Efficient solution concepts Mathematical Programming Models For Portofolio Selections.

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2.2 Efficient solution concepts

Consider a model in which the designdecision associated with a system is specified via vector x. Under uncertainty, the system operates in an environment in which there are uncontrollable parameters which are modeled using random variables. Consequently, the performance of such a system can also be viewed as a random variable. Accordingly, stochastic programming models provide a framework in which designs x can be chosen to optimize some measure of the performance random variable. It is therefore natural to consider measures such as the worst case performance, expectation and other moments of performance, or even the probability of attaining a predetermined performance goal. Let us consider the stochastic multi-objective programming problem Caballero, et al [21] ~ , ,..., ~ , min 1 c x z c x z q D x ∈ , 2.1 where the following notations and assumptions • there is a compact set n D R ⊆ of feasible actions; • n x R ∈ is thevector of decision variables of the problem and c~ is a random vector whose components are random continous variables, defined on the set n R E ∈ . We assume given the family F of events that is, subset of E and the distribution of probability P defined on F so that, for any subset of E , E ⊂ A , F ∈ A , the probability PA is known. Also, we assume that the distribution of probability P is independent of the decision variables n x x ,..., 1 ; • there are q objective functions } { ⋅ k f with + ∈ R x f k for all D x ∈ and c~ is a random vector whose components are random continuous variable; • it is required to find members of the efficient vector minimal set E of D with respect to the order relation ≤ on q R , where, by definition, } , : { x f y f x f y f D y D x E = → ≤ ∈ ∈ = 2.2 Let x z k is the expected value of the kth objective function, and let x k σ be its standard deviation, } ,..., 1 { q k ∈ . Let us assume that, for every } ,..., 1 { q k ∈ and for every feasible vector x of the stochastic multi-objective programming problem, the standard deviation x k σ is finite. In this section we will shows the definitions and 11 relations between expected value standard deviation efficient solution and efficient solutions. Next the following definitions by Caballero, et al [21] , Definition 2.1 [21] Expected-Value Efficient Solution. The point D x ∈ is an expected- value efficient solution of the stochastic multi-objective problem if it is Pareto efficient to the following problem : ,..., min : 1 x z x z PE q D x ∈ . Let PE E be the set of expected-value efficient solution of the stochastic multi-objective problem. Definition 2.2 [21] Minimum-Variance Efficient Solution. The point D x ∈ is a minimum-variance efficient solution for the stochastic multi-objective problem if it is a Pareto efficient solution for the problem : ,..., min : 2 2 1 2 x x P q D x σ σ σ ∈ . Let 2 σ P E be the set of efficient solutions of the problem 2 σ P . Definition 2.3 [21] Expected-Value Standard-Deviation Efficient Solution or σ E Efficient Solution. The point D x ∈ is an expected-value standard-deviation efficient solution for the stochastic multi-objective programming problem if it is a Pareto efficient solution to the problem ,..., , ,..., min : 1 1 x x x z x z PE q q D x σ σ σ ∈ . Let σ PE E be the set of expected-value standard-deviation efficient solutions of the stochastic multi-objective programming problem 2.1. Now, we give the concepts of efficiency for two criteria of maximum probability. As we will see next, in order to define these two concepts, the minimum-risk criterion concept of minimum-risk efficiency and Kataoka criterion efficiency in probability are applied respectively to