ON SOLUTIONS OF PERTURBED OPTIMIZATION PROBLEMS · Mitrofan M. Choban Tiraspol State University,...

ON SOLUTIONS OF PERTURBED OPTIMIZATIONPROBLEMS

Mitrofan M. Choban

Tiraspol State University, Republic of Moldovae-mail: [email protected]

13-th WORKSHOP ON WELL-POSEDNESSOF OPTIMIZATION PROBLEMS AND RELATED TOPICS,

September 12-16, 2011, Borovets, Bulgaria

September 15, 2011

Mitrofan M. Choban ON SOLUTIONS OF PERTURBED OPTIMIZATION PROBLEMS

1. Introduction

By a space we understand a completely regular topological Hausdorff space. We usethe terminology from: R.Engelking [1], L. Gillman and M. Jerison [2], K. Kuratowski [3],R. Lucchetti and J. Revalski (eds.) [4].

Let R be the space of reals, R+ = x ∈ R : x ≥ 0,N = 1,2, ... and I = [0,1] be thesubspaces of the space R, R∞ = R ∪ +∞. The sets R∞t = x ∈ R∞ : t < x form abase of the space R∞ at the point +∞.Let X be a topological space.Denote by C(X) the Banach space of all bounded continuous functions f : X −→ Rwith the sup-norm ‖ f ‖∞ = sup|f(x)|; x ∈ X .


Our principal topological notions are the following:

Definitions 1.1. (a) A sequence Un : n ∈N of subsets of a space X is called aq-sequence if:(i) any sequence xn ∈ Un : n ∈N has an accumulation point in X;(ii) the sets X \ Un and Un+1 form a pair of completely separated subsets of the spaceX for any n ∈N.

(b) A sequence Un : n ∈N of subsets of a space X is called a k-sequence if it is aq-sequence and ∩Un : n ∈N is a compact subset.

(c) A sequence Un : n ∈N of subsets of a space X is called a p-sequence if:(i) any sequence xn ∈ Un : n ∈N is convergent in X;(ii) clX Un+1 ⊆ Un for any n ∈N.

(d) A point x ∈ X is called:- a q-point if there exists a countably compact subset F of X of countable character inX such that x ∈ F;- a k-point or a point of countable type if there exists a compact subset F of countablecharacter in X such that x ∈ F.

(e) A space X is called a complete M-space if there exists a continuous closedmapping g : X −→ Y onto some complete metric space Y such that the fibers f−1(y),y ∈ Y, are countable compact.

(f) if the sets A and X \ B are completely separated, then we put A B. In this caseA ⊆ B.


Distinct classes of spaces were studied in the works of: A. V. Arhangel’skii [16, 17], A.V. Arhangel’skii, A. V. Arhangel’skii and M.M. Choban A. V. Arhangel’skii, M.M. Chobanand P. S. Kenderov [21], A. V. Arhangelskii and M. G. Tkachenko [22], J. Chaber,M.M.Coban and K. Nagami [23], M. Choban [24], M. Choban, P. Kenderov andJ.Revalski [4, 6, 7, 8], P. S. Kenderov, I. S. Kortezov and W. B. Moors [25], E. Michael[26, 27, 28], K.Morita [29], H.H. Wicke [30], etc.


2. Topological optimization problems

A function f : X −→ R∞ is called lower semi-continuous if and only if the setx ∈ X : f(x) > t is an open set for every t ∈ R.Many optimization problems can be formulated in the following way:Given a space X and a lower semi-continuous function f : X −→ R∞. Sought a pointx0 ∈ X such that f(x0) ≤ f(x) for each x ∈ X . In this case we say that the minimizationproblem min(X , f) is given.

For any function f : X −→ R∞ we put infX (f) = inf f(x) : x ∈ X ,MX (f) = x ∈ X : f(x) = infX (f) and dom(f) = x ∈ X : f(x) < +∞.The set MX (f) is the set of all solutions of the minimization problem min(X , f).The function f is called proper if its domain dom(f) is non-empty.

A sequence yn ∈ X : n ∈N is called a minimizing sequence of the problem min(X , f)if limf(yn) = infX (f).


3. Well-posed minimization problemsFix a space X and a proper bounded from below lower semi-continuous functionf : X −→ R∞ on a space X .

