E as both positive and negative eigenvalues the

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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 J-dimensional 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 semi-definite programs (SPD), second-order 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 so-called 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 non-negativity constraints in the standard form of a linear program with so-called conic inclusion constraints we obtain the conic optimization problem min c ¢x x s.t . Ax = b x ÎC where c is an N-dimensional vector, A is a J×N matrix, b is a J-dimensional 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 Second-Order Cone Program (SOCP) One important class of cone programs is second-order 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 N-dimensional vector, A is a J×N matrix, b is a J-dimensional 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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