L-shaped algorithm for two stage problems of stochastic convex programming

In this paper we study two stage problems of stochastic convex programming. Solving the problems is very hard. A L-shaped method for it is given. The implement of the algorithm is simple, so less computation work is needed. The result of computation shows that the algorithm is effective.     

求解随机凸规划概率约束问题的对偶算法

1引言随机规划中的概率约束问题在工程和管理中有广泛的应用.因为问题中包含非线性的概率约束,它们的求解非常困难.如果目标函数是线性的,问题的求解就比较容易.给出了一个求解随机线性规划概率约束问题的综述.原-对偶算法和切平面算法是比较有效的.在本文中,我们讨论随机凸规划概率约束问题:     

Deep Learning for Constrained Utility Maximisation

This paper proposes two algorithms for solving stochastic control problems with deep learning, with a focus on the utility maximisation problem. The first algorithm solves Markovian problems via the Hamilton Jacobi Bellman (HJB) equation. We solve this highly nonlinear partial differential equation (PDE) with a second order backward stochastic differential equation (2BSDE) formulation. The convex structure of the problem allows us to describe a dual problem that can either verify the original primal approach or bypass some of the complexity. The second algorithm utilises the full power of the duality method to solve non-Markovian problems, which are often beyond the scope of stochastic control solvers in the existing literature. We solve an adjoint BSDE that satisfies the dual optimality conditions. We apply these algorithms to problems with power, log and non-HARA utilities in the Black-Scholes, the Heston stochastic volatility, and path dependent volatility models. Numerical experiments show highly accurate results with low computational cost, supporting our proposed algorithms.

     

A stochastic quasigradient algorithm with variable metric

This paper deals with a new variable metric algorithm for stochastic optimization problems. The essence of this is as follows: there exist two stochastic quasigradient algorithms working simultaneously — the first in the main space, the second with respect to the matrices that modify the space variables. Almost sure convergence of the algorithm is proved for the case of the convex (possiblynonsmooth) objective function.     

Convex inequalities in stochastic control

The purpose of this paper is to show the relation between the search for the minimal solution of certain systems of convex inequalities in potential theory and the solution of a very large class of stochastic control problems. Special attention is given to the control of alternating processes, and impulse control problems. The dual problems—in the sense of convex programming—are studied as well; and their solutions are interpreted in the framework of stochastic control theory.     

An optimal method for stochastic composite optimization

This paper considers an important class of convex programming (CP) problems, namely, the stochastic composite optimization (SCO), whose objective function is given by the summation of general nonsmooth and smooth stochastic components. Since SCO covers non-smooth, smooth and stochastic CP as certain special cases, a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming. Note however that the optimization algorithms that can achieve this lower bound had never been developed. In this paper, we show that the simple mirror-descent stochastic approximation method exhibits the best-known rate of convergence for solving these problems. Our major contribution is to introduce the accelerated stochastic approximation (AC-SA) algorithm based on Nesterov’s optimal method for smooth CP (Nesterov in Doklady AN SSSR 269:543–547, 1983; Nesterov in Math Program 103:127–152, 2005), and show that the AC-SA algorithm can achieve the aforementioned lower bound on the rate of convergence for SCO. To the best of our knowledge, it is also the first universally optimal algorithm in the literature for solving non-smooth, smooth and stochastic CP problems. We illustrate the significant advantages of the AC-SA algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems.     

A Progressive Hedging based branch-and-bound algorithm for mixed-integer stochastic programs

Progressive Hedging (PH) is a well-known algorithm for solving multi-stage stochastic convex optimization problems. Most previous extensions of PH for mixed-integer stochastic programs have been implemented without convergence guarantees. In this paper, we present a new framework that shows how PH can be utilized while guaranteeing convergence to globally optimal solutions of mixed-integer stochastic convex programs. We demonstrate the effectiveness of the proposed framework through computational experiments.     

AN ALGORITHM FOR L-SHAPED CONVEX PROGRAMMING

In this paper we study L-shaped convex programming. An algorithm for itis given. The result of computation shows that the algorithm is effective. The algorithmcan be applied to two stage problem of stochastic convex programming.     

