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Papers by Andrzej Ruszczynski

Siam Journal on Optimization, 2008

ABSTRACT For stochastic optimization problems with second order stochastic dominance constraints ... more ABSTRACT For stochastic optimization problems with second order stochastic dominance constraints we develop a new form of the duality theory featuring measures on the product of the probability space and the real line. We present two formulations involving small numbers of variables and exponentially many constraints: primal and dual. The dual formulation reveals connections between dominance constraints, generalized transportation problems, and the theory of measures with given marginals. Both formulations lead to two classes of cutting plane methods. Finite convergence of both methods is proved in the case of finitely many events. Numerical results for a portfolio problem are provided.

Society for Industrial and Applied Mathematics eBooks, Jul 7, 2021

arXiv (Cornell University), Jun 18, 2022

We consider a Markov decision process subject to model uncertainty in a Bayesian framework, where... more We consider a Markov decision process subject to model uncertainty in a Bayesian framework, where we assume that the state process is observed but its law is unknown to the observer. In addition, while the state process and the controls are observed at time t, the actual cost that may depend on the unknown parameter is not known at time t. The controller optimizes total cost by using a family of special risk measures, that we call risk filters and that are appropriately defined to take into account the model uncertainty of the controlled system. These key features lead to non-standard and non-trivial risk-averse control problems, for which we derive the Bellman principle of optimality. We illustrate the general theory on two practical examples: optimal investment and clinical trials.

IFAC Proceedings Volumes, 1977

In this paper practical aspects of on-line coordination of large dynamical systems are discussed.... more In this paper practical aspects of on-line coordination of large dynamical systems are discussed. The Feasible Price Coordination (FPC) method (Ref.4) is used for step-by-step improvement of batch prooesses. It is assumed that for the purpose of control approximate mathematical models of the systems are evaluated. Two examples of systems are considered. The first example is a linear dynamical system and a second one is a chemical plant described by highly monlinear differential equations.

Society for Industrial and Applied Mathematics eBooks, Jul 1, 2015

In this paper, we provide a theory of time-consistent dynamic risk measures for finite-state part... more In this paper, we provide a theory of time-consistent dynamic risk measures for finite-state partially observable Markov decision problems. By employing our new concept of stochastic conditional time consistency, we show that such dynamic risk measures have a special structure, given by transition risk mappings as risk measures on the space of functionals on the observable state space only. Moreover, these mappings enjoy a strong monotonicity with respect to first order stochastic dominance.

Operations Research Letters, Oct 1, 1992

An augmented Lagrangian method is proposed for handling the common rows in large scale linear pro... more An augmented Lagrangian method is proposed for handling the common rows in large scale linear programming problems with block-diagonal structure and linking constraints. Using a diagonal quadratic approximation of the augmented Lagrangian one obtains subproblems that can be readily solved in parallel by a nonlinear primal-dual barrier method for convex separable programs. The combined augmented Lagrangian/barrier method applies in a natural way to stochastic programming and multicommodity networks.

arXiv (Cornell University), Mar 24, 2012

We use Markov risk measures to formulate a risk-averse version of the undiscounted total cost pro... more We use Markov risk measures to formulate a risk-averse version of the undiscounted total cost problem for a transient controlled Markov process. Using the new concept of a multikernel, we derive conditions for a system to be risk-transient, that is, to have finite risk over an infinite time horizon. We derive risk-averse dynamic programming equations satisfied by the optimal policy and we describe methods for solving these equations. We illustrate the results on an optimal stopping problem and an organ transplant problem.

arXiv (Cornell University), Jan 28, 2020

We propose a single timescale stochastic subgradient method for constrained optimization of a com... more We propose a single timescale stochastic subgradient method for constrained optimization of a composition of several nonsmooth and nonconvex functions. The functions are assumed to be locally Lipschitz and differentiable in a generalized sense. Only stochastic estimates of the values and generalized derivatives of the functions are used. The method is parameter-free. We prove convergence with probability one of the method, by associating with it a system of differential inclusions and devising a nondifferentiable Lyapunov function for this system. For problems with functions having Lipschitz continuous derivatives, the method finds a point satisfying an optimality measure with error of order N −1/2 , after executing N iterations with constant stepsize.

CRC Press eBooks, Jan 9, 2014

Mathematical Programming, May 10, 2014

Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik eBooks, Oct 16, 2009

Society for Industrial and Applied Mathematics eBooks, 2009

Econometrica, Jul 1, 2003

Springer eBooks, Dec 12, 2007

Multistage stochastic programs (MSP) are optimization problems with a dynamic (multiperiod) struc... more Multistage stochastic programs (MSP) are optimization problems with a dynamic (multiperiod) structure and uncertain parameters which are modeled as random variables. Multistage stochastic programs are among the most intractable in numericM computations. Not only does the problem size grow fast when the number of time periods and scenarios increases, but also the problem's structure is difficult to take advantage of due to numerical instability. As a consequence, few actual implcmentations of MSPs have occurred. It seems unlikely that direct solvers will be able to handle MSPs in the foreseeable future. Decomposition is the only real Mternative. But decomposition gencrally has an unpleasant tendency - the number of iterations can become unmanageable, especiMly as the dcgrceof decomposition increases and the subproblcms bccome a smaller part of tile original model. This argues against a massively parallel approach.

John Wiley & Sons, Inc. eBooks, 2004

arXiv (Cornell University), Sep 3, 2016

We prove that the bundle method for nonsmooth optimization achieves solution accuracy ε in at mos... more We prove that the bundle method for nonsmooth optimization achieves solution accuracy ε in at most O ln(1/ε)/ε iterations, if the function is strongly convex. The result is true for the versions of the method with multiple cuts and with cut aggregation.