each stochastic objective. Definition 2.4 [21] Minimum-Risk Efficient Solution for the Levels q u u ,..., 1 . See Stancu-Minasian and Tigan 180. The point D x ∈ is a minimum risk vectorial solution for levels q u u ,..., 1 if it is a Pareto efficient solution to the problem: 12 ~ , ~ ,..., ~ , ~ max : 1 1 q q D x u c x z P u c x z P u PRM ≤ ≤ ∈ , Let u PRM E be the set of efficient solution for the problem PMRu. Definition 2.5 [21] Efficient Solution with Probabilities q β β ,..., 1 or β -Efficient Solution. The point D x ∈ is an efficient solution with probabilities q β β ,..., 1 if there exist q u R ∈ such that t t t u x , is a Pareto efficient solution to problem: β PP D x q k u c x z P u u k k k q u x ∈ = ≥ ≤ , , 1 , } ~ , ~ { ,..., min 1 , β Let β PP E n R ⊂ be the set of efficient solutions with probabilities q β β ,..., 1 for the stochastic multi-objective programming problem 2.1. It may be noted that these definitions of efficient solution are obtained by applying the same transformation criterion to each one of the objectives separately expected value, minimum variance, etc., and by building after word the resulting deterministic multiobjective problem. In this sense, it is necessary to the following results. Remark 2.1 The concepts of expected value, minimum variance, etc., weak and properly efficient solution can be defined in a natural way. Remark 2.2 The concepts of minimum-risk efficiency and β -efficiency require setting a priori a vector of aspiration levels u or a probability vector β . This implies that, in both cases, the efficient set obtained depends on the predetermined vectors in such a way that, in general, different level and proba bility vector give rise to different efficient sets, . , β β β β PP PP PRM PRM E E u E u E u u ≠ ⇒ ≠ ≠ ⇒ ≠ Remark 2.3 The concept of expected standard-deviation efficient solution is an extention to multiobjective case of the concept of the mean-variance efficient solution that Markowitz [114] defines for the stochastic single objective problem of portfolio selection. In this way, we have the two statistical moments corresponding to each stochastic objective in the same measuring units. Since the square root function is 13 strictly increasing, the set of efficient solutions does not vary in problem if we substitute standard deviation for variance, White [209]. Remark 2.46 The efficiency in probability criterion is a generalization of the one presented by Goicoechea, Hansen, and Duckstein [63], who define the same concept taking the same probability β for all the stochastic objectives and with the probabilistic equality constraints taking the form β = ≤ } ~ , ~ { k k u c x z P . This notion was introduced by Stancu-Minasian [179], considering the Kataoka problem in the case of multiple criteria. 2.3 Relations between the efficient sets of several of deterministic multiobjective programming problems We present some relations between the efficient sets of several problems of deterministic multi-objective programming problems. These results will be used later for analysis of concepts of efficient solutions for multi-objective stochastic problems. Considered f and g be vectorial functions defined on the same set n H R ⊂ with n H f R ⊂ : q R → and n H g R ⊂ : q R → and let γ α, be nonnull vectors with q real components, that is, q R ∈ γ α, and , ≠ γ α . Let us consider the following multiobjective problems: PD 1 ,..., , ,..., min 1 1 1 x g x g x f x f q q q D x γ γ ∈ 2.3 PD 2 ,..., min 1 x f x f q D x ∈ 2.4 PD 3 ,..., min 2 2 1 1 x g x g q q D x γ γ ∈ 2.5 with, q R ∈ γ , ≠ γ . Let 3 2 1 , , E E E be the sets of weakly efficient, efficient, and proper efficient points of problem i PD , respectively. The following theorem relates these problems PD 1 , PD 2 and PD 3 problems to each other. 