3, we first establish the closeness, Berge-lower semicontinuity and Painlevé-Kuratowski convergence of constraint set mapping. Then, under the assumption of continuous convergence of the objective function, we obtain some sufficient conditions of the upper Painlevé-Kuratowski stability of efficient solution mappings to the semi-infinite vector optimization problem (SIO). We also give some examples to illustrate our main results. 2
3 2 Preliminaries In this section, we give some basic definitions and preliminary results which are needed in the sequel. Throughout this paper, unless specified otherwise, X, Y, Z, D, K and T are as mentioned above. Let D (resp. K) be closed, convex and pointed cone in Y (resp. Z) with nonempty interior intd (resp. intk). The vector ordering relations in Y associated with the cone D are defined as follows: for any y 1, y 2 Y, y 1 D y 2 y 2 y 1 D; y 1 D y 2 y 2 y 1 / D; y 1 D y 2 y 2 y 1 D \ {0}; y 1 D y 2 y 2 y 1 / D \ {0}; y 1 < D y 2 y 2 y 1 intd; y 1 Dy 2 y 2 y 1 / intd, and the vector ordering relations in Z associated with the cone K are similar as above. For the semi-infinite vector optimization problem (1.1), we call the set-valued mapping M : USC[A T, Z] A (given in (1.2)) the constraint set mapping of (SIO). A vector x M(h) is said to be a strictly efficient solution of (SIO), if and only if for any y M(h), y x, f(y) f(x) / D. A vector x M(h) is said to be a efficient solution of (SIO), if {f(x)} = (f(x) D) f(m(h)). A vector x M(h) is said to be a weakly efficient solution of (SIO), if (f(x) intd) f(m(h)) =. For each p = (f, h) G 0, let SSol(M(h), f), ESol(M(h), f) and WESol(M(h), f) denote the sets of strictly efficient solutions, efficient solutions and the set of weakly efficient solutions of (SIO), respectively. Now, we give Example 2.1 to illustrate efficient solutions of (SIO) in Banach space. Example 2.1 Let X = Y = l 1 = {(x 1,, x n, ) : n=1 x n < }, A = cl co{{ en n } n=1 {0 X }}, where e 1 = (1, 0, 0, ), e 2 = (0, 1, 0, ), e 1 = (0, 0, 1, 0, ),. Let Z = R 2, K = R 2 +, T = [0, 1] R, D = {x = (x 1,, x n, ) l 1 : x n 0, n = 1, 2, }. Then, we can observe that A is a compact convex set in X. We consider h : A T Z, f : A Y by h(x, t) = ( y n x n + t 2 +1, x n + 2) 1, x = (x1,, x n, ), y = (y 1,, y n, ) A, n=1 n=1 f(x) = x 3, x = (x 1,, x n, ) A. From a direct computation, we can get that M(h) = A and ESol(M(h), f) = {(0, 0,, 0, )}. 3
4 Definition 2.1 Let A be a nonempty convex subset of X. Let f be a mapping from A to Y. We say that f is D-convex on A, if for any x 1, x 2 A and λ [0, 1], f(λx 1 + (1 λ)x 2 ) λf(x 1 ) + (1 λ)f(x 2 ) D. Definition 2.2 Let A be a nonempty convex subset of X. Let f be a mapping from A to Y. We say that (i) f is properly quasi D-convex on A, if for any x 1, x 2 A and λ [0, 1], either f(λx 1 + (1 λ)x 2 ) f(x 1 ) D or f(λx 1 + (1 λ)x 2 ) f(x 2 ) D. (ii) f is semistrictly (strictly) properly quasi D-convex on A, if for any x 1, x 2 A with f(x 1 ) f(x 2 ) (x 1 x 2 ) and λ (0, 1), either f(λx 1 + (1 λ)x 2 ) f(x 1 ) intd or f(λx 1 + (1 λ)x 2 ) f(x 2 ) intd. In [21], Luc gave the following definition of C-upper semicontinuity. Definition 2.3 Let E be a nonempty subset of X. Let f be a mapping from E to Y. f is said to be D-upper semicontinuous at x 0 E, if for any neighborhood W of 0 in Y, there is a neighborhood U of x 0 such that for each x U E, f(x) f(x 0 ) + W D. Definition 2.4 Let E be a nonempty convex subset of X. Let f be a mapping from E to Z. We say that f is K-quasiconvex on E, if for any z Z, x 1, x 2 E with x 1 x 2 and λ [0, 1], f(x 1 ), f(x 2 ) z K implies f(λx 1 + (1 λ)x 2 ) z K. Remark 2.1 We call f is K-quasiconcave on E if f is K-quasiconvex on E. Definition 2.5 [1, 3] Let X and Y be topological vector spaces, F : X 2 Y be a setvalued