A DC programming framework for portfolio selection by minimizing the - PowerPoint PPT Presentation

A DC programming framework for portfolio selection by minimizing the transaction costs Pham Viet Nga Department of Mathematics, Hanoi University of Agriculture Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 1 / 21

Table of contents Mathematical formulation 1 Methodology 2 DC programming and DCA Branch-and-Bound (BB) Solving ( P ) by DCA 3 Method 1: Solving ( P ) by DCA Method 2: Solving ( P ) by DCA-B&B Numerical results 4 Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 2 / 21

Mathematical formulation Table of contents Mathematical formulation 1 Methodology 2 DC programming and DCA Branch-and-Bound (BB) Solving ( P ) by DCA 3 Method 1: Solving ( P ) by DCA Method 2: Solving ( P ) by DCA-B&B Numerical results 4 Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 3 / 21

Mathematical formulation Problem description n assets: the return on asset i is a i (random variable) E ( a − a )( a − a ) T = Σ . a = ( a 1 , . . . , a n ) , E ( a ) = a , w = ( w i , . . . , w n ) T : current holdings x i : amount transacted in asset i , x i > 0 for buying, x i < 0 for selling, x = ( x 1 , . . . , x n ) T : portfolio selection. Adjusted portfolio w + x ⇒ The wealth at the end of the period: W = a T ( w + x ), = E ( W − E W ) 2 = ( w + x ) T Σ( w + x ) E W = a T ( w + x ) , Shortselling constraints: w i + x i ≥ − s i , ∀ i where s i ≥ 0 Constraint on Expected return: a ( w + x ) ≥ r min Constraint on Variance: ( w + x ) T Σ( w + x ) ≤ σ 2 max Diversification constraints: w i + x i ≤ λ i 1 T ( w + x ) , ∀ i , λ i ≥ 0 Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 4 / 21

Mathematical formulation Mathematical formulation Problem ( P )  n �    min φ ( x ) = φ i ( x i )     i =1   � �  s.t. a ( w + x ) ≥ r min � n φ i ( x i ) | x ∈ C or min φ ( x ) =  w i + x i ≥ − s i , ∀ i  i =1     w i + x i ≤ λ i 1 T ( w + x ) , ∀ i     ( w + x ) T Σ( w + x ) ≤ σ 2 max φ i is the transaction cost function for asset i given by   0 , x i = 0  β i − α 1 ( β i , α 1 i , α 2 φ i ( x i ) = i ≥ 0 , ∀ i ) i x i , x i < 0   β i + α 2 i x i , x i > 0 Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 5 / 21

Mathematical formulation Mathematical formulation Problem ( P )  n �    min φ ( x ) = φ i ( x i )     i =1   � �  s.t. a ( w + x ) ≥ r min � n φ i ( x i ) | x ∈ C or min φ ( x ) =  w i + x i ≥ − s i , ∀ i  i =1     w i + x i ≤ λ i 1 T ( w + x ) , ∀ i     ( w + x ) T Σ( w + x ) ≤ σ 2 max φ i is the transaction cost function for asset i given by   0 , x i = 0  β i − α 1 ( β i , α 1 i , α 2 φ i ( x i ) = i ≥ 0 , ∀ i ) i x i , x i < 0   β i + α 2 i x i , x i > 0 Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 5 / 21

Methodology Table of contents Mathematical formulation 1 Methodology 2 DC programming and DCA Branch-and-Bound (BB) Solving ( P ) by DCA 3 Method 1: Solving ( P ) by DCA Method 2: Solving ( P ) by DCA-B&B Numerical results 4 Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 6 / 21

Methodology DC programming and DCA DC programming and DCA: DC program DC = Difference of Convex functions DC program inf { f ( x ) = g ( x ) − h ( x ) | x ∈ R n } ( P dc ) ( g , h : lower semicontinuous proper convex functions on R n ) g − h : a DC decomposition of f . g , h : convex DC components of f . g ∗ , h ∗ : conjugate functions of g , h . g ∗ ( y ) := sup {� x , y � − g ( x ) | x ∈ R n } , y ∈ R n Subdifferential of a convex function θ at x 0 ∈ dom f (dom f := { x ∈ R n : θ ( x ) ≤ + ∞} ) ∂θ ( x 0 ) := { y ∈ R n : θ ( x ) ≥ θ ( x 0 ) + � x − x 0 , y � , ∀ x ∈ R n } Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 7 / 21

Methodology DC programming and DCA DC programming and DCA: DC program DC = Difference of Convex functions DC program inf { f ( x ) = g ( x ) − h ( x ) | x ∈ R n } ( P dc ) ( g , h : lower semicontinuous proper convex functions on R n ) g − h : a DC decomposition of f . g , h : convex DC components of f . g ∗ , h ∗ : conjugate functions of g , h . g ∗ ( y ) := sup {� x , y � − g ( x ) | x ∈ R n } , y ∈ R n Subdifferential of a convex function θ at x 0 ∈ dom f (dom f := { x ∈ R n : θ ( x ) ≤ + ∞} ) ∂θ ( x 0 ) := { y ∈ R n : θ ( x ) ≥ θ ( x 0 ) + � x − x 0 , y � , ∀ x ∈ R n } Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 7 / 21

