Decomposition Mathematical Programming Models For Portofolio Selections.

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4.5 Decomposition

It follows that the dual functional can be expressed as a weighted sum of 2 + T functions 4.30–4.32. In order to analyze their properties and to develop a numerical method we need to find a proper representation of the utility function u . We represent the function u by its slopes between break points. Let us denote the values of u at its break points by k l k l y u u = , υ , 1 , , 1 = = k m l . We introduce the slope variables k l k l y u − = β , υ , 1 , , 1 = = k m l The vectors ,.... ,..., , ,..., 2 2 1 1 1 1 m m β β β β β = is nonnegative, because u is nondecreasing. As u is concave, k l k l 1 + ≥ β β , υ , 1 , 1 , 1 = − = k m l . We can represent the values of u at break points as follows ∑ − − − = l l k l k l k l k l y y u 1 β , 1 , 1 − = m l . The function 4.24 takes on the form ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − = − = ∑ − ≤ ≤ ≤ ≤ l l k l k t k l k l k l m l k l k l k l m l k t k t y y y y u u D max ] [ max , 1 1 1 θ β θ θ . In this way we have expressed , k t k t u D θ as a functions of the slope vector m R ∈ β and of + ∈R k t θ . We denote ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − = ∑ − ≤ ≤ l l k l k t k l k l k l m l k t k t y y y B max , 1 1 θ β θ β . 4.34 Observe that k t B is the maximum of finitely many linear functions in its domain. The domain is a convex polyhedron defined by k l k t β θ ≤ ≤ . 65 Consequently, k t B is a convex polyhedral function. Therefore its subgradient at a point , k t θ β of the domain can be calculated as the gradient of the linear function at which the maximum in 4.34 is attained. Let l be the index of this linear function. Denoting by l δ the l th unit vector in m R we obtain that: ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − − ∑ − , 1 l l k l k l k l l y y y δ is the subgradient of , k t k t B θ β . Similarly, function 3.32 can be represented as a function k T B 1 + of the slope vectors β : ∑ ∑ = − + − = m l l l k l k l k l k l k T y y B 1 1 1 β π β . It is linear in β and its gradient has the form 1 1 k l k l l l k l n l l y y − = − ∑ ∑ π δ . Finally, denoting by j the index at which the maximum in 4.30 is attained, we see that the vector with coordinates k t j k t r p , υ , 1 , , 1 = = k T t , 4.35 is a subgradient of k D . Summing up, with our representation of the utility function by its slopes, the dual function is a sum of T + 2 convex polyhedral functions with known domains. Moreover, their subgradients are readily available. Therefore the dual problem can be solved by nonsmooth optimization methods see Dentcheva and Ruszczynski [41], Beale [8] and Bonnans and Shapiro [18]. We have developed a specialized version of the regularized decomposition method described in Dentcheva and Ruszczy´nski [41] and Ogryezak and Ruszczy’nski [127]. This approach is particularly suitable, because the dual function is a sum of very many polyhedral functions. 66 After the dual problem is solved, we obtain not only the optimal dual solution ˆ , ˆ θ β , but also a collection of active cutting planes for each component of the dual function. Let us denote by k j the collection of active cuts for k D . Each cutting plane for k D provides a subgradient 4.35 at the optimal dual solution. A convex combination of these subgradients provides the subgradient of k D that enters the optimality conditions for the dual problem. The coefficients of this convex combination are also identified by the regularized decomposition method. Let k g denote this subgradient and let , J j v k j ∈ the corresponding coefficients. Then ∑ ∑ ∈ = = J j 1 k j k jt k t T t k t k v r p g δ , where ≥ k j v , ∑ ∈ = J j 1 k j v . For each t the subgradient of k t B with respect to k t θ entering the optimality conditions is } 34 . 4 in maximizer is : { ˆ l y conv v k l k t ∈ . Therefore ˆ 1 = − ∑ = T k t k t k v p g l , Using these relations we can verify that vˆ are the vector of optimal portfolio returns in scenarios T t , 1 = . Thus the optimal portfolio has the weights ⎩ ⎨ ⎧ ∉ = ∈ = . , ˆ , , ˆ J j x J j v x j j j 67 68 C HAPTER 5 A FUZZY APPROACH TO PORTFOLIO OPTIMIZATION

5.1 Introduction