mapping. (i) F is said to be Berge-lower semicontinuous at x 0 X, if for any open set V with F (x 0 ) V, there exists a neighborhood U of x 0 in X such that F (x) V for all x U; (ii) F is said to be Berge-lower semicontinuous on X, iff it is Berge-lower semicontinuous at each x X; (iii) F is closed if Graph(F ) is a closed set in X Y. Now, we recall the well known notion of set-convergence, namely Painlevé-Kuratowski set-convergence. A sequence of sets {B n X : n N} is said to converge in the sense of Painlevé- P.K. Kuratowski (P.K.) to B (denoted as B n B) if lim sup B n B lim inf B n 4
5 with lim inf B n := {x X (x n ), x n B n, n N, x n x}, lim sup B n := {x X (n k ), (x nk ), x nk B nk, k N, x nk x}. When lim sup B n B holds, the relation is referred as upper Painlevé-Kuratowski convergence (u.p.k, for briefness). When K lim inf K n holds, the relation is referred as lower Painlevé-Kuratowski convergence (l.p.k, for briefness). A set-valued mapping ψ : X 2 Y is said to be Painlevé-Kuratowski convergent at x domψ := {x X ψ(x) } if and only if for any sequence x n in domψ converging to x one has lim sup ψ(x n ) ψ(x) lim inf ψ(x n). Definition 2.6 [20, 26] Let f n, f : X Y be vector-valued mappings and A X. We c.c say that f n continuous converges to f (denoted as f n f), iff for every x A and for every sequence {x n } in A, f n (x n ) f(x) for all x n x. In [1], Aubin et al. also gave the following properties for Berge-lower semicontinuous. Lemma 2.1 Let X and Y be topological vector spaces, F : X 2 Y be a set-valued mapping. F is Berge-lower semicontinuous at x 0 X if and only if for any sequence {x α } X with x α x 0 and any y 0 F (x 0 ), there exists y α F (x α ) such that y α y 0. Lemma 2.2 [3] Let Y be topological vector spaces. For each zero neighborhood U in Y, there exist zero neighborhood U 1 and U 2 in Y such that U 1 + U 2 U. 3 Main results In this section, we aim to establish the Painlevé-Kuratowski stability of efficient solution mappings to the semi-infinite vector optimization problem. We first give sufficient conditions for closeness, Berge-lower semicontinuity and Painlevé- Kuratowski convergence of the constraint set mapping M : USC[A T, Z] A as follows. Theorem 3.1 Let p := (f, h) be any given point in G 0. (i) for each t T, x h(x, t) is K-quasiconcave on A, then M( ) is convex at h. ρ (ii) If h n (, t) h(, t) for any t T, then the constraint set mapping M( ) is closed at h; 5
6 Proof (i) Getting x 1, x 2 M(h), one has and h(x 1, t) K 0, t T h(x 2, t) K 0, t T. Then, for each t [0, 1], tx 1 + (1 t)x 2 A as A is convex. It follows from the K- quasiconcavity of h(, t) on A and equations above that h(tx 1 + (1 t)x 2, t) K, t T. This means tx 1 + (1 t)x 2 M(h), i.e., M(h) is a convex set. (ii) Let {(h n, x n )} Graph(M), h n Since x n M(h n ), for every n N, ρ h, x n x. Then x A as A is compact. h n (x n, t) K 0, t T. (3.1) Now, we verify that x M(h). If not, there exists t T such that h(x, t ) K. By the openness of Y \ K, there exists a open neighborhood U of 0 Y in Y such that h(x, t ) + U Y \ K. (3.2) From Lemma 2.2, for above U, there exist three neighborhoods U 1 and U 2 of 0 Y in Y such that U 1 + U 2 U. (3.3) By the K-upper semicontinuity of h(, t ) at x for above U 1, there exists a neighborhood U(x ) of x, such that h(x, t ) h(x, t ) + U 1 K, x U(x ) A. Since x n x, there exists n 1 N such that for any n n 1, one has It follows that As h n x n U(x 0 ) A. h(x n, t ) h(x, t ) + U 1 K. (3.4) ρ h, there exists n 2 N such that for any n n 2, From (3.2)-(3.5), for n max{n 1, n 2, n 3 }, we have h n (x n, t ) h(x n, t ) U 2. (3.5) h n (x n, t ) = h n (x n, t ) h(x n, t ) + h(x n, t ) U 2 + h(x, t ) + U 1 K K + Y \ K Y \ K, which contradicts (3.1). Then x M(h). This implies that M is closed at h. 6