Sto chastic Maximum Principle for Optimal Control of Forward-backward Sto chastic Pantograph Systems with Regime Switching

In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stochastic pantograph equation and modulated by a continuous-time finite-state Markov chain. By virtue of classical variational approach, duality method, and convex analysis, we obtain a stochastic maximum principle for the optimal control.     

简单补偿随机规划的一种异步并行算法

本文利用凸规划的近似分解方法，给出了求解具有简单补偿随机规划问题的一种异步并行算法．     

Comparison of two algorithms for solving a two‐stage bilinear stochastic programming problem with quantile criterion

The paper is devoted to solving the two‐stage problem of stochastic programming with quantile criterion. It is assumed that the loss function is bilinear in random parameters and strategies, and the random vector has a normal distribution. Two algorithms are suggested to solve the problem, and they are compared. The first algorithm is based on the reduction of the original stochastic problem to a mixed integer linear programming problem. The second algorithm is based on the reduction of the problem to a sequence of convex programming problems. Performance characteristics of both the algorithms are illustrated by an example. A modification of both the algorithms is suggested to reduce the computing time. The new algorithm uses the solution obtained by the second algorithm as a starting point for the first algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

Stochastic programming with simple integer recourse

Stochastic integer programs are notoriously difficult. Very few properties are known and solution algorithms are very scarce. In this paper, we introduce the class of stochastic programs with simple integer recourse, a natural extension of the simple recourse case extensively studied in stochastic continuous programs.Analytical as well as computational properties of the expected recourse function of simple integer recourse problems are studied. This includes sharp bounds on this function and the study of the convex hull. Finally, a finite termination algorithm is obtained that solves two classes of stochastic simple integer recourse problems.Supported by the National Operations Research Network in the Netherlands (LNMB).     

A gaussian upper bound for gaussian multi-stage stochastic linear programs

This paper deals with two-stage and multi-stage stochastic programs in which the right-hand sides of the constraints are Gaussian random variables. Such problems are of interest since the use of Gaussian estimators of random variables is widespread. We introduce algorithms to find upper bounds on the optimal value of two-stage and multi-stage stochastic (minimization) programs with Gaussian right-hand sides. The upper bounds are obtained by solving deterministic mathematical programming problems with dimensions that do not depend on the sample space size. The algorithm for the two-stage problem involves the solution of a deterministic linear program and a simple semidefinite program. The algorithm for the multi-stage problem invovles the solution of a quadratically constrained convex programming problem.     

Globally and superlinearly convergent trust-region algorithm for convex SC1-minimization problems and its application to stochastic programs

A function mapping from ⁿ to is called an SC¹-function if it is differentiable and its derivative is semismooth. A convex SC¹-minimization problem is a convex minimization problem with an SC¹-objective function and linear constraints. Applications of such minimization problems include stochastic quadratic programming and minimax problems. In this paper, we present a globally and superlinearly convergent trust-region algorithm for solving such a problem. Numerical examples are given on the application of this algorithm to stochastic quadratic programs.This work was supported by the Australian Research Council.We are indebted to Dr. Xiaojun Chen for help in the computation. We are grateful to two anonymous referees for their comments and suggestions, which improved the presentation of this paper.     

On Stochastic Mirror-prox Algorithms for Stochastic Cartesian Variational Inequalities: Randomized Block Coordinate and Optimal Averaging Schemes

Motivated by multi-user optimization problems and non-cooperative Nash games in uncertain regimes, we consider stochastic Cartesian variational inequality problems where the set is given as the Cartesian product of a collection of component sets. First, we consider the case where the number of the component sets is large and develop a randomized block stochastic mirror-prox algorithm, where at each iteration only a randomly selected block coordinate of the solution vector is updated through implementing two consecutive projection steps. We show that when the mapping is strictly pseudo-monotone, the algorithm generates a sequence of iterates that converges to the solution of the problem almost surely. When the maps are strongly pseudo-monotone, we prove that the mean-squared error diminishes at the optimal rate. Second, we consider large-scale stochastic optimization problems with convex objectives and develop a new averaging scheme for the randomized block stochastic mirror-prox algorithm. We show that by using a different set of weights than those employed in the classical stochastic mirror-prox methods, the objective values of the averaged sequence converges to the optimal value in the mean sense at an optimal rate. Third, we consider stochastic Cartesian variational inequality problems and develop a stochastic mirror-prox algorithm that employs the new weighted averaging scheme. We show that the expected value of a suitably defined gap function converges to zero at an optimal rate.     

