Penalized sample average approximation methods for stochastic programs in economic and secure dispatch of a power system

In this paper, we develop a stochastic programming model for economic dispatch of a power system with operational reliability and risk control constraints. By defining a severity-index function, we propose to use conditional value-at-risk (CVaR) for measuring the reliability and risk control of the system. The economic dispatch is subsequently formulated as a stochastic program with CVaR constraint. To solve the stochastic optimization model, we propose a penalized sample average approximation (SAA) scheme which incorporates specific features of smoothing technique and level function method. Under some moderate conditions, we demonstrate that with probability approaching to 1 at an exponential rate with the increase of sample size, the optimal solution of the smoothing SAA problem converges to its true counterpart. Numerical tests have been carried out for a standard IEEE-30 DC power system.     

Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization

In this paper, we consider the stochastic mathematical programs with linear complementarity constraints, which include two kinds of models called here-and-now and lower-level wait-and-see problems. We present a combined smoothing implicit programming and penalty method for the problems with a finite sample space. Then, we suggest a quasi-Monte Carlo approximation method for solving a problem with continuous random variables. A comprehensive convergence theory is included as well. We further report numerical results with the so-called picnic vender decision problem.     

Generalized solution in singular stochastic control: The nondegenerate problem

This paper is concerned with singular stochastic control for non-degenerate problems. It generalizes the previous work in that the model equation is nonlinear and the cost function need not be convex. The associated dynamic programming equation takes the form of variational inequalities. By combining the principle of dynamic programming and the method of penalization, we show that the value function is characterized as a unique generalized (Sobolev) solution which satisfies the dynamic programming variational inequality in the almost everywhere sense. The approximation for our singular control problem is given in terms of a family of penalized control problems. As a result of such a penalization, we obtain that the value function is also the minimum cost available when only the admissible pairs with uniformly Lipschitz controls are admitted in our cost criterion.     

Efficient sample sizes in stochastic nonlinear programming

We consider a class of stochastic nonlinear programs for which an approximation to a locally optimal solution is specified in terms of a fractional reduction of the initial cost error. We show that such an approximate solution can be found by approximately solving a sequence of sample average approximations. The key issue in this approach is the determination of the required sequence of sample average approximations as well as the number of iterations to be carried out on each sample average approximation in this sequence. We show that one can express this requirement as an idealized optimization problem whose cost function is the computing work required to obtain the required error reduction. The specification of this idealized optimization problem requires the exact knowledge of a few problems and algorithm parameters. Since the exact values of these parameters are not known, we use estimates, which can be updated as the computation progresses. We illustrate our approach using two numerical examples from structural engineering design.     

A smoothing SAA method for a stochastic mathematical program with complementarity constraints

A smoothing sample average approximation (SAA) method based on the log-exponential function is proposed for solving a stochastic mathematical program with complementarity constraints (SMPCC) considered by Birbil et al. (S. I. Birbil, G. Gürkan, O. Listes: Solving stochastic mathematical programs with complementarity constraints using simulation, Math. Oper. Res. 31 (2006), 739–760). It is demonstrated that, under suitable conditions, the optimal solution of the smoothed SAA problem converges almost surely to that of the true problem as the sample size tends to infinity. Moreover, under a strong second-order sufficient condition for SMPCC, the almost sure convergence of Karash-Kuhn-Tucker points of the smoothed SAA problem is established by Robinson’s stability theory. Some preliminary numerical results are reported to show the efficiency of our method.     

Optimality functions in stochastic programming

Optimality functions define stationarity in nonlinear programming, semi-infinite optimization, and optimal control in some sense. In this paper, we consider optimality functions for stochastic programs with nonlinear, possibly nonconvex, expected value objective and constraint functions. We show that an optimality function directly relates to the difference in function values at a candidate point and a local minimizer. We construct confidence intervals for the value of the optimality function at a candidate point and, hence, provide a quantitative measure of solution quality. Based on sample average approximations, we develop an algorithm for classes of stochastic programs that include CVaR-problems and utilize optimality functions to select sample sizes.     

Multi-stage portfolio selection problem with dynamic stochastic dominance constraints

We study the multi-stage portfolio selection problem where the utility function of an investor is ambiguous. The ambiguity is characterized by dynamic stochastic dominance constraints, which are able to capture the dynamics of the random return sequence during the investment process. We propose a multi-stage dynamic stochastic dominance constrained portfolio selection model, and use a mixed normal distribution with time-varying weights and the K-means clustering technique to generate a scenario tree for the transformation of the proposed model. Based on the scenario tree representation, we derive two linear programming approximation problems, using the sampling approach or the duality theory, which provide an upper bound approximation and a lower bound approximation for the original nonconvex problem. The upper bound is asymptotically tight with infinitely many samples. Numerical results illustrate the practicality and efficiency of the proposed new model and solution techniques.

     

Sample approximation technique for mixed-integer stochastic programming problems with expected value constraints

This paper deals with the theory of sample approximation techniques applied to stochastic programming problems with expected value constraints. We extend the results of Branda (Optimization 61(8):949–968, 2012c) and Wang and Ahmed (Oper Res Lett 36:515–519, 2008) on the rates of convergence to the problems with a mixed-integer bounded set of feasible solutions and several expected value constraints. Moreover, we enable non-iid sampling and consider Hölder-calmness of the constraints. We derive estimates on the sample size necessary to get a feasible solution or a lower bound on the optimal value of the original problem using the sample approximation. We present an application of the estimates to an investment problem with the Conditional Value at Risk constraints, integer allocations and transaction costs.     

Two-stage stochastic variational inequalities: an ERM-solution procedure

We propose a two-stage stochastic variational inequality model to deal with random variables in variational inequalities, and formulate this model as a two-stage stochastic programming with recourse by using an expected residual minimization solution procedure. The solvability, differentiability and convexity of the two-stage stochastic programming and the convergence of its sample average approximation are established. Examples of this model are given, including the optimality conditions for stochastic programs, a Walras equilibrium problem and Wardrop flow equilibrium. We also formulate stochastic traffic assignments on arcs flow as a two-stage stochastic variational inequality based on Wardrop flow equilibrium and present numerical results of the Douglas–Rachford splitting method for the corresponding two-stage stochastic programming with recourse.     

Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications

Inspired by a recent work by Alexander et al. (J Bank Finance 30:583–605, 2006) which proposes a smoothing method to deal with nonsmoothness in a conditional value-at-risk problem, we consider a smoothing scheme for a general class of nonsmooth stochastic problems. Assuming that a smoothed problem is solved by a sample average approximation method, we investigate the convergence of stationary points of the smoothed sample average approximation problem as sample size increases and show that w.p.1 accumulation points of the stationary points of the approximation problem are weak stationary points of their counterparts of the true problem. Moreover, under some metric regularity conditions, we obtain an error bound on approximate stationary points. The convergence result is applied to a conditional value-at-risk problem and an inventory control problem.