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Unformatted text preview: tive and negative
eigenvalues) the problem can have several stationary points and local
solutions and then becomes more difficult to solve RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 12 Convex Set
Convex set: A set X is convex if, for all x 1, x 2 Î X and all t Î [0,1] tx 1 + (1  t )x 2 Î X
That is, every point on the line segment between x 1 and x 2 is in X as depicted
below Convex set Not a convex set RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 13 Convex Function
Convex function:4 Let X be a convex set. A function f : X is convex if for
any two points x 1, x 2 Î X and t Î [0,1] f (tx 1 + (1  t )x 2 ) £ tf (x 1 ) + (1  t )f (x 2 ) RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 14 Convex Programming (CP) A convex program (CP) in standard form is given by min f (x )
x s.t . gi (x ) £ 0, i = 1,..., I
Ax = b
where f and gi are convex functions, A is a J×N matrix, and b is a Jdimensional
vector RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 15 Comments: An important result: For convex programs (unlike general nonlinear programs) local optimal solutions = global optimal solutions Convex programming is a large class of nonlinear programming that contains subclasses such as semidefinite programs (SPD), secondorder cones
programs (SOCP), geometric programs (GP), least squares (LS), convex
quadratic programming (QS), and linear programming (LP) Unfortunately, checking that a given optimization problem is convex is far from straightforward and can even be more difficult than solving the
problem itself Many efficient algorithms for these types of problems are available. In particular, during the last decade or so the development of socalled interior
point methods for convex programming has been tremendous RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 16 Conic Optimization (CO) By replacing the nonnegativity constraints in the standard form of a linear
program with socalled conic inclusion constraints we obtain the conic
optimization problem min c ¢x
x s.t . Ax = b
x ÎC where c is an Ndimensional vector, A is a J×N matrix, b is a Jdimensional
vector, and C is a closed convex cone5
Note: Any convex program can be represented as a conic optimization problem by appropriately specifying C When C = N this problem reduces to the linear programming problem in
+ standard form RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 17 SecondOrder Cone Program (SOCP) One important class of cone programs is secondorder cone programs (SOCP): min c ¢x
s.t. Ax = b x C i x + di £ ci¢x + ei , i = 1,..., I where c is an Ndimensional vector, A is a J×N matrix, b is a Jdimensional
vector, C i are I i ´ N matrices, di are I i dimensional vectors, and ei are scalars RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 18 Comments: This problem is general enough to contain a large class of optimization problems such as linear programs, convex quadratic programs and
quadratically constrained convex quadratic programs At the same time this problem shares many of the same mathematical and numerical properties as linear programs, making optimization algorithms
very efficient and highly scalable Many robust portfolio allocation problems can be formulated as SOCPs. This has led to a tremend...
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