Siam Journal on Optimization, 2008

ABSTRACT For stochastic optimization problems with second order stochastic dominance constraints ... more ABSTRACT For stochastic optimization problems with second order stochastic dominance constraints we develop a new form of the duality theory featuring measures on the product of the probability space and the real line. We present two formulations involving small numbers of variables and exponentially many constraints: primal and dual. The dual formulation reveals connections between dominance constraints, generalized transportation problems, and the theory of measures with given marginals. Both formulations lead to two classes of cutting plane methods. Finite convergence of both methods is proved in the case of finitely many events. Numerical results for a portfolio problem are provided.

Society for Industrial and Applied Mathematics eBooks, Jul 7, 2021

arXiv (Cornell University), Jun 18, 2022

We consider a Markov decision process subject to model uncertainty in a Bayesian framework, where... more We consider a Markov decision process subject to model uncertainty in a Bayesian framework, where we assume that the state process is observed but its law is unknown to the observer. In addition, while the state process and the controls are observed at time t, the actual cost that may depend on the unknown parameter is not known at time t. The controller optimizes total cost by using a family of special risk measures, that we call risk filters and that are appropriately defined to take into account the model uncertainty of the controlled system. These key features lead to non-standard and non-trivial risk-averse control problems, for which we derive the Bellman principle of optimality. We illustrate the general theory on two practical examples: optimal investment and clinical trials.

IFAC Proceedings Volumes, 1977

In this paper practical aspects of on-line coordination of large dynamical systems are discussed.... more In this paper practical aspects of on-line coordination of large dynamical systems are discussed. The Feasible Price Coordination (FPC) method (Ref.4) is used for step-by-step improvement of batch prooesses. It is assumed that for the purpose of control approximate mathematical models of the systems are evaluated. Two examples of systems are considered. The first example is a linear dynamical system and a second one is a chemical plant described by highly monlinear differential equations.

Society for Industrial and Applied Mathematics eBooks, Jul 1, 2015

In this paper, we provide a theory of time-consistent dynamic risk measures for finite-state part... more In this paper, we provide a theory of time-consistent dynamic risk measures for finite-state partially observable Markov decision problems. By employing our new concept of stochastic conditional time consistency, we show that such dynamic risk measures have a special structure, given by transition risk mappings as risk measures on the space of functionals on the observable state space only. Moreover, these mappings enjoy a strong monotonicity with respect to first order stochastic dominance.

Operations Research Letters, Oct 1, 1992

An augmented Lagrangian method is proposed for handling the common rows in large scale linear pro... more An augmented Lagrangian method is proposed for handling the common rows in large scale linear programming problems with block-diagonal structure and linking constraints. Using a diagonal quadratic approximation of the augmented Lagrangian one obtains subproblems that can be readily solved in parallel by a nonlinear primal-dual barrier method for convex separable programs. The combined augmented Lagrangian/barrier method applies in a natural way to stochastic programming and multicommodity networks.

arXiv (Cornell University), Mar 24, 2012

We use Markov risk measures to formulate a risk-averse version of the undiscounted total cost pro... more We use Markov risk measures to formulate a risk-averse version of the undiscounted total cost problem for a transient controlled Markov process. Using the new concept of a multikernel, we derive conditions for a system to be risk-transient, that is, to have finite risk over an infinite time horizon. We derive risk-averse dynamic programming equations satisfied by the optimal policy and we describe methods for solving these equations. We illustrate the results on an optimal stopping problem and an organ transplant problem.

arXiv (Cornell University), Jan 28, 2020

We propose a single timescale stochastic subgradient method for constrained optimization of a com... more We propose a single timescale stochastic subgradient method for constrained optimization of a composition of several nonsmooth and nonconvex functions. The functions are assumed to be locally Lipschitz and differentiable in a generalized sense. Only stochastic estimates of the values and generalized derivatives of the functions are used. The method is parameter-free. We prove convergence with probability one of the method, by associating with it a system of differential inclusions and devising a nondifferentiable Lyapunov function for this system. For problems with functions having Lipschitz continuous derivatives, the method finds a point satisfying an optimality measure with error of order N −1/2 , after executing N iterations with constant stepsize.

CRC Press eBooks, Jan 9, 2014

Mathematical Programming, May 10, 2014

Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik eBooks, Oct 16, 2009

Society for Industrial and Applied Mathematics eBooks, 2009

Econometrica, Jul 1, 2003

Springer eBooks, Dec 12, 2007

Multistage stochastic programs (MSP) are optimization problems with a dynamic (multiperiod) struc... more Multistage stochastic programs (MSP) are optimization problems with a dynamic (multiperiod) structure and uncertain parameters which are modeled as random variables. Multistage stochastic programs are among the most intractable in numericM computations. Not only does the problem size grow fast when the number of time periods and scenarios increases, but also the problem's structure is difficult to take advantage of due to numerical instability. As a consequence, few actual implcmentations of MSPs have occurred. It seems unlikely that direct solvers will be able to handle MSPs in the foreseeable future. Decomposition is the only real Mternative. But decomposition gencrally has an unpleasant tendency - the number of iterations can become unmanageable, especiMly as the dcgrceof decomposition increases and the subproblcms bccome a smaller part of tile original model. This argues against a massively parallel approach.

John Wiley & Sons, Inc. eBooks, 2004

arXiv (Cornell University), Sep 3, 2016

We prove that the bundle method for nonsmooth optimization achieves solution accuracy ε in at mos... more We prove that the bundle method for nonsmooth optimization achieves solution accuracy ε in at most O ln(1/ε)/ε iterations, if the function is strongly convex. The result is true for the versions of the method with multiple cuts and with cut aggregation.