14 Theorem 2.1 We assume that g for every D x ∈ ,. Then: i 1 1 3 2 E E E ⊂ ∩ i 2 w E E E 1 3 2 ⊂ ∪ i 3 w w w E E E 1 3 2 ⊂ ∪ Proof: 1 i 3 2 E E x ∩ ∈ Let us show that 1 E x ∈ by reductio ad absurdum. We assume that 1 E x ∉ . Then, there exist an D x ∈ such that x f x f k k ≤ and x g x g k k k k γ γ ≤ , for every } ,..., 1 { q k ∈ , there being an } ,..., 1 { q s ∈ for which the inequality is strict, x f x f s s or x g x g s s s s γ γ . Therefore, 2 E x ∉ or 3 E x ∉ , since x g x g k k k k γ γ ≤ , implies ≤ x g s k k γ x g s k k γ , contrary to 3 2 E E x ∩ ∈ . i 2 w E E E 1 3 2 ⊂ ∪ Let 3 2 E E x ∪ ∈ . Let us see that w E x 1 ∈ by reductio de absurdum. We assume that w E x 1 ∉ . Then, there exist a vector D x ∈ that weakly dominates x and so verifies x f x f k k and x g x g k k k k γ γ , for every } ,..., 1 { q k = . Thus, 2 E x ∉ and, since x g x g k k k k γ γ , implies x g s k k γ x g s k k γ , 3 E x ∉ , contrary to 3 2 E E x ∪ ∈ . i 3 w w w E E E 1 3 2 ⊂ ∪ Let w w E E x 3 2 ∪ ∈ . Let us see that w E x 1 ∈ by reductio de absurdum. We assume that w E x 1 ∉ . Then, there exist a vector D x ∈ that weakly dominates the vector x and therefore verifies that x f x f k k and x g x g k k k k γ γ , for every } ,..., 1 { q k ∈ . Thus, w E x 2 ∉ and, since x g x g k k k k γ γ , implies x g s k k γ x g s k k λ , w E x 3 ∉ , contrary to w w E E x 3 2 ∪ ∈ . 15 Thus, i 2 can be deduced from i 3 □ Now we consider the following problem ,..., min 1 1 1 x g x f x g x f q q q D x α α + + ∈ 2.6 where q 1 : ,..., R R → = + q α α α . Let 4 α E and 4 α G E denote the efficient solutions set and the properly efficient solutions set respectively for problem 2.6. We will now present some relations between these sets and the set of efficient solutions and properly efficient solutions for problem PD 1 . Theorem 2.2 [21] For q q R R → = + : ,..., 1 γ γ γ , q 1 : ,..., R R → = + q α α α , with , ≠ k k γ α and q k sign sign k k , 1 , = = γ α , the following relation holds : 1 4 E E ⊂ α . Proof: Let 4 α E x ∈ . We assume that 1 E x ∉ . In this case, there is a solution x that dominates the solution x, that is, x f x f k k ≤ and x g x g k k k k γ γ ≤ , for every } ,..., 1 { q k ∈ , and there exist at least one } ,..., 1 { q s ∈ for which the inequality is strict, that is, x f x f s s or x g x g s s s s γ γ From this point onward, since x f x f k k ≤ , x g x g k k k k γ γ ≤ , implies ≤ x g k k α x g k k α , the following inequalities are verified: x g x f x g x f k k k k k k λ α + ≤ + , for every } ,..., 1 { q k ∈ , 2.7 x g x f x g x f k k k k k k α α + ≤ + , for every } ,..., 1 { q k ∈ . 2.8 From 2.7 and 2.8, we obtain x g x f x g x f k k k k k k α α + ≤ + , for every } ,..., 1 { q k ∈ . In particular, for s k = , we have the results bellow: a if x f x f s s , x g x f x g x f s s s s s s α α + + , 16 and the following inequality is obtained from 2.8: x g x f x g x f s s s s s s α α + + ; b if x g x g s s s s α α , x g x f x g x f s s s s s s α α + + , and since x f x f s s ≤ , we obtain x g x f x g x f s s s s s s α α + + . Therefore, for every } ,..., 1 { q k ∈ , x g x f x g x f k k k k k k α α + ≤ + , and there is at least a subscript } ,..., 1 { q s ∈ for which x g x f x g x f s s s s s s α α + + , which implies that the solution x dominates the solution x; therefore, we reach a contradiction with the hypothesis of x being the efficient solution to problem 2.6. ■ Next, we prove that, in some conditions, this relationship is hold for the set of properly efficient solution. For this purpose, we define problems , , μ λ γ g f P and ξ α P , obtained by applying the weighting method to problems 2.3-2.6 respectively as follows: ∑ = ∈ + q k k k k t D x g f x g x f P 1 , min : , γ μ λ μ λ γ , min : 1 x g x f P k k k q k k D x α ξ ξ α + ∑ = ∈ . We use the results available in the literature about the relationships between