7,076 research outputs found
Leveraging Decision Diagrams to Solve Two-stage Stochastic Programs with Binary Recourse and Logical Linking Constraints
Two-stage stochastic programs with binary recourse are challenging to solve
and efficient solution methods for such problems have been limited. In this
work, we generalize an existing binary decision diagram-based (BDD-based)
approach of Lozano and Smith (Math. Program., 2018) to solve a special class of
two-stage stochastic programs with binary recourse. In this setting, the
first-stage decisions impact the second-stage constraints. Our modified problem
extends the second-stage problem to a more general setting where logical
expressions of the first-stage solutions enforce constraints in the second
stage. We also propose a complementary problem and solution method which can be
used for many of the same applications. In the complementary problem we have
second-stage costs impacted by expressions of the first-stage decisions. In
both settings, we convexify the second-stage problems using BDDs and
parametrize either the arc costs or capacities of these BDDs with first-stage
solutions depending on the problem. We further extend this work by
incorporating conditional value-at-risk and we propose, to our knowledge, the
first decomposition method for two-stage stochastic programs with binary
recourse and a risk measure. We apply these methods to a novel stochastic
dominating set problem and present numerical results to demonstrate the
effectiveness of the proposed methods
A decomposition algorithm for two-stage stochastic programs with nonconvex recourse
In this paper, we have studied a decomposition method for solving a class of
nonconvex two-stage stochastic programs, where both the objective and
constraints of the second-stage problem are nonlinearly parameterized by the
first-stage variable. Due to the failure of the Clarke regularity of the
resulting nonconvex recourse function, classical decomposition approaches such
as Benders decomposition and (augmented) Lagrangian-based algorithms cannot be
directly generalized to solve such models. By exploring an implicitly
convex-concave structure of the recourse function, we introduce a novel
decomposition framework based on the so-called partial Moreau envelope. The
algorithm successively generates strongly convex quadratic approximations of
the recourse function based on the solutions of the second-stage convex
subproblems and adds them to the first-stage master problem. Convergence under
both fixed scenarios and interior samplings is established. Numerical
experiments are conducted to demonstrate the effectiveness of the proposed
algorithm
A comment on "computational complexity of stochastic programming problems"
Although stochastic programming problems were always believed to be computationally challenging, this perception has only recently received a theoretical justification by the seminal work of Dyer and Stougie (Math Program A 106(3):423–432, 2006). Amongst others, that paper argues that linear two-stage stochastic programs with fixed recourse are #P-hard even if the random problem data is governed by independent uniform distributions. We show that Dyer and Stougie’s proof is not correct, and we offer a correction which establishes the stronger result that even the approximate solution of such problems is #P-hard for a sufficiently high accuracy. We also provide new results which indicate that linear two-stage stochastic programs with random recourse seem even more challenging to solve
Sampling-Based Algorithms for Two-Stage Stochastic Programs and Applications
In this dissertation, we present novel sampling-based algorithms for solving two-stage stochastic programming problems. Sampling-based methods provide an efficient approach to solving large-scale stochastic programs where uncertainty is possibly defined on continuous support. When sampling-based methods are employed, the process is usually viewed in two steps - sampling and optimization. When these two steps are performed in sequence, the overall process can be computationally very expensive. In this dissertation, we utilize the framework of internal-sampling where sampling and optimization steps are performed concurrently. The dissertation comprises of two parts. In the first part, we design a new sampling technique for solving two-stage stochastic linear programs with continuous recourse. We incorporate this technique within an internal-sampling framework of stochastic decomposition. In the second part of the dissertation, we design an internal-sampling-based algorithm for solving two-stage stochastic mixed-integer programs with continuous recourse. We design a new stochastic branch-and-cut procedure for solving this class of optimization problems. Finally, we show the efficiency of this method for solving large-scale practical problems arising in logistics and finance
Design and Implementation of a Stochastic Programming Optimizer with Recourse and Tenders
This paper serves two purposes, to which we give equal emphasis. First, it describes an optimization system for solving large-scale stochastic linear programs with simple (i.e. decision-free in the second stage) recourse and stochastic right-hand-side elements. Second, it is a study of the means whereby large-scale Mathematical Programming Systems may be readily extended to handle certain forms of uncertainty, through post-optimal options akin to sensitivity on parametric analysis, which we term "recourse analysis". This latter theme (implicit throughout the paper) is explored in a proselytizing manner, in the concluding section
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