ON LOWER LIPSCHITZ CONTINUITY OF MINIMAL POINTS. Ewa M. Bednarczuk

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1 Disussiones Mathematiae Differential Inlusions, Control and Optimization ) ON LOWER LIPSCHITZ CONTINUITY OF MINIMAL POINTS Ewa M. Bednarzuk Systems Researh Institute, PAS Warsaw, Newelska 6, Poland bednarz@ibspan.waw.pl Abstrat In this paper we investigate the lower Lipshitz ontinuity of minimal points of an arbitrary set A depending upon a parameter u. Our results are formulated with the help of the modulus of minimality. The ruial requirement whih allows us to derive suffiient onditions for lower Lipshitz ontinuity of minimal points is that the modulus of minimality is at least linear. The obtained results an be diretly applied to stability analysis of vetor optimization problems. Keywords: minimal points, Lipshitz ontinuity, vetor optimization Mathematis Subjet Classifiation: 90C29, 90C48. 1 Introdution Let Y, ) be a normed spae and let K Y be a losed onvex pointed one in Y. Let A Y be a subset of Y. We say that y A is a minimal point of A with respet to K if y K) A = {y} see [12). By MinA K) we denote the set of all minimal points of A with respet to K. We say that the domination property DP ) holds for A if A MinA K) + K see [12, 15). Let U = U, ) be a normed spae and let Γ : U Y be a set-valued mapping. Define a set-valued mapping M : U Y as follows Mu) = MinΓu) K). The set-valued mapping M is alled the minimal point multifuntion.

2 246 E.M. Bednarzuk In the present paper we give suffiient onditions whih ensure that M is lower Lipshitz ontinuous and/or loally Lipshitz at a given u 0 U. Lipshitz behaviour of solutions to optimization problems is one of entral topis of stability analysis in optimization. For salar optimization it was investigated by many authors, see e.g. [2, 20, 16, 21, 11, 13, 14, 23, 18, 19, 24, 1 and many others. In vetor optimization the results on Lipshitz ontinuity of solutions are not so numerous, and onern some lasses of problems, for linear problems see e.g. [7, 8, 9, for onvex problems see e.g. [6, 10. We say that a multivalued mapping F : U Y is loally Lipshitz at u 0, [2, if there exist a neighbourhood U 0 domf of u 0 and a positive onstant l suh that F u 1 ) F u 2 ) + l u 1 u 2 for u 1, u 2 U. We say that F : U Y is lower Lipshitz ontinuous at u 0 U if there exist a onstant L and a neighbourhood U 0 of u 0 suh that F u 0 ) F u) + L u u 0 for u U 0. F : U Y is upper Lipshitz ontinuous at u 0 U if there exist a onstant L and a neighbourhood U 0 of u 0 suh that F u) F u 0 ) + L u u 0 for u U 0. 2 Modulus of minimality Let Y, ) be a normed spae and let K be a losed onvex and pointed one in Y. By Ba, r) we denote the open ball of entre a and radius r, B0, 1) = B. It was shown in [4, and [5 that for the lower ontinuity of minimal point multifuntion M at u 0 U the ruial requirement is that stritly minimal points are dense in MinΓu 0 ) K). Some onditions assuring this kind of density are given in [5. Let A Y be a subset of Y. Definition 21 [4, 5). Strit minimality) We say that x MinA K) is a stritly minimal point, x SMA K), if for eah ε > 0 there exists δ > 0 suh that [A \ Bx, ε) x + δb) K =.

3 On lower Lipshitz ontinuity of minimal points 247 Clearly, eah stritly minimal point is minimal. Other properties of stritly minimal points were investigated in [4, 5. For minimality notions of similar type, see e.g. [17, 22. To derive our ontinuity results we introdue the modulus of minimality of a set A. Definition 22. Modulus of minimality) The modulus of minimality of a set A Y is the funtion m : R + R, defined as 1) mε) = inf νε, x) x SMA K) where ν : R + A R, is the modulus of minimality of x A defined as 2) νε, x) = sup{δ : A \ Bx, ε)) [x + δb K = }. For eah x A, νε, x) ε, and for x SMA K), 0 < νε, x) ε. Clearly, [A \ Bx, ε) x + νε, x) B) K = for x SMA \ K). 3 Lower Lipshitz ontinuity We start with suffiient onditions for lower Hausdorff ontinuity of minimal point multifuntion M. By SMu) we denote the set of stritly minimal points of the set Γu), l ) stands for the losure. Theorem 31. Let Y be a normed spae and let K Y be a losed onvex pointed one. Assume that Γ : U Y is a set-valued mapping defined on a normed spae U, u 0 U. If i) Mu 0 ) lsmu 0 )), ii) DP ) holds for all Γu) in some neighbourhood U 1 of u 0, iii) Γ is Hausdorff ontinuous at u 0, i.e., for eah ε > 0 there exists a neighbourhood U 2 of u 0 suh that Γu) Γu 0 ) + ε B, and Γu 0 ) Γu) + ε B, for u U 2,

