Optimization

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Unformatted text preview: ous interest in SOCPs in finance RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 19 Integer and Combinatorial Programming So far our discussion has focused on optimization problems where the variables are continuous When the variables are only allowed to take on discrete values such as binary values (0, 1) or integer values (…,-2, -1, 0, -1, 2,…) we refer to resulting mathematical programming problem as a combinatorial, discrete or integer programming problem (ILP) If some variables are continuous and others are discrete the resulting optimization problem is called a mixed-integer programming problem (MILP) RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 20 Example 1 Minimum holding constraints: wi ³ Lw di i i = 1,..., N where Lw is the smallest holding size allowed and i ì1, if w ¹ 0 ï i di = ï í ï0, if wi = 0 ï î Transaction size constraints: x i ³ Lx di i i = 1,..., N where Lx is the smallest transaction size permitted and i ì1, if x ¹ 0 ï i di = ï í ï0, if x i = 0 ï î RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 21 Example 2 The M-V optimization with cardinality constraints6 min w'Sw s.t . w' m ³ m0 w'i = 1 w N åd i =1 i =K 0 £ wi £ di , i = 1, , N and ì1, if w ¹ 0 ï i di = ï í ï0, if wi = 0 ï î [For you: How can this be useful?] RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 22 Necessary Conditions for Optimality In calculus we learn how to solve unconstrained optimization problems (of smooth functions). Here is a short review: 1-Dimensional Case Local minima, x * , of min f (x ) x satisfy the derivative condition f ¢(x * ) = 0 N-dimensional Case Local minima, x * , of min f (x ) x satisfy the gradient condition æ¶ ö ¶ * *÷ ç ÷ f (x ) = ç ç x f (x ),..., ¶x f (x )÷ = 0 ÷ ç¶ è ø * 1 N RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 23 How Do We Handle Constraints? Equality constraints are easy to handle – we learned how to do that in calculus using the technique of Lagrange multipliers With inequality constraints (or equality and inequality constraints) require a bit more work. Here the necessary conditions are referred to as the KarushKuhn-Tucker conditions (KKT) RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 24 N-dimensional Case With Equality Constraints Local minima, x * , of min f (x ) x s.t . h j (x ) = 0, j = 1,..., J satisfy the gradient condition J f (x * ) + ålj h j (x * ) = 0 j =1 and h j (x * ) = 0, j = 1,..., J [For you: Verify this result.] RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 25 N-dimensional Case With Equality and Inequality Constraints Local minima, x * , of min f (x ) x s.t. gi (x ) £ 0 i = 1,..., I h j (x ) = 0 j = 1,..., J satisfy the Karush-Kuhn-Tucker conditions7 J I f (x ) + ålj h j (x ) + åmi gi (x * ) = 0 * * j =1 i =1 h j (x * ) = 0, j = 1,..., J g...
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