11212012 p kolm 41 the sequential quadratic approach

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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 General-purpose integer programming routines are based upon a procedure called “branch-and-bound” An optimal integer solution is arrived at by solving a sequence of so-called 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.

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