We put X(f , t) = f−1(−∞, t) for each t ∈ R∞. We say that U is a f -open subset of thespace X if U is open in X , or there exists t ∈ R∞ such that U ⊆ X(f , t) and U isrelatively open in X(ψ, t). All sets X(f , t) are f -open. The intersection of a finite numberof f -open subsets is f -open. The family of all f -open subsets is a base of a newTychonoff topology Tf on X . That is the minimal topology on X which contains theinitial topology T of the space X and the function f is continuous. Denote by Xf the setX with the new topology Tf .Consider the mapping ef : X −→ βX ×R∞, where ef (x) = (x , f(x)) for each x ∈ X . Bycf X denote the closure of the set ef (X) in βX ×R∞. Then cf X is the f -compactificationof the space Xf . The mapping cf : cf X −→ βX , where cf (x , y) = x for each (x , y) ∈ cf X ,is continuous.


For any r > 0 and g ∈ C(X) we put:- M(X ,f ,r)(g) = clX x ∈ X : f(x) + g(x) ≤ r + infX (f + g);- M(βX ,f ,r)(g) = M(βX ,r)(f + g) = clβX M(X ,f ,r)(g);- M(X ,f)(g) = ∩M(X ,f ,r)(g) : r > 0 = MX (f + g);- M(βX ,f)(g) = ∩M(βX ,f ,r)(g) : r > 0 = MβX (f + g);- M(cf X ,f ,r)(g) = M(cf X ,r)(f + g) = clcf X M(X ,f ,r)(g);- M(cf X ,f)(g) = ∩M(cf X ,f ,r)(g) : r > 0 = Mcf X (f + g).


By construction, M(βX ,f) is a set-valued mapping of C(X) in βX , M(cf X ,f) is a set-valuedmapping of C(X) in cf X and M(X ,f ,r), M(X ,f) are set-vaued mappings of C(X) in X .

Obviously, M(X ,f)(g) = x ∈ X : f(x) + g(x) = infX (f + g) andcf (M(cf X ,f)(g)) = M(βX ,f)(g) for each g ∈ C(X).

The mapping M(X ,f) is called the solution mapping of the continuous perturbations ofthe function f , the mapping M(X ,f ,r) is the r-solution mapping of the continuousperturbations of the function f , where r ≥ 0. We may say that M(βX ,f)(g) \M(X ,f)(g) isthe set of the virtual solutions of the minimization problem min(X , f + g). We say thatM(cf X ,f) is the extension solution mapping of the minimization problem min(X , f + g).


Fix a space X and a proper bounded from below lower semi-continuous functionψ : X −→ R∞ on the space X .

Definition 3.1. A minimization problem min(X , ψ) is called:

(T) Tychonoff well-posed if every minimizing sequence xn ∈ X : n ∈N of the functionψ is convergent (to a point from MX (ψ));

(AT) almost Tychonoff well-posed if every minimizing sequence xn ∈ X : n ∈N of thefunction ψ has a cluster point in X;

(WT) weakly Tychonoff well-posed if MX (ψ) = MβX (ψ) is a compact set and everyminimizing sequence xn ∈ X : n ∈N of the function ψ has a cluster point in X.

The following implications are obvious(T) −→ (WT) −→ (AT).


4. Global conditions of well-posedness

For a proper bounded from below lower semi-continuous function ψ : X −→ R∞ on aspace X consider the following sets of continuous perturbations:

- SM(ψ,X) = f ∈ C(X) : MX (f + ψ) is a singleton;

- TWP(ψ,X) = f ∈ C(X) : the minimization problem min(X , f + ψ) is Tychonoffwell-posed;

- aTWP(ψ,X) = f ∈ C(X) : the minimization problem min(X , f +ψ) is almost Tychonoffwell-posed;

- wTWP(ψ,X) = f ∈ C(X) : the minimization problem min(X , f + ψ) is weaklyTychonoff well-posed.


Let S ∈ SM(ψ,X),TWP(ψ,X),aTWP(ψ,X),TWP(ψ,X).

The following questions are natural:Q1. Under which conditions the set S is non-empty?Q2. Under which conditions the set S is dense in the space C(X)?Q2. Under which conditions the set S contains a dense Gδ-subset of the space C(X)?

These questions are typical for the distinct variational principles in optimization. Someoptimization problems in topological spaces were studied in [4] and in the works of: J.M. Borwein and D. Preiss [5], M. M. Choban, P. S. Kenderov and J. P. Revalski [6, 7, 8],R.Deville, G. Godefroy and V. Zizler [9], I. Ekeland [10, 11], P. S. Kenderov and J.Revalski [12], R. R. Phelps [13], J. Saint-Raymond [14], C. Stegall [15].

Our aim is to examined the above questions for some concrete properties of sets. Wepresent conditions under which the set of continuous perturbations of a given lowersemi-continuous function attains minimum on a subset with concrete properties is ”big”in a topological sense.


