Linear Recourse Research Articles

Improving constants of strong convexity in linear stochastic programming

We derive formulas for constants of strong convexity (CSCs) of expectation functions encountered in two-stage stochastic programs with linear recourse. One of them yields a CSC as the optimal value of a certain quadratically constrained quadratic program, another one in terms of the thickness of the feasibility polytope of the dual problem associated to the recourse problem. CSCs appear in Hoelder-type estimates relating the distance of optimal solution sets of stochastic programs to a suitable distance of underlying probability distributions.

Economic Dispatch of Integrated Energy Systems With Robust Thermal Comfort Management

The heat and electricity integrated energy systems (HE-IESs) have a bright prospect in future energy systems owing to their excellent economic and environmental performance. Many uncertain factors in the HE-IESs, such as renewable energy sources, electrical load, and weather factors, threaten the secure and economical operation of systems as well as the thermal comfort of end-users. In this article, the uncertainties that affect the thermal comfort of end-users are modeled comprehensively, including building parameters, weather factors, and human behavior. Then, a two-stage adaptive robust economic dispatch (ARED) model with a robust thermal comfort management strategy is proposed for the operation of HE-IESs, where different policies are adopted to deal with multiple uncertainties based on their different properties. The ARED model is four-level programming with a static robust inner problem, making it computationally intractable. Hence, a sampling-based approximation method is proposed for the robust inner problem to convert the ARED model into a two-stage robust optimization with a linear recourse problem, and then the column-and-constraint generation method is used to solve it. Simulation results demonstrate the effectiveness of the proposed approach.

Adaptive Robust Dispatch of Integrated Energy System Considering Uncertainties of Electricity and Outdoor Temperature

The integrated energy system (IES) has bright prospects in engineering applications for its excellent performance in energy efficiency and renewable energy consumption. In the IES, the thermal comfort is affected by the heating power, buildings parameters, and outdoor temperatures simultaneously. Therefore, the uncertainty of the outdoor temperature will bring some adverse effects on thermal comfort, which need to be considered in the dispatch decision of the IES. In this article, we propose a day-ahead adaptive robust dispatch model (ARDM) for the IES to make a dispatch plan under the uncertainties of the net electrical-load and outdoor temperature, with the aim of guaranteeing the safe operation of IES and the thermal comfort of end-users. The thermal dynamic characteristics of the district heating network and buildings are utilized to provide operational flexibility and improve economic performance. To decrease the conservatism of dispatch results, the multi-interval uncertainty set is introduced to model the uncertainties. The ARDM model is a two-stage robust optimization with a linear recourse problem, and the column-and-constraint generation method is used to solve it. Two cases of different scale are studied to verify the effectiveness and advantages of the proposed method.

Strong convexity in risk-averse stochastic programs with complete recourse

We give sufficient conditions for the expected excess and the mean-upper-semideviation of recourse functions to be strongly convex. This is done in the setting of two-stage stochastic programs with complete linear recourse and random right-hand side. This work extends results on strong convexity of risk-neutral models.

A model of distributionally robust two-stage stochastic convex programming with linear recourse

We consider distributionally robust two-stage stochastic convex programming problems, in which the recourse problem is linear. Other than analyzing these new models case by case for different ambiguity sets, we adopt a unified form of ambiguity sets proposed by Wiesemann, Kuhn and Sim, and extend their analysis from a single stochastic constraint to the two-stage stochastic programming setting. It is shown that under a standard set of regularity conditions, this class of problems can be converted to a conic optimization problem. Numerical results are presented to show the efficiency of the distributionally robust approach.

Lipschitzian Properties and Stability of a Class of First-Order Stochastic Dominance Constraints

Considering first-order stochastic dominance constraints for random variables arising as optimal values of stochastic programs with linear recourse, verifiable sufficient conditions for metric regularity are presented. A growth condition developed in [R. Henrion and W. Römisch, Math. Program., 84 (1999), pp. 55--88] has a crucial role in the analysis of the present paper. Implications regarding stability and sensitivity of optimal values and optimal solutions of stochastic optimization problems involving the dominance constraints considered conclude the paper.