7 Theorem 3.2 Let p := (f, h) be any given point in G 0, for each t T, x h(x, t) is K-quasiconcave on A, then the constraint set mapping M is Berge-lower semicontinuous at h. Proof Let W be an open convex set such that W M(h). Since M(h), there exists an element x M(h) satisfying h( x, t) K 0, t T. Taking any x 0 W M(h), there exists r (0, 1] such that x r := x 0 + r( x x 0 ) W, then x r W M(h) as M(h) is convex by Theorem 3.1. Since x 0 M(h), we have h(x 0, t) K 0, t T, and x r := x 0 + r( x x 0 ) A by the convexity of A. It follows from two equations above and the K-quasiconcavity of h(, t) that This means that h(x r, t) K. g(x r, t) K 0, t T. (3.6) For h USC[A T, Z] satisfies ρ( h, h)) < δ 2 (δ > 0), we clarify that x r M( h). On the contrary, there exists t T such that h(x r, t) K 0. By the openness of Y \ K, there exists a zero neighborhood U in Y such that h(x r, t) + U Y \ K. (3.7) It follows from ρ( h, h) < δ 2 that for above U, h(x r, t) h(x r, t) U. (3.8) Combining (3.7)-(3.8), we otain h(x r, t) = h(x r, t) h(x r, t) + h(x r, t) U + h(x r, t) Y \ K. This contradicts to (3.6). Then we have x r M( h) and W M( h). This means M is Berge-lower semicontinuous at p. The proof is complete. 7
8 Theorem 3.3 Let p := (f, h) be any given point in G 0. Suppose that (i) for each t T, x h(x, t) is K-quasiconcave on A, (ii) If h n (, t) Then ρ h(, t) for any t T. M(h n ) P.K. M(h). Proof. Take any x lim sup n M(h n ). Then, there exists a subsequence {x nk } M(h nk ) such that x nk x. By Theorem 3.1 (ii), M is closed at h, then we get x M(h). Hence, we have lim sup n M(h n ) M(h). Next, we prove M(h) lim inf n M(h n ). Take any x M(h), then by Theorem 3.2 (M Berge-lower semicontinuous), there exists x n M(h n ) such that x n x. From the definition of lower Painlevé-Kuratowski convergence, we have x lim inf n M(h n ), which means that M(h) lim inf n M(h n ) as x M(h) is arbitrary. This completes the proof. Now, we establish the upper Painlevé-Kuratowski stability of solution mappings for the semi-infinite vector optimization problem (SIO). Theorem 3.4 Let p := (f, h) be any given point in G 0. Assume that the conditions (i) c.c and (ii) of Theorem 3.3 are satisfied and f n f. Then lim sup WESol(M(h n ), f n ) WESol(M(h), f). Proof. Take any x limsup n WESol(M(h n ), f n ). Then, there exists a subsequence {x nk } in WESol(M(h nk )), f nk ) such that x nk x. From Theorem 3.3, we conclude x M(h). For any y M(h), there exists y n M(h n ) such that y nk y since M(h n ) P.K. M(h). It follows from {x nk } WESol(M(h nk ), f nk ) and y nk M(h nk ), that Since f n f nk (y nk ) f nk (x nk ) / intd. (3.9) c.c f, there exists N N for any n k > N f nk (y nk ) f(y) and f nk (x nk ) f(x). (3.10) Together (3.9) (3.10) and the closedness of Y \ -intd, we have f(y) f(x) / intd. As y M(h) is arbitrary, we conclude that x WESol(M(h), f). Thus, lim sup WESol(M(h n ), f n ) WESol(M(h), f). This completes the proof. 8
9 Lemma 3.5 Let p := (f, h) be any given point in G 0. (i) If x f(x) is semistrictly proper quasi-d-convex on A. Then ESol(M(h), f) = WESol(M(h), f); (ii) If x f(x) is strictly proper quasi-d-convex on A. Then SSol(M(h), f) = ESol(M(h), f). Proof. (i) By the definition, ESol(M(h), f) WESol(M(h), f). We need only to prove WESol(M(h), f) ESol(M(h), f). Suppose to the contrary, there exists x 0 WESol(M(h), f) such that x 0 / ESol(M(h), f). Hence, there exists y 0 M(h) such that f(y 0 ) f(x 0 ) D \ {0}. (3.11) It follows from semistrictly proper quasi-d-convexity of f( ) on A and (3.11), for every λ (0, 1) that λx 0 + (1 λ)y 0 A as A is convex, and f(λx 0 + (1 λ)y 0 ) f(x 0 ) intd, which contradicts x 0 WSol(M(h), f). Then we get WESol(M(h), f) ESol(M(h), f). (ii) From the definition of strictly proper quasi-d-convexity, by using the same method