Methodology DC programming and DCA DC programming and DCA: DC program DC = Difference of Convex functions DC program inf { f ( x ) = g ( x ) − h ( x ) | x ∈ R n } ( P dc ) ( g , h : lower semicontinuous proper convex functions on R n ) g − h : a DC decomposition of f . g , h : convex DC components of f . g ∗ , h ∗ : conjugate functions of g , h . g ∗ ( y ) := sup {� x , y � − g ( x ) | x ∈ R n } , y ∈ R n Subdifferential of a convex function θ at x 0 ∈ dom f (dom f := { x ∈ R n : θ ( x ) ≤ + ∞} ) ∂θ ( x 0 ) := { y ∈ R n : θ ( x ) ≥ θ ( x 0 ) + � x − x 0 , y � , ∀ x ∈ R n } Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 7 / 21

Methodology DC programming and DCA DC programming and DCA: DC Algorithm DCA = DC algorithm Constructing two sequences { x k } et { y k } , candidates to be solutions of ( P dc ) and its dual program respectively Generic DCA scheme Initial point x 0 ∈ dom g , k ← − 0 Repeat: x k − → y k ∈ ∂ h ( x k ) ւ x k +1 ∈ ∂ g ∗ ( y k ) − → y k +1 ∈ ∂ h ( x k +1 ) Until: convergence of { x k } . x k +1 ∈ ∂ g ∗ ( y k ) ⇐ ⇒ x k +1 ∈ arg min { g ( x ) − � x , y k � | x ∈ R n } Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 8 / 21

Methodology Branch-and-Bound (BB) BB: methods for global optimization for nonconvex problems Lower bounds: Found from convex relaxation, duality, Lipschitz or other bounds,... Upper bounds: can be found by choosing any point in the region, or by a local optimization method. Basic idea: partition feasible set into convex sets, and find lower/upper bounds for each form global lower and upper bounds; quit if close enough else, refine partition and repeat Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 9 / 21

Solving ( P ) by DCA Table of contents Mathematical formulation 1 Methodology 2 DC programming and DCA Branch-and-Bound (BB) Solving ( P ) by DCA 3 Method 1: Solving ( P ) by DCA Method 2: Solving ( P ) by DCA-B&B Numerical results 4 Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 10 / 21

Solving ( P ) by DCA Method 1: Solving ( P ) by DCA Method 1: Solving ( P ) by DCA  φ i ( x i )  0 , x i = 0  β i − α 1 φ i ( x i ) = i x i , x i < 0  β i  β i + α 2 i x i , x i > 0 � n φ ( x ) = φ i ( x i ) i =1 x i 0   β i − α 1 x i ≤ − ǫ i i x i ,  f i ( x i )    − c 1 i x i , − ǫ i ≤ x i ≤ 0 f i ( x i ) = c 2  β i i x i , 0 ≤ x i ≤ ǫ i     β i + α 2 i x i , x i ≥ ǫ i � n ( c j i = β i ǫ i + α j f ( x ) = f i ( x i ) i ) i =1 0 x i ǫ i − ǫ i Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 11 / 21

Solving ( P ) by DCA Method 1: Solving ( P ) by DCA Method 1: Solving ( P ) by DCA  φ i ( x i )  0 , x i = 0  β i − α 1 φ i ( x i ) = i x i , x i < 0  β i  β i + α 2 i x i , x i > 0 � n φ ( x ) = φ i ( x i ) i =1 x i 0   β i − α 1 x i ≤ − ǫ i i x i ,  f i ( x i )    − c 1 i x i , − ǫ i ≤ x i ≤ 0 f i ( x i ) = c 2  β i i x i , 0 ≤ x i ≤ ǫ i     β i + α 2 i x i , x i ≥ ǫ i � n ( c j i = β i ǫ i + α j f ( x ) = f i ( x i ) i ) i =1 0 x i ǫ i − ǫ i Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 11 / 21

Solving ( P ) by DCA Method 1: Solving ( P ) by DCA Method 1: Solving ( P ) by DCA  φ i ( x i )  0 , x i = 0  β i − α 1 φ i ( x i ) = i x i , x i < 0  β i  β i + α 2 i x i , x i > 0 � n φ ( x ) = φ i ( x i ) i =1 x i 0   β i − α 1 x i ≤ − ǫ i i x i ,  f i ( x i )    − c 1 i x i , − ǫ i ≤ x i ≤ 0 f i ( x i ) = c 2  β i i x i , 0 ≤ x i ≤ ǫ i     β i + α 2 i x i , x i ≥ ǫ i � n ( c j i = β i ǫ i + α j f ( x ) = f i ( x i ) i ) i =1 0 x i ǫ i − ǫ i Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 11 / 21

Solving ( P ) by DCA Method 1: Solving ( P ) by DCA Method 1: Solving ( P ) by DCA � n f ( x ) = f i ( x i ) i =1 f is a polyhedral DC function f ( x ) ≤ φ ( x ) , ∀ x ∈ C A DC approximation program of ( P ) is � n min { f ( x ) = f i ( x i ) = g ( x ) − h ( x ) | x ∈ C ∩ R 0 } ( P dc ) i =1 Solving ( P dc ) by DCA to find a solution for ( P ) (DCA has a finite convergence for polyhedral DC programs) Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 12 / 21

Solving ( P ) by DCA Method 2: Solving ( P ) by DCA-B&B Method 2: Solving ( P ) by DCA-B&B  φ i ( x i ) 0 , x i = 0   β i − α 1 φ i ( x i ) = i x i , x i < 0  β i  β i + α 2 i x i , x i > 0 l 0 i = min { x i | x ∈ C } u 0 i = max { x i | x ∈ C } x i 0 � φ i ( x i )  � � β i  i − α 1 β i x i , x i ≤ 0 l 0 i � � � φ i ( x i ) =  β i i + α 2 x i ≥ 0 x i , u 0 i l 0 u 0 x i 0 i i Pha .m Viˆ e .t Nga (HUA) HUA 04/11/2013 13 / 21