the optimal solution to the weighting problem and the efficient solutions to the multi- objective problem. Some results, see Chankong and Haimes [29], applied to problem 2.3 and its associated weighted problem , , μ λ γg f P , are as follow : 17 a If f and t q q g g ,..., 1 1 γ γ are convex functions, D is convex, and x is a properly efficient solution for the multi-objective problem 2.3, there exist some weight vector μ λ, with strictly positive components such that x is the optimal solution for weighted problem , , μ λ γ g f P . b For each vector of weights with strictly positive components, the optimal solution to the weighted problem , , μ λ γg f P is properly efficient for the multi-objective problem P 1 . Proposition 2.1 If f and ,..., 1 1 q q g g γ γ are convex functions, D is a convex set and there exist q q R R → = + : ,..., 1 α α α , k k sign sign γ α = , for every } ,..., 1 { q k ∈ then G G E E 1 4 ⊂ α . Proof: If f and ,..., 1 1 q q g g γ γ are convex functions and if D is a convex set, then the set of properly efficient solutions to problems PD 1 and 2.6 are obtained from the associated weighted problems for strictly positive weight vectors. We will prove that any solutions to the optimization problem ξ α P , with ξ , is a solution to problem , , μ λ γg f P for some vector , μ λ . Let 4 α G E x ∈ . Then, given the established hypotheses, there exist a vector ξ for which x is the solution for problem ξ α P . Let us assume that, for every , }, ,..., 1 { ≠ ∈ k k q k γ α . Then, we take k k ξ λ = , , , = k k k k k k μ λ γ α ξ μ , Since ξ , we obtain that x is an optimal solution to problem , , μ λ λ g f P . For some } ,..., 1 { q i ∈ if = = i i γ α , then the proof would be the same, since in problem 2.3 the function i g is not involved and since in problem 2.6 the function ith objective would be i f . ■ In general, the inverse inclution does not hold, as it’s shown by the following example. 18 Example 2.1. Let us consider the following problem: , , , 1 9 . , , max 2 2 , ≥ ≤ + y x y x t s y x y x with 1 , , , , = = = u y y x g x y x f . The set of efficient points for this problem is { } , , 1 4 , 2 2 2 = + ∈ y x y x y x t R and is represented in Fig. 2.1. We outline the solution of the problem , , , 1 9 . , max 2 2 , ≥ ≤ + + y x y x t s y x y x α with α . For each fixed α , the optimal solution of the resulting problem is one of property efficient solutions to the original becriterion problem. y 1 ε D 3 x Figura 2.1 Proposition 2.2 If f and ,..., 1 1 q q g g γ γ are convex functions, then U Ω ∈ ⊂ α α 4 1 G G E E , with } , 1 , : ,..., { 1 q k sign sign R k k q q = = → = = Ω + γ α α α α R . Proof: As the previous case, the proof of the proposition is carried out by demonstrating that any solution to the problem , , μ λ γ g f P is a solution to the problem ξ α P for some vector n R ∈ α , with } ,..., 1 { , q k sign sign k k ∈ = γ α , and for some ξ . 19 Consider G E x 1 ∈ . Since f and t q q g g ,..., 1 1 γ γ are convex functions, there exist vector , μ λ such that x is a solution to problem , μ λ fug P . Because , μ ξ we put k k λ ξ = , k k k k ξ γ μ α = , since , μ ξ , therefore we obtain that x is also a solution to the problem ξ α P . □ From Proposition 2.1 and Proposition 2.2, if f and t q q g g ,..., 1 1 γ γ are convex functions and if k k sign sign γ α = , . t t k k γ α , for every } ,..., 1 { q k ∈ , the sets of properly efficient solutions to problem 2.3 and 2.6 verify the following properties: a. Every properly efficient solution to problem 2.6 is properly efficient for problem 2.3; b. Setting q R ∈ γ , with nonnull components, the set of properly efficient solutions to problem 2.3 is a subset of the union in α of the set of properly efficient solutions for problem 2.6.

2.4 Some relation between expected-value efficient solution, minimum-