A Gradient-Based Method for Joint Chance-Constrained Optimization with Continuous Distributions
(2024)
The input parameters of an optimization problem are often affected by uncertainties. Chance constraints are a common way to model stochastic uncertainties in the constraints. Typically, algorithms for solving chance-constrained problems require convex functions or discrete probability distributions. In this work, we go one step further and allow non-convexities as well as continuous distributions. We propose a gradient-based approach to approximately solve joint chance-constrained models. We approximate the original problem by smoothing indicator functions. Then, the smoothed chance constraints are relaxed by penalizing their violation in the objective function. The approximation problem is solved with the Continuous Stochastic Gradient method that is an enhanced version of the stochastic gradient descent and has recently been introduced in the literature. We present a convergence theory for the smoothing and penalty approximations. Under very mild assumptions, our approach is applicable to a wide range of chance-constrained optimization problems. As an example, we illustrate its computational efficiency on difficult practical problems arising in the operation of gas networks. The numerical experiments demonstrate that the approach quickly finds nearly feasible solutions for joint chance-constrained problems with non-convex constraint functions and continuous distributions, even for realistically-sized instances.
Branch-and-cut for mixed-integer robust chance-constrained optimization with discrete distributions
(2025)
We study robust chance-constrained problems with mixed-integer design variables and ambiguity sets consisting of discrete probability distributions. Allowing general non-convex constraint functions, we develop a branch-and-cut framework using scenario-based cutting planes to generate lower bounds. The cutting planes are obtained by exploiting the classical big-M reformulation of the chance-constrained problem in the case of discrete distributions. Furthermore, we include the calculation of initial feasible solutions based on a bundle method applied to an approximation of the original problem into the branch-and-cut procedure. We conclude with a detailed discussion about the practical performance of the branch-and-cut framework with and without initial feasible solutions. In our experiments we focus on gas transport problems under uncertainty and provide a comparison of our method with solving the classical reformulation directly for various real-world sized instances.
Typically, probability distributions that generate uncertain parameters cannot be measured exactly in practice. As a remedy, distributional robustness determines optimized decisions that are protected in a robust fashion against all probability distributions in some appropriately chosen ambiguity set. In this work, we consider robust joint chance-constrained optimization problems and focus on discrete probability distributions. Many methods for this kind of problems study convex or even linear constraint functions. In contrast, we introduce a practically efficient scenario-based bundle method without convexity assumptions on the constraint functions. We start by deriving an approximation problem to the original robust chance-constrained version by using smoothing and penalization techniques that build on our former work on chance-constrained optimization. Our convergence results with respect to the smoothing approximation and well-known results for penalty approximations suggest replacing the original problem with the approximation problem for large smoothing and penalty parameters. Our scenario-based bundle method starts by solving the approximation problem with a bundle method, and then uses the bundle solution to decide which scenarios to include in a scenario-expanded formulation. This formulation is a standard nonlinear optimization problem. Our approach is guaranteed to find feasible solutions. Furthermore, in the numerical experiments on real-world gas transport problems with uncertain demands, we mostly find globally optimal solutions. Comparing these results to the classical robust reformulations for ambiguity sets consisting of confidence intervals and Wasserstein balls, we observe that the scenario-based bundle method typically outperforms solving the classical reformulation directly.
Constructing ambiguity sets in distributionally robust optimization is difficult and currently receives increased attention. In this paper, we focus on mixture models with finitely many reference distributions. We present two different solution concepts for robust joint chance-constrained optimization problems with these ambiguity sets and non-convex constraint functions. Both concepts rely on solving an approximation problem that is based on well-known smoothing and penalization techniques. On the one side, we consider a classical bundle method together with an approach for finding good starting points. On the other side, we integrate the Continuous Stochastic Gradient method, a variant of the stochastic gradient descent that is able to exploit regularity in the data. On the example of gas networks we compare the two algorithmic concepts for different topologies and two types of mixture ambiguity sets with Gaussian reference distributions and polyhedral and ϕ-divergence based feasible sets for the mixing coefficients. The results show that both solution approaches are well-suited to solve this difficult problem class. Based on the numerical results we provide some general advices for choosing the more efficient algorithm depending on the main challenges of the considered optimization problem. We give an outlook for the applicability of the method in a wider context.