above, with appropriate modification, we can get the result. The proof is complete. Theorem 3.6 Let p := (f, h) be any given point in G 0. Assume that the conditions (i) c.c and (ii) of Theorem 3.3 are satisfied, f n f and x f(x) is semistrictly proper quasi-d-convex on A. Then lim sup ESol(M(h n ), f n ) ESol(M(h), f). Proof. Combing Theorem 3.4 and Lemma 3.5, we can get the result easily. Now, we give an example to illustrate that Theorem 3.6 is applicable. Example 3.1 Let X := R 2, Z := R, Y := R 2, T = [0, 1] R and A := {(x 1, x 2 ) R 2 : 1 x 1 1, 1 x 2 1}, K := R +, D := R 2 +. Consider h, h n : A T Z, f n = f : A Z, which are given by: f(x) := ( f 1 (x), f 2 (x) ), x = (x 1, x 2 ) A, where f 1 (x) := 1 3 x 1 1 4, f 2 (x) := 1 5 x ; 9
10 h(x, t) := 1 2 x , h n(x, y) := 1 2 x t 12n 2, x = (x 1, x 2 ), y = (y 1, y 2 ) A. Let p := (f, h), p n := (f n, h n ) G 0. It is easy to verify that all conditions of Theorem 3.6 are satisfied. By a direct computation, we get { M(h) = (x 1, x 2 ) R 2 : 2 } 5 x 1 1, 1 x 2 1, { } ( 2 ESol(M(h), f) = 5, x ) 2 R 2 1 x 2 1 ; { M(h n ) = (x 1, x 2 ) R 2 : 2 5 t } 6n x 2 1 1, 1 x 2 1, { ( 2 ESol(M(h n ), f n ) = 5 t } 6n, x ) 2 2 R 2 1 x 2 1. Obviously, lim sup ESol(M(h n ), f n ) ESol(M(h), f). Thus, Theorem 3.6 is applicable. Corollary 3.7 Let p := (f, h) be any given point in G 0. Assume that the conditions (i) c.c and (ii) of Theorem 3.3 are satisfied, f n f and x f(x) is strictly proper quasi-dconvex on A. Then SSol(M(h n ), f n ) u.p.k. SSol(M(h), f). Proof. By virtue of Lemma 3.5 and Theorem 3.6, we can get the result. Acknowledgments. The work of the first author was completed during his visit to the Department of Mathematics, University of British Columbia, Kelowna, Canada, to which he is grateful to the hospitality received. His work was partially supported by the National Natural Science Foundation of China ( ), the Basic and Advanced Research Project of Chongqing (2015jcyjA00025) and the China Postdoctoral Science Foundation funded project (2015M580774). The second author was partially supported by the National Outstanding Youth Science Fund Project of China ( ) and the National Natural Science Foundation of China ( ). References [1] Aubin, J. P., Ekeland, I.: Applied Nonlinear Analysis. John Wiley and Sons, New York, (1984). [2] Attouch, H, Riahi, H.: Stability results for Ekeland s ɛ-variational principle and cone extremal solution. Math Oper Res. 18, (1993). 10
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12 [20] López, R.: Variational convergence for vector-valued functions and its applications to convex multiobjective optimization. Math. Methods Oper. Res., 78, 1-34 (2013). [21] Luc, D. T.: Theory of Vector Optimization, Lecture Notes in Econom. and Math. Systems, vol. 319, Springer-Verlag, Berlin, (1989). [22] Long, X. J., Peng, Z. Y., Wang, X. F.: Characterizations of the solution set for nonconvex semi-infinite programming problems. J. Nonl. Convex Anal., (2016), to appear. [23] Mishra, S. K., Jaiswal, M., Le Thi, H. A.: Nonsmooth semi-infinite programming problem using limiting subdifferentials. J. Global Optim., 53, (2012). [24] Tanaka, T.: Generalized quasiconvexities, cone saddle points and minimax theorems for vector valued functions. J. Optim. Theory Appl. 81, (1994). [25] Peng, Z.Y., Yang, X.M.: Painlevé-Kuratowski Convergences of the solution sets for perturbed vector equilibrium problems without monotonicity. Acta Math Appl Sin-E. 30, (2014). [26] Rockafellar, R.T., Wets, R.J.: Variational analysis. Springer-Verlag, Berlin (1998). [27] Zhao, Y., Peng, Z.Y., Yang, X.M.: Painlevé-Kuratowski Convergences of the solution sets for perturbed generalized systems. J. Nonlinear Conv. Anal. 15, (2014). 12
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