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Unformatted text preview: TFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 41 The sequential quadratic approach solves the nonlinear problem as a sequence of
QP problems, that converges to the solution of the original problem
Main idea: Assume we have calculated an approximate solution x k to the
nonlinear programming problem. Define a subproblem by approximating the objective function with a quadratic function and linearizing the inequality and
equality constraints11 1
min d ¢Bkd + f (x k )d
d
2
s.t . hi (x k )d + hi (x k ) = 0, i = 1,..., I
g j (x k )d + g j (x k ) £ 0, j = 1,..., J
where Bk = 2 f (x k ) is the Hessian of the objective function at x k
The new approximate solution is then defined as x k +1 = x k + h RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 42 Combinatorial and Integer Programming The nonlinear discrete or integer programming problem has the same form as the
nonlinear programming problem with the additional requirement that all
variables can only take on discrete or integer values min f (z )
z s.t . gi (z ) £ 0 i = 1,..., I z: integer One solution approach: exhaustive search by calculating the value of the
objective function for all feasible combinations of the “allowed” integer values.
However, this approach is only possible for very small problems RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 43 Branch and Bound Generalpurpose integer programming routines are based upon a procedure called “branchandbound” An optimal integer solution is arrived at by solving a sequence of socalled continuous relaxations organized in an enumeration tree with two branches
at each node
Starting at the root, we would solve the optimization problem removing the
requirement that variables take on integer values min f (z )
z s.t . gi (z ) £ 0 i = 1,..., I Note: The solution to the root problem, x, will not have all integer components RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 44 In the next step we will perform a branching in which we partition the problem
into two mutually exclusive problems. First, we choose some noninteger component x j of x and round this to the closest integer, I j = [x j ] . Then, we define the two subproblems (also referred to as children)
1. min f (z )
z s.t . gi (z ) £ 0 i = 1,..., I zj £ Ij
2. min f (z )
z s.t . gi (z ) £ 0 i = 1,..., I zj ³ Ij + 1 RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 45 These two subproblems with the additional constraints are now solved and a new
branching is performed. In this way, each of the subproblems leads to two new
children. If we repeat this process, sooner or later, when enough bounds have
been introduced, integer solutions to the different subproblems are obtained
We need to keep track of the best integer solution, z * , that so far has given the
smallest value of the objective function. Doing so allows us to prune the binary
enumeration tree. For example, if another subproblem at another branch has been solved and its final objective value is greater than f (z * ) , then all its childre...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.
 Fall '14

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