An SDP approach for multiperiod mixed 0–1 linear programming models with stochastic dominance constraints for risk management

In this paper we consider multiperiod mixed 0–1 linear programming models under uncertainty. We propose a risk averse strategy using stochastic dominance constraints (SDC) induced by mixed-integer linear recourse as the risk measure. The SDC strategy extends the existing literature to the multistage case and includes both the first-order and second-order constraints. We propose a stochastic dynamic programming (SDP) solution approach, where one has to overcome the negative impact of the cross-scenario constraints on the decomposability of the model. In our computational experience we compare our SDP approach against a commercial optimization package, in terms of solution accuracy and elapsed time. We use supply chain planning instances, where procurement, production, inventory, and distribution decisions need to be made under demand uncertainty. We confirm the hardness of the testbed, where the benchmark cannot find a feasible solution for half of the test instances while we always find one, and show the appealing tradeoff of SDP, in terms of solution accuracy and elapsed time, when solving medium-to-large instances.

An algorithm for stochastic programs with first-order dominance constraints induced by linear recourse

We derive a cutting plane decomposition method for stochastic programs with first-order dominance constraints induced by linear recourse models with continuous variables in the second stage.

A note on second-order stochastic dominance constraints induced by mixed-integer linear recourse

We introduce stochastic integer programs with second-order dominance constraints induced by mixed-integer linear recourse. Closedness of the constraint set mapping with respect to perturbations of the underlying probability measure is derived. For discrete probability measures, large-scale, block-structured, mixed- integer linear programming equivalents to the dominance constrained stochastic programs are identified. For these models, a decomposition algorithm is proposed and tested with instances from power optimization.

Stochastic Programs with First-Order Dominance Constraints Induced by Mixed-Integer Linear Recourse

We propose a new class of stochastic integer programs whose special features are dominance constraints induced by mixed-integer linear recourse. For these models, we establish closedness of the constraint set mapping with the underlying probability measure as a parameter. In the case of finite probability spaces, the models are shown to be equivalent to large-scale, block-structured, mixed-integer linear programs. We propose a decomposition algorithm for the latter and discuss computational results.

Applying the minimax criterion in stochastic recourse programs

We consider an optimization problem in which some uncertain parameters are replaced by random variables. The minimax approach to stochastic programming concerns the problem of minimizing the worst expected value of the objective function with respect to the set of probability measures that are consistent with the available information on the random data. Only very few practicable solution procedures have been proposed for this problem and the existing ones rely on simplifying assumptions. In this paper, we establish a number of stability results for the minimax stochastic program, justifying in particular the approach of restricting attention to probability measures with support in some known finite set. Following this approach, we elaborate solution procedures for the minimax problem in the setting of two-stage stochastic recourse models, considering the linear recourse case as well as the integer recourse case. Since the solution procedures are modifications of well-known algorithms, their efficacy is immediate from the computational testing of these procedures and we do not report results of any computational experiments.

Scenario formulation in an algebraic modelling language

Algebraic modelling languages have simplified management of many types of large linear programs but have not specifically supported stochastic modelling. This paper considers modelling language support for multistage stochastic linear recourse problems with finite distributions. We describe basic language requirements for formulation of finite event trees in algebraic modelling languages and show representative problems in AMPL using three commonly used scenario types.

When Is a Multistage Stochastic Programming Problem Well-Defined?

Certain measure-theoretic issues raise the possibility that (a) the optimal value of a multistage stochastic programming problem may be ill-defined and (b) the recursion defining the problem may fail, so the problem itself is not even defined. The first difficulty is illustrated by example. A rigorous definition of multistage stochastic programming with fixed linear recourse is shown to avoid this difficulty. In the context of the new definition, certain measurability, convexity, and lower-semicontinuity assumptions on the objective function preclude the second possibility.