4 248 E.M. Bednarzuk then M is lower Hausdorff semiontinuous at u 0, i.e. for eah ε > 0 for u U 1 U 2. Mu 0 ) Mu) + ε B P roof. If Mu 0 ) =, then, by the assumptions, Γu) =, and, onsequently, Mu) =, for u U 0 U 1. Hene, we an suppose that Mu 0 ). Take any ε > 0, and y Mu 0 ). By i) there exists y 1 SMu 0 ) suh that y 1 y + 1 4ε B, and Γu 0 ) \ y )) 1 ) 2 ε B + ν 2 ε, y 1 B y 1 K) =. Hene, Γu 0 )\ y )) 2 ε B ) 2 ν 2 ε, y 1 B y ) ) 3) 2 ν 2 ε, y 1 B K =. I. Consider first the ase where νε, y 1 ) 1 2ε. By the upper Hausdorff semiontinuity of Γ 4) Γu) Γu 0 ) ) 2 ν 2 ε, y 1 B Γu 0 ) \ y )) 2 ε B ) 2 ν 2 ε, y 1 [ 1 1 ) y ν 2 ε, y ) 2 ε B, B for u U 2, and by the lower Hausdorff semiontinuity of Γ, for u U 1 there exists y 2 Γu) suh that y 2 y ) 2 ν 2 ε, y 1 B and By 3) y 2 K y ) 2 ν 2 ε, y 1 B K. y 2 K) Γu) \ y )) 2 ε B ) 2 ν 2 ε, y 1 B =. Now, by 4) for u U ) y 2 K) Γu) y ν 2 ε, y ) 2 ε B.

5 On lower Lipshitz ontinuity of minimal points 249 Sine DP ) holds for Γu), for u U 1 U 2 there exists η 2 Mu) suh that 1 1 ) η 2 y 2 K) Γu) y ν 2 ε, y ) 2 ε B, and sine νε, y 1 ) 1 2 ε, η 2 y ε B y + ε B. This means that for u U 1 U 2 Mu 0 ) Mu) + ε B whih ompletes the proof in the ase I. II. Consider now the ase where νε, y 1 ) > 1 2ε. By the upper Hausdorff semiontinuity of Γ we have for u U 2 5) Γu) Γu 0 ) ε B Γu 0 ) \ y ε B )) ε B [ 1 y ε + 1 ) 2 ε B, and by the lower Hausdorff semiontinuity of Γ there exists y 2 Γu), u U 2 suh that y 2 y ) 2 ν 2 ε, y 1 B. In onsequene, y 2 K y ν 1 2 ε, y 1 ) B K, and by 3), y 2 K) Γu 0 ) \ y )) 2 ε B ) 2 ν 2 ε, y 1 B =. Sine 1 2 ν 1 2 ε, y 1) > 1 8ε the latter implies that y 2 K) Γu 0 ) \ y )) 2 ε B ε B =. Now, by 5) y 2 K) Γu) y ε B.

6 250 E.M. Bednarzuk Sine DP ) holds for Γu), u U 1, there exists η 2 Mu), u U 1 U 2 suh that η 2 y 2 K) Γu) y ε B and η 2 y ε B y + ε B. This means that for u U 1 U 2 whih ompletes the proof. Mu 0 ) Mu) + ε B Now, by strengthening the assumption i) of Theorem 31 we prove suffiient onditions for lower Lipshitz ontinuity of M at u 0. Theorem 32. Let Y be a normed spae and let K Y be a losed onvex pointed one. Assume that Γ : U Y is a set-valued mapping defined on a normed spae U, u 0 U. If i) Mu 0 ) lsmu 0 )), and the modulus of minimality mε) of Γu 0 ), satisfies the ondition mε) ε, where R, > 0, ii) DP ) holds for all Γu) in some neighbourhood U 0 of u 0, iii) Γ is upper and lower Lipshitz at u 0, i.e. Γu) Γu 0 ) + L u u 0 B, for u in a neighbourhood U 1 of u 0, Γu 0 ) Γu) L u u 0 B then M is lower Lipshitz at u 0, i.e. for u U 0 U 1 Mu 0 ) Mu) ) L u u 0. P roof. As previously, we an assume that Mu 0 ). Let u U 0 U 1 and y Mu 0 ). By i) there exists y 1 SMu 0 ) suh that y 1 y + 1 L u u 0 B. Sine y 1 SMu 0 ) Γu 0 ) \ y )) 1 ) L u u 0 B + m L u u 0 B y 1 K) =,

