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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 Ndimensional 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 Ndimensional 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. 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 semidefinite 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.
 Fall '14

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