A stochastic gradient type algorithm for closed-loop problems

We focus on the numerical solution of closed-loop stochastic problems, and propose a perturbed gradient algorithm to achieve this goal. The main hurdle in such problems is the fact that the control variables are infinite-dimensional, due to, e.g., the information constraints. Alternatively said, control variables are feedbacks, i.e., functions. Such controls have hence to be represented in a finite way in order to solve the problem numerically. In the same way, the gradient of the criterion is itself an infinite-dimensional object. Our algorithm replaces this exact (and unknown) gradient by a perturbed one, which consists of the product of the true gradient evaluated at a random point and a kernel function which extends this gradient to the neighbourhood of the random point. Proceeding this way, we explore the whole space iteration after iteration through random points. Since each kernel function is perfectly known by a small number of parameters, say N, the control at iteration k is perfectly known as an infinite-dimensional object by at most N × k parameters. The main strength of this method is that it avoids any discretization of the underlying space, provided that we can sample as many points as needed in this space. Moreover, our algorithm can take into account the possible measurability constraints of the problem in a new way. Finally, the randomized strategy implemented by the algorithm causes the most probable parts of the space to be the most explored ones, which is a priori an interesting feature. In this paper, we first prove two convergence results of this algorithm in the strongly convex and convex cases, and then give some numerical examples showing the interest of this method for practical stochastic optimization problems. In Memoriam: Jean-Sébastien Roy passed away July 04, 2007. He was 33 years old.     

A simple randomised algorithm for convex optimisation

We consider maximising a concave function over a convex set by a simple randomised algorithm. The strength of the algorithm is that it requires only approximate function evaluations for the concave function and a weak membership oracle for the convex set. Under smoothness conditions on the function and the feasible set, we show that our algorithm computes a near-optimal point in a number of operations which is bounded by a polynomial function of all relevant input parameters and the reciprocal of the desired precision, with high probability. As an application to which the features of our algorithm are particularly useful we study two-stage stochastic programming problems. These problems have the property that evaluation of the objective function is #P-hard under appropriate assumptions on the models. Therefore, as a tool within our randomised algorithm, we devise a fully polynomial randomised approximation scheme for these function evaluations, under appropriate assumptions on the models. Moreover, we deal with smoothing the feasible set, which in two-stage stochastic programming is a polyhedron.     

Maximum principle for optimal control problems involving impulse controls with nonsmooth data

This paper is concerned with the stochastic maximum principle for impulse optimal control problems of forward–backward systems, where the coefficients of the forward part are Lipschitz continuous. The domain of the regular controls is not necessarily convex. We establish a Pontryagins maximum principle for this control problem by applying Ekelands variational principle to a sequence of approximated control problems with smooth coefficients of the initial problems.     

Computational Experience with a New Class of Convex Underestimators: Box-constrained NLP Problems

In Akrotirianakis and Floudas (2004) we presented the theoretical foundations of a new class of convex underestimators for C ² nonconvex functions. In this paper, we present computational experience with those underestimators incorporated within a Branch-and-Bound algorithm for box-conatrained problems. The algorithm can be used to solve global optimization problems that involve C ² functions. We discuss several ways of incorporating the convex underestimators within a Branch-and-Bound framework. The resulting Branch-and-Bound algorithm is then used to solve a number of difficult box-constrained global optimization problems. A hybrid algorithm is also introduced, which incorporates a stochastic algorithm, the Random-Linkage method, for the solution of the nonconvex underestimating subproblems, arising within a Branch-and-Bound framework. The resulting algorithm also solves efficiently the same set of test problems.     

Maximum Principle for Partially-Observed Optimal Control of Fully-Coupled Forward-Backward Stochastic Systems

This paper is concerned with partially-observed optimal control problems for fully-coupled forward-backward stochastic systems. The maximum principle is obtained on the assumption that the forward diffusion coefficient does not contain the control variable and the control domain is not necessarily convex. By a classical spike variational method and a filtering technique, the related adjoint processes are characterized as solutions to forward-backward stochastic differential equations in finite-dimensional spaces. Then, our theoretical result is applied to study a partially-observed linear-quadratic optimal control problem for a fully-coupled forward-backward stochastic system and an explicit observable control variable is given.