Stochastic Programming

A Branch-Reduce-Cut Algorithm for the Global Optimization of Probabilistically Constrained Linear Programs

  • Shabbir Ahmed
  • Faiz Al-Khayyal
  • Myun-Seok Cheon
    • Categories Global Optimization, Stochastic Programming

    We consider probabilistic constrained linear programs with general distributions for the uncertain parameters. These problems generally involve non-convex feasible sets. We develop a branch and bound algorithm that searches for a global solution to this problem by successively partitioning the non-convex feasible region and by using bounds on the objective function to fathom inferior partitions. … Read more

    Stochastic Programming with Equilibrium Constraints

  • Alexander Shapiro
    • Categories Complementarity and Variational Inequalities, Stochastic Programming Tags , , , , , ,

    In this paper we discuss here-and-now type stochastic programs with equilibrium constraints. We give a general formulation of such problems and study their basic properties such as measurability and continuity of the corresponding integrand functions. We also discuss consistency and rates of convergence of sample average approximations of such stochastic problems. CitationSchool of Industrial and … Read more

    Solving Multistage Asset Investment Problems by the Sample Average Approximation Method

  • Jorgen Blomvall
  • Alexander Shapiro
    • Categories Finance and Economics, Stochastic Programming

    The vast size of real world stochastic programming instances requires sampling to make them practically solvable. In this paper we extend the understanding of how sampling affects the solution quality of multistage stochastic programming problems. We present a new heuristic for determining good feasible solutions for a multistage decision problem. For power and log-utility functions … Read more

    Portfolio Investment with the Exact Tax Basis via Nonlinear Programming

  • Victor DeMiguel
  • Raman Uppal
    • Categories Finance and Economics, Stochastic Programming Tags , ,

    Computing the optimal portfolio policy of an investor facing capital gains tax is a challenging problem: because the tax to be paid depends on the price at which the security was purchased (the tax basis), the optimal policy is path dependent and the size of the problem grows exponentially with the number of time periods. … Read more

    Mean-risk objectives in stochastic programming

  • Shabbir Ahmed
    • Categories Nonsmooth Optimization, Stochastic Programming Tags , , ,

    Traditional stochastic programming is risk neutral in the sense that it is concerned with the optimization of an expectation criteria. A common approach to addressing risk in decision making problems is to consider a weighted mean-risk criterion, where some dispersion statistic is used as a measure of risk. We investigate the computational suitability of various … Read more

    Solving Nonlinear Portfolio Optimization Problems with the Primal-Dual Interior Point Method

  • Jacek Gondzio
  • Andreas Grothey
    • Categories Finance and Economics, Nonlinear Optimization, Stochastic Programming Tags , , ,

    Stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning. The deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets, time stages and scenarios per time stage. So far models treated by mathematical programming approaches have been limited to simple linear … Read more

    Conditional Risk Mappings

  • Andrzej Ruszczyński
  • Alexander Shapiro
    • Categories Finance and Economics, Stochastic Programming Tags , , , , ,

    We introduce an axiomatic definition of a conditional convex risk mapping and we derive its properties. In particular, we prove a representation theorem for conditional risk mappings in terms of conditional expectations. We also develop dynamic programming relations for multistage optimization problems involving conditional risk mappings. CitationPreprintArticleDownload View PDF

    Convexification of Stochastic Ordering

  • Darinka Dentcheva
  • Andrzej Ruszczyński
    • Categories Convex and Nonsmooth Optimization, Finance and Economics, Stochastic Programming Tags ,

    We consider sets defined by the usual stochastic ordering relation and by the second order stochastic dominance relation. Under fairy general assumptions we prove that in the space of integrable random variables the closed convex hull of the first set is equal to the second set. ArticleDownload View PDF

    Portfolio Optimization with Stochastic Dominance Constraints

  • Darinka Dentcheva
  • Andrzej Ruszczyński
    • Categories Convex Optimization, Finance and Economics, Stochastic Programming Tags , , ,

    We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for these models. We construct equivalent optimization models with utility functions. Numerical illustration … Read more

    Optimization of Convex Risk Functions

  • Andrzej Ruszczyński
  • Alexander Shapiro
    • Categories Finance and Economics, Stochastic Programming Tags , , , ,

    We consider optimization problems involving convex risk functions. By employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models, and optimality and duality theory for problems involving risk functions. CitationPreprintArticleDownload View PDF