Low order-value approach for solving VaR-constrained optimization problems

In Low Order-Value Optimization (LOVO) problems the sum of the r smallest values of a finite sequence of q functions is involved as the objective to be minimized or as a constraint. The latter case is considered in the present paper. Portfolio optimization problems with a constraint on the admissible Value at Risk (VaR) can be modeled in terms of a LOVO problem with constraints given by Low order-value functions. Different algorithms for practical solution of this problem will be presented. Using these techniques, portfolio optimization problems with transaction costs will be solved.

[1]  José Mario Martínez,et al.  On Augmented Lagrangian Methods with General Lower-Level Constraints , 2007, SIAM J. Optim..

[2]  Georg Ch. Pflug,et al.  FINDING OPTIMAL PORTFOLIOS WITH CONSTRAINTS ON VALUE AT RISK , 2000 .

[3]  M. Teboulle,et al.  AN OLD‐NEW CONCEPT OF CONVEX RISK MEASURES: THE OPTIMIZED CERTAINTY EQUIVALENT , 2007 .

[4]  W. Marsden.  I and J , 2012 .

[5]  A. Neumaier,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs — I. Theoretical advances , 1998 .

[6]  José Mario Martínez,et al.  Global minimization using an Augmented Lagrangian method with variable lower-level constraints , 2010, Math. Program..

[7]  Zengxin Wei,et al.  On the Constant Positive Linear Dependence Condition and Its Application to SQP Methods , 1999, SIAM J. Optim..

[8]  Hubertus Th. Jongen,et al.  Slopes of shadow prices and Lagrange multipliers , 2008, Optim. Lett..

[9]  Stephen P. Boyd,et al.  Portfolio optimization with linear and fixed transaction costs , 2007, Ann. Oper. Res..

[10]  José Mario Martínez,et al.  Nonlinear-programming reformulation of the order-value optimization problem , 2005, Math. Methods Oper. Res..

[11]  Henry Wolkowicz,et al.  Large scale portfolio optimization with piecewise linear transaction costs , 2008, Optim. Methods Softw..

[12]  Jaroslava Hlouskova,et al.  Portfolio Selection and Transactions Costs , 2003, Comput. Optim. Appl..

[13]  Ronald L. Rivest,et al.  Introduction to Algorithms, Second Edition , 2001 .

[14]  Xiaoguang Yang,et al.  Optimal portfolio allocation under the probabilistic VaR constraint and incentives for financial innovation , 2008 .

[15]  Panos M. Pardalos,et al.  Encyclopedia of Optimization , 2006 .

[16]  R. Rockafellar.  Augmented Lagrange Multiplier Functions and Duality in Nonconvex Programming , 1974 .

[17]  Reiner Horst,et al.  Introduction to Global Optimization (Nonconvex Optimization and Its Applications) , 2002 .

[18]  R. Rockafellar,et al.  Conditional Value-at-Risk for General Loss Distributions , 2001 .

[19]  G. Pflug,et al.  Value-at-Risk in Portfolio Optimization: Properties and Computational Approach ⁄ , 2005 .

[20]  José Mario Martínez,et al.  Order-Value Optimization: Formulation and solution by means of a primal cauchy method , 2003, Math. Methods Oper. Res..

[21]  Anthony Man-Cho So,et al.  Stochastic Combinatorial Optimization with Controllable Risk Aversion Level , 2009, Math. Oper. Res..

[22]  Christodoulos A. Floudas,et al.  αBB: A global optimization method for general constrained nonconvex problems , 1995, J. Glob. Optim..

[23]  Rima Febrian,et al.  Optimal execution of portfolio transaction with linear cost model , 2009 .

[24]  Panos M. Pardalos,et al.  Handbook of applied optimization , 2002 .

[25]  C. Adjiman,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs—II. Implementation and computational results , 1998 .

[26]  Mario Mart,et al.  ORDER-VALUE OPTIMIZATION AND NEW APPLICATIONS , 2011 .

[27]  Christodoulos A. Floudas,et al.  A global optimization method, αBB, for process design , 1996 .

[28]  Lonnie Hamm,et al.  GLOBAL OPTIMIZATION METHODS , 2002 .

[29]  J. M. Martínez,et al.  Quasi-Newton methods for Order-value optimization and value-at-risk calculations , 2006 .

[30]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[31]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[32]  José Mario Martínez,et al.  Continuous optimization methods for structure alignments , 2007, Math. Program..

[33]  M. Hestenes.  Multiplier and gradient methods , 1969 .

[34]  Natasa Krejic,et al.  VaR optimal portfolio with transaction costs , 2011, Appl. Math. Comput..

[35]  J. M. Martínez,et al.  On sequential optimality conditions for smooth constrained optimization , 2011 .

[36]  José Mario Martínez,et al.  Low Order-Value Optimization and applications , 2009, J. Glob. Optim..

[37]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[38]  F. Lillo,et al.  Econophysics: Master curve for price-impact function , 2003, Nature.

[39]  Panos M. Pardalos,et al.  Introduction to Global Optimization , 2000, Introduction to Global Optimization.

[40]  Jean-Philippe Bouchaud,et al.  Fluctuations and Response in Financial Markets: The Subtle Nature of 'Random' Price Changes , 2003 .

[41]  José Mario Martínez,et al.  Order-value optimization and new applications , 2009 .

[42]  Erhard Reschenhofer.  Robust tests of the random walk hypothesis , 2004 .

[43]  M. J. D. Powell,et al.  A method for nonlinear constraints in minimization problems , 1969 .

[44]  Phhilippe Jorion.  Value at Risk: The New Benchmark for Managing Financial Risk , 2000 .

[45]  Nils Tönshoff,et al.  Implementation and Computational Results , 1997 .

[46]  R. Andreani,et al.  On the Relation between Constant Positive Linear Dependence Condition and Quasinormality Constraint Qualification , 2005 .

[47]  R. Mansini,et al.  An exact approach for portfolio selection with transaction costs and rounds , 2005 .

[48]  Robert Almgren,et al.  Optimal execution with nonlinear impact functions and trading-enhanced risk , 2003 .