Optimization

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Unformatted text preview: © P. KOLM. 7 Comments: Most optimization algorithms available today attempt to find only a local solution. In general, finding the global optimal solution can be very difficult In principle, finding the global optimal solution requires an exhaustive search that first locates all local optimal solutions and then chooses the best one among those. There is no general efficient algorithm for the global optimization problem available at present time, but rather specialized algorithms that rely upon unique properties of the objective function and constraints Although a vast set of problems can be formulated as nonlinear programs, in practice, many problems possess further structure and have properties that if taken into account will deliver stronger mathematical results as well as more efficient algorithms. Therefore, it makes sense to categorize optimization problems based upon their properties. Typically, problems are classified according to the form of the objective function and the functions defining the constraints. We will take a look at a few standardized formulations, standard forms, next RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 8 Linear Programming (LP) Linear programming (LP) refers to the problem of minimizing a linear function subject to linear equality and inequality constraints. The standard form of a linear program is given by min c ¢x x s.t . Ax = b x ³0 where c is an N-dimensional vector, A is an J×N matrix, and b is an Jdimensional vector RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 9 Comments: The linear programming problem is maybe the best known and the most frequently solved optimization problem in the real world Some examples of when linear programming arises in financial applications are for determining whether there exist static arbitrage opportunities in current market prices1, calculating the smallest cost hedging portfolio, pricing of American options2, and solving portfolio optimization problems with linear risk measures such as mean absolute deviation (MAD) or portfolio shortfall3 RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 10 Quadratic Programming (QP) Minimizing a quadratic objective function subject to linear equality and inequality constraints is referred to as quadratic programming (QP). This problem is represented in standard form as: 1 min x ¢Qx + c ¢x x 2 s.t . Ax = b x ³0 where Q is an N×N matrix, c is an N-dimensional vector, A is an J×N matrix, and b is an J-dimensional vector RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 11 Comments: We can assume that Q is a symmetric. If this is not the case we can replace 1 (Q + Q ¢) without changing the value of the objective function since 2 x ¢Qx = x ¢Q ¢x [For you: Do you understand why?] Q by If the matrix Q is positive semi-definite or positive definite, then this becomes a convex programming problem. In this case any local optimum is a global optimum, and the problem can be solved by many of the standard algorithms for convex quadratic programming When the matrix Q is indefinite (i.e. as both posi...
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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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