5. Main properties of the solution mappings

Fix a space X and a proper bounded from below lower semi-continuous functionf : X −→ R∞ on a space X .Unfortunately, since we may to have M(X ,f)(g) = ∅ for some g ∈ C(X), the mappingM(X ,f) does not describe the real situation. The mappings M(βX ,f), M(cf X ,f) and M(X ,f ,r)

are more sensible in this sense.

For any subset H of X we put M≈(X ,f)(H) = g ∈ C(X) : M(X ,f ,r)(g) H for some r > 0.

For each y ∈ cf X we put f(y) = inf r ∈ R∞ : y ∈ clcf X f−1(−∞, r]. Then f is thecontinuous extension of the function f on cf X . If Bf1 = c−1

f (U) : U is open in βX andBf2 = f(−∞, r) : r ∈ R, then Bf = U ∩ V : U ∈ Bf1,V ∈ Bf2 is an open base of thespace cf X . The set f−1(+∞) is closed in cf X . Thus Dom(f) is open in cf X . Byconstruction, Xf is the set X as a subspace of the space cf X .


We say that the set-valued mapping θ : Y −→ Z is d-open if for any open subset V ofY there exists an open subset W of Z such that W ⊆ clZ (W ∩ θ(V)).

Theorem 5.1 Under the above conditions the next assertions are true:1. The mapping M(cf X ,f) is upper semi-continuous.2. The mapping M(cf X ,f) is d-open and for each open subset V of the space C(X) theset W = X ∩M(cf X ,f)(V) is open in Xf and M(cf X ,f)(V) ⊆ clcf X W.3. If U is an open subset of cf X and U ∩ dom(f) , ∅, then M≈

(X ,f)(U) , ∅.

4. The set M≈(cf X ,f)

(U) is dense in the set M−1(cf X ,f)

(U) for any open subset U of cf X.

Corollary 5.2 Let γ be a family of open subsets of a space X and ∪U∩dom(f) : U ∈ γ= dom(f). Then ∪M≈

(βX ,f)(U) : U ∈ γ is an open dense subset of the space C(X).

Corollary 5.3 Let γ be a family of open subsets of a space Xf and∪U ∩ dom(f) : U ∈ γ is dense in dom(f). Then ∪M≈

(βX ,f)(U) : U ∈ γ is an open densesubset of the space C(X).


6. Solutions of minimization problemsand topological games

Fix a space X and a proper bounded from below lower semicontinuous functionψ : X −→ R∞ on a space X .We mention that the structure of the sets TWP(X , ψ), aWP(ψ,X), wTWP(ψ,X)depend as of properties of the space X , as of properties of the function ψ.

Fix a property P of the sequences An : n ∈N of subsets of the space X . We considerthe next properties:∅: any sequence;p: any sequence xn ∈ An : n ∈N is convergent in X ;q: any sequence xn ∈ An : n ∈N has any cluster point in X ;k : any sequence xn ∈ An : n ∈N has any cluster point in X and the set∩clX An : n ∈N is a compact subset of X .


Given two players Σ and Ω which play the following infinite topological game in X : theplayers select alternatively (Σ starts) non-empty ψ-open subsets An (for Σ) and Bn (forΩ) of X so that An+1 ⊆ Bn ⊆ An for each n ∈N. The so obtained sequence of setsAn ,Bn : n ∈N is called a ψ-play in this game. Denote this game by G(X , ψ,P). Theplayer Ω wins the ψ-play An ,Bn : n ∈N if ∩clX An : n ∈N = ∩clX Bn : n ∈N , ∅and the sequence Bn : n ∈N has the property P. In this game one may assume thatthe sets An ,Bn : n ∈N are open in the space Xψ.


Using the methods developed in [6, 7, 8, 4], we prove the following facts.

Theorem 6.1. Let Y = dom(ψ) and Yψ is the set Y as a subspace of the space Xψ.The following assertions are equivalent:1. The player Ω have a winning strategy in the game G(X , ψ, ∅).2. The player Ω have a winning strategy in the Banach-Mazur game BM(Yψ) on thespace Yψ.3. The set f ∈ C(X) : MX (ψ+ f) , ∅ contains a dense Gδ-subset of C(X).

Theorem 6.2. The following assertions are equivalent:1. The player Ω have a winning strategy in the game G(X , ψ,p).2. The set TWP(ψ,X) contains a dense Gδ-subset of C(X).3. The space Xψ contains a dense complete metrizable subspace.


