Regularized Decomposition of Stochastic Programs: Algorithmic Techniques and Numerical Results

A finitely convergent non-simplex method for large scale structured linear programming problems arising in stochastic programming is presented. The method combines the ideas of the Dantzig-Wolfe decomposition principle and modern nonsmooth optimization methods. Algorithmic techniques taking advantage of properties of stochastic programs are described and numerical results for large real world problems reported

An interior-point and decomposition approach to multiple stage stochastic programming

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Separable approximations and decomposition methods for the augmented Lagrangian

In this paper we study decomposition methods based on separable approximations for minimizing the augmented Lagrangian. In particular, we study and compare the Diagonal Quadratic Approximation Method (DQAM) of Mulvey and Ruszczyński [13] and the Parallel Coordinate Descent Method (PCDM) of Richtárik and Takáč [23]. We show that the two methods are equivalent for feasibility problems up to the selection of a single step-size parameter. Furthermore, we prove an improved complexity bound for PCDM under strong convexity, and show that this bound is at least 8(L ′ / ¯ L)(ω − 1) 2 times better than the best known bound for DQAM, where ω is the degree of partial separability and L ′ and ¯ L are the maximum and average of the block Lipschitz constants of the gradient of the quadratic penalty appearing in the augmented Lagrangian.

Volumetric center method for stochastic convex programs using sampling

We develop an algorithm for solving the stochastic convex program (SCP) by combining Vaidya's volumetric center interior point method (VCM) for solving non-smooth convex programming problems with the Monte-Carlo sampling technique to compute a subgradient. A near-central cut variant of VCM is developed, and for this method an approach to perform bulk cut translation, and adding multiple cuts is given. We show that by using near-central VCM the SCP can be solved to a desirable accuracy with any given probability. For the two-stage SCP the solution time is independent of the number of scenarios

An interior-point and decomposition approach to multiple stage stochastic programming

There is no abstract of this repor

Multistage quadratic stochastic programming

Multistage stochastic programming is an important tool in medium to long term planning where there are uncertainties in the data. In this thesis, we consider a special case of multistage stochastic programming in which each subprogram is a convex quadratic program. The results are also applicable if the quadratic objectives are replaced by convex piecewise quadratic functions. Convex piecewise quadratic functions have important application in financial planning problems as they can be used as very flexible risk measures. The stochastic programming problems can be used as multi-period portfolio planning problems tailored to the need of individual investors. Using techniques from convex analysis and sensitivity analysis, we show that each subproblem of a multistage quadratic stochastic program is a polyhedral piecewise quadratic program with convex Lipschitz objective. The objective of any subproblem is differentiable with Lipschitz gradient if all its descendent problems have unique dual variables, which can be guaranteed if the linear independence constraint qualification is satisfied. Expression for arbitrary elements of the subdifferential and generalized Hessian at a point can be calculated for quadratic pieces that are active at the point. Generalized Newton methods with linesearch are proposed for solving multistage quadratic stochastic programs. The algorithms converge globally. If the piecewise quadratic objective is differentiable and strictly convex at the solution, then convergence is also finite. A generalized Newton algorithm is implemented in Matlab. Numerical experiments have been carried out to demonstrate its effectiveness. The algorithm is tested on random data with 3, 4 and 5 stages with a maximum of 315 scenarios. The algorithm has also been successfully applied to two sets of test data from a capacity expansion problem and a portfolio management problem. Various strategies have been implemented to improve the efficiency of the proposed algorithm. We experimented with trust region methods with different parameters, using an advanced solution from a smaller version of the original problem and sorting the stochastic right hand sides to encourage faster convergence. The numerical results show that the proposed generalized Newton method is a highly accurate and effective method for multistage quadratic stochastic programs. For problems with the same number of stages, solution times increase linearly with the number of scenarios

Decomposition and duality based approaches to stochastic integer programming

Stochastic Integer Programming is a variant of Linear Programming which incorporates integer and stochastic properties (i.e. some variables are discrete, and some properties of the problem are randomly determined after the first-stage decision). A Stochastic Integer Program may be rewritten as an equivalent Integer Program with a characteristic structure, but is often too large to effectively solve directly. In this thesis we develop new algorithms which exploit convex duality and scenario-wise decomposition of the equivalent Integer Program to find better dual bounds and faster optimal solutions. A major attraction of this approach is that these algorithms will be amenable to parallel computation

Modelling dynamic stochastic user equilibrium for urban road networks

In this study a dynamic assignment model is developed which estimates travellers' route and departure time choices and the resulting time varying traffic patterns during the morning peak. The distinctive feature of the model is that it does not restrict the geometry of the network to specific forms. The proposed framework of analysis consists of a travel time model, a demand model and a demand adjustment mechanism. Two travel time models are proposed. The first is based on elementary relationships from traffic flow theory and provides the framework for a macroscopic simulation model which calculates the time varying flow patterns and link travel times given the time dependent departure rate distributions; the second is based on queueing theory and models roads as bottlenecks through which traffic flow is either uncongested or fixed at a capacity independent of traffic density. The demand model is based on the utility maximisation decision rule and defines the time dependent departure rates associated with each reasonable route connecting, the O-D pairs of the network, given the total utility associated with each combination of departure time and route. Travellers' choices are assumed to result from the trade-off between travel time and schedule delay and each individual is assumed to first choose a departure time t, and then select a reasonable route, conditional on the choice of t. The demand model has therefore the form of a nested logit. The demand adjustment mechanism is derived from a Markovian model, and describes the day-to-day evolution of the departure rate distributions. Travellers are assumed to modify their trip choice decisions based on the information they acquire from recent trips. The demand adjustment mechanism is used in order to find the equilibrium state of the system, defined as the state at which travellers believe that they cannot increase their utility of travel by unilaterally changing route or departure time. The model outputs exhibit the characteristics of real world traffic patterns observed during the peak, i. e., time varying flow patterns and travel times which result from time varying departure rates from the origins. It is shown that increasing the work start time flexibility results in a spread of the departure rate distributions over a longer period and therefore reduces the level of congestion in the network. Furthermore, it was shown that increasing the total demand using the road network results in higher levels of congestion and that travellers tend to depart earlier in an attempt to compensate for the increase in travel times. Moreover, experiments using the queueing theory based travel time model have shown that increasing the capacity of a bottleneck may cause congestion to develop downstream, which in turn may result in an increase of the average travel time for certain O-D pairs. The dynamic assignment model is also applied to estimate the effects that different road pricing policies may have on trip choices and the level of congestion; the model is used to demonstrate the development of the shifting peak phenomenon. Furthermore, the effect of information availability on the traffic patterns is investigated through a number of experiments using the developed dynamic assignment model and assuming that guided drivers form a class of users characterised by lower variability of preferences with respect to route choice

Complementarity and related problems

In this thesis, we present results related to complementarity problems. We study the linear complementarity problems on extended second order cones. We convert a linear complementarity problem on an extended second order cone into a mixed complementarity problem on the non-negative orthant. We present algorithms for this problem, and exemplify it by a numerical example. Following this result, we explore the stochastic version of this linear complementarity problem. Finally, we apply complementarity problems on extended second order cones in a portfolio optimisation problem. In this application, we exploit our theoretical results to find an analytical solution to a new portfolio optimisation model. We also study the spherical quasi-convexity of quadratic functions on spherically self-dual convex sets. We start this study by exploring the characterisations and conditions for the spherical positive orthant. We present several conditions characterising the spherical quasi-convexity of quadratic functions. Then we generalise the conditions to the spherical quasi-convexity on spherically self-dual convex sets. In particular, we highlight the case of spherical second order cones