7 On lower Lipshitz ontinuity of minimal points 251 and hene [ Γu 0 ) \ 6) y ) L u 1 u 2 B y m L u 1 u 2 By the upper Lipshitz ontinuity of Γ 7) ) 2 m L u 1 u 2 ) ) B K =. B Γu) Γu 0 ) + L u u 0 B Γu 0 ) \ y )) L u u 0 B + L u u 0 B [ y ) L u u 0 B and sine y 1 Γu 0 ), by the lower Lipshitz ontinuity there exists y 2 Γu) suh that y 2 y L u u 0 B, and, sine 1 2 L u u m 1 L u u 0 ) y 2 K y L u u 0 B K y ) 2 m L u u 0 B K. By 6) y 2 K) [ Γu 0 ) \ y ) L u u 0 B ) 2 m L u u 0 B =, and sine L u u 0 m 1 L u u 0 ) [ y 2 K) Γu 0 ) \ y ) L u u 0 B + L u u 0 B =., Now, by 7) y 2 K) Γu) y ) L u u 0 B. Sine DP ) holds for Γu) there exists η 2 Mu) suh that η 2 y 2 K) Γu) y ) L u u 0 B y ) L u u 0. This means that for u U 0 U 1 Mu 0 ) Mu) L u u 0 B whih ompletes the proof.

8 252 E.M. Bednarzuk Theorem 33. Let Y be a normed spae and let K Y be a losed onvex pointed one. Assume that Γ : U Y is a set-valued mapping defined on a normed spae U, u 0 U. If i) Mu) lsmu)), in some neighbourhood U 2 of u 0 and for any ε > 0, mε) = inf u U 2 m u ε) 2ε > 0, where m u ) is the modulus of minimality of Γu), R, > 0, ii) DP ) holds for all Γu) in some neighbourhood U 0 of u 0, iii) Γ is loally Lipshitz at u 0, i.e. Γu 1 ) Γu 2 ) + L u 1 u 2 B for u 1, u 2 in a neighbourhood U 1 of u 0, then M is loally Lipshitz at u 0, i.e. for eah u 1, u 2 U 0 U 1 U 2 Mu 1 ) Mu 2 ) ) L B. P roof. By i), for any ε > 0, u U 2, and any z SMu), Γu) \ z + εb)) + mε)b z K =. Let u 1, u 2 U 0 U 1 U 2, and y Mu 1 ). By i) there exists y 1 SMu 1 ) suh that y 1 y + 1 L u 1 u 2 B. Sine y 1 SMu 1 ) Γu 1 ) \ y )) 1 ) L u 1 u 2 B + m L u 1 u 2 B y 1 K) =, and hene, 8) [ Γu 1 ) \ y ) L u 1 u 2 B ) 2 m L u 1 u 2 B y ) ) 2 m L u 1 u 2 B K =. By loal Lipshitz ontinuity of Γ 9) Γu 2 ) Γu 1 ) + L u 1 u 2 B Γu 1 ) \ y )) L u 1 u 2 B + L u 1 u 2 B [ y ) L u 1 u 2 B,

9 On lower Lipshitz ontinuity of minimal points 253 and, sine y 1 Γu 1 ) there exists y 2 Γu 2 ) suh that y 2 y 1 + L u 1 u 2 B and, sine L u 1 u m 1 L u 1 u 2 ), y 2 K y 1 + L u 1 u 2 B K y ) 2 m L u 1 u 2 B K. By 8) [ y 2 K) Γu 1 ) \ y ) L u 1 u 2 B ) 2 m L u 1 u 2 B = and sine L u 1 u m 1 L u 1 u 2 ) [ y 2 K) Γu 1 ) \ y ) L u 1 u 2 B + L u 1 u 2 B =. Now, by 9) y 2 K) Γu 2 ) y ) L u 1 u 2 B. Sine DP ) holds for Γu 2 ) there exists η 2 Mu 2 ) suh that η 2 y 2 K) Γu 2 ) y ) L u 1 u 2 B y ) L u 1 u 2 B. This means that for u 1, u 2 U 0 U 1 U 2 whih ompletes the proof. Mu 1 ) Mu 2 ) ) L u 1 u 2 B 4 Final remarks Parametri vetor optimization problem K min{fx) x Au)} onsists of finding all x Au) suh that fx) MinfAu)) K), where f : X Y is a mapping defined on a spae X to be minimized and A : U X is a feasible set multifuntion. By taking Γu) = fau)) Theorems 31, 32,

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