Theorem 6.3. The following assertions are equivalent:1. The player Ω have a winning strategy in the game G(X , ψ, k).2. The set wTWP(ψ,X) contains a dense Gδ-subset of C(X).3. The space Xψ contains a dense Cech-complete subspace.

Theorem 6.4. The following assertions are equivalent:1. The player Ω have a winning strategy in the game G(X , ψ,q).2. The set aWP(ψ,X) contains a dense Gδ-subset of C(X).3. The space Xψ contains a dense complete M-subspace.Corollary 6.5 Let ψ : X −→ Y be an open continuous mapping of a space X onto aspace Y and P ∈ ∅,p,q, k . If for any proper bounded from below lowersemicontinuous function f : X −→ R∞ the player Ω have a winning strategy in the gameG(X , f ,P), then for any proper bounded from below lower semicontinuous functiong : Y −→ R∞ the player Ω have a winning strategy in the game G(Y ,g,P).


Example 6.6. In the Euclidian space R2 with the distanced((x , y), (u, v)) = max|x − u|, |y − v | consider the subspace X = Y ∪ B, whereY = (x , y) : 0 ≤ x ≤ 1,0 < y ≤ 1 and B = (t ,0) : 0 ≤ t ≤ 1, t is a rational number.On the space X consider the bounded lower semicontinuous function ψ, whereψ(x) = 0 for each x ∈ B and ψ(x) = 2 for each x ∈ Y . The subspace Y is open anddense in X . Moreower, the subspace Y is locally compact and complete metrizable.The function ψ is continuous in any point of the set Y and discontinuous in the points ofthe set B. But ψ|B is continuous on B. Obviously, dom(ψ) = X .Since the space X is metrizable, the set TWP(X , ψ) is dense in C(X).There exists an open non-empty subset U of C(X) such that the sets U ∩ TWP(X , ψ),U ∩ wTWP(X , ψ) and U ∩ aTWP(X , ψ) are of the first category in C(X).The player Σ have a winning strategy in the game G(X , ψ,p).


Example 6.7. Let the space X and their subspaces Y and B be as in Example 6.6.Assume that B = bn : n ∈N. On the space X consider the bounded lowersemicontinuous function φ, where φ(x) = 1 for each x ∈ Y and φ(bn) = 1 − n−1 foreach n ∈N.The function φ is continuous in any point of the set Y and discontinuous in the points ofthe set B. Moreover, the function φ|B is not continuous in each point of the space B.Obviously, dom(φ) = X .The set TWP(X , ψ) = TWP(X , φ,Y) ∪ (∪TWP(X , φ,b) : b ∈ B) is a dense Gδ-subsetof the space C(X) and contains a dense open subset of C(X).The player Ω have a winning strategy in the game G(X , φ,p).


Example 6.8. Let the space X and their subspaces Y and B be as in Example 6.6.Assume that B = bn : n ∈N. On the space X consider the bounded lowersemicontinuous functions fn : n ∈N, where fn(x) = 2 for each x ∈ Y , fn(bn) = 0 andfn(x) = 1 for each x ∈ B \ bn for each n ∈N.The functions fn are continuous in any point of the set Y and discontinuous in thepoints of the set B. Obviously, dom(fn) = X .The set ∩TWP(X , fn) : n ∈N is dense in C(X). For each n ∈N there exists an opennon-empty subset Un of C(X) such that the set Un ∩ TWP(X , fn) is of the first categoryin C(X).The player Σ have winning strategies in the games G(X , fn ,p).If F = fn : n ∈N and TF = supTfn : n ∈N, then (X ,TF ) is a complete metrizablespace and that space is favorable for the player Ω.


Example 6.9. Let Y = dom(ψ), Yψ be the set Y as a subspace of the space Xψ and Ybe a space with Gδ-diagonal. If the player Ω have a winning strategy in the gameG(X , ψ, ∅), then the set f ∈ C(X) : MX (ψ+ f) is a singleton contains a denseGδ-subset of C(X).

Now let X be the Sorgenfrey line, ψ ∈ C(X). Then:1. The set f ∈ C(X) : MX (ψ+ f) is a singleton contains a dense Gδ-subset of C(X).2. The set TWP(ψ,X) is dense in C(X) and does not contain a dense Gδ-subset ofC(X).

By virtue of above examples, the structure of the sets TWP(X , ψ), aWP(ψ,X),wTWP(ψ,X) depend as of properties of the space X , as of properties of the function ψ.


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ON SOLUTIONS OF PERTURBED OPTIMIZATION PROBLEMS · Mitrofan M. Choban Tiraspol State University,...

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