Optimization with a class of multivariate integral stochastic order constraints

We study convex optimization problems with a class of multivariate integral stochastic order constraints defined in terms of parametrized families of increasing concave functions. We show that utility functions act as the Lagrange multipliers of the stochastic order constraints in this general setting, and that the dual problem is a search over utility functions. Practical implementation issues are discussed.

[1]  A. Müller,et al.  Comparison Methods for Stochastic Models and Risks , 2002 .

[2]  T. P. Dinh,et al.  Convex analysis approach to d.c. programming: Theory, Algorithm and Applications , 1997 .

[3]  Fabrizio Dabbene,et al.  A Randomized Cutting Plane Method with Probabilistic Geometric Convergence , 2010, SIAM J. Optim..

[4]  Jian Hu,et al.  Sample Average Approximation for Stochastic Dominance Constrained Programs , 2009 .

[5]  Gautam Mitra,et al.  Portfolio construction based on stochastic dominance and target return distributions , 2006, Math. Program..

[6]  Darinka Dentcheva,et al.  Optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints , 2004, Math. Program..

[7]  René Henrion,et al.  Stability and Sensitivity of Optimization Problems with First Order Stochastic Dominance Constraints , 2007, SIAM J. Optim..

[8]  Kenneth O. Kortanek,et al.  Semi-Infinite Programming: Theory, Methods, and Applications , 1993, SIAM Rev..

[9]  Darinka Dentcheva,et al.  Inverse stochastic dominance constraints and rank dependent expected utility theory , 2006, Math. Program..

[10]  Rüdiger Schultz,et al.  A note on second-order stochastic dominance constraints induced by mixed-integer linear recourse , 2011, Math. Program..

[11]  Gautam Mitra,et al.  Processing second-order stochastic dominance models using cutting-plane representations , 2011, Math. Program..

[12]  Giuseppe Carlo Calafiore,et al.  Uncertain convex programs: randomized solutions and confidence levels , 2005, Math. Program..

[13]  Tito Homem-de-Mello,et al.  Risk-adjusted budget allocation models with application in homeland security , 2011 .

[14]  Marco A. López,et al.  A New Exchange Method for Convex Semi-Infinite Programming , 2010, SIAM J. Optim..

[15]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

[16]  Tito Homem-de-Mello,et al.  Optimal Path Problems with Second-Order Stochastic Dominance Constraints , 2012 .

[17]  Darinka Dentcheva,et al.  Optimization with multivariate stochastic dominance constraints , 2008, SIAM J. Optim..

[18]  A. Ruszczynski,et al.  Portfolio optimization with stochastic dominance constraints , 2006 .

[19]  Darinka Dentcheva,et al.  Optimization with Stochastic Dominance Constraints , 2003, SIAM J. Optim..

[20]  Gábor Rudolf,et al.  Relaxations of linear programming problems with first order stochastic dominance constraints , 2006, Oper. Res. Lett..

[21]  Sanjay Mehrotra,et al.  A Cutting-Surface Method for Uncertain Linear Programs with Polyhedral Stochastic Dominance Constraints , 2009, SIAM J. Optim..

[22]  Rüdiger Schultz,et al.  An algorithm for stochastic programs with first-order dominance constraints induced by linear recourse , 2010, Discret. Appl. Math..

[23]  Darinka Dentcheva,et al.  Robust stochastic dominance and its application to risk-averse optimization , 2010, Math. Program..

[24]  Le Thi Hoai An,et al.  The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems , 2005, Ann. Oper. Res..

[25]  R. Phelps.  Lectures on Choquet's Theorem , 1966 .