Combining penalty‐based and Gauss–Seidel methods for solving stochastic mixed‐integer problems

Abstract

In this paper, we propose a novel decomposition approach (named PBGS) for stochastic mixed‐integer programming (SMIP) problems, which is inspired by the combination of penalty‐based Lagrangian and block Gauss–Seidel methods. The PBGS method is developed such that the inherent decomposable structure that SMIP problems present can be exploited in a computationally efficient manner. The performance of the proposed method is compared with the progressive hedging (PH) method, which also can be viewed as a Lagrangian‐based method for obtaining solutions for SMIP problems. Numerical experiments performed using instances from the literature illustrate the efficiency of the proposed method in terms of computational performance and solution quality.

A Fast Penalty-Based Gauss-Seidel Method for Stochastic Unit Commitment With Uncertain Load and Wind Generation

“…Problem ( 1)-( 8) is also known as the extensive form (EF) of the twostage S-NCUC, in which even a moderate number of scenarios can result in a computational burden that quickly exceeds the capability of any state-of-the-art MIP solver. In addition, the computational burden increases exponentially with the size of the problem using the branch-and-cut method [26]. This is why scenario-based decomposition frameworks such as PHA are used to solve a large-scale S-NCUC problem [21].…”

“…An important question is how to construct an exact augmenting function . We employ an l-1-norm-like function based on a semi-Lagrangian approach [26]. Unlike the semi-Lagrangian approach wherein an equality constraint is reformulated as a pair of inequality constraints and the Lagrangian relaxation is applied to either of the pair [30], here we reformulate NAC (9) as two inequality constraints and then relax both constraints using two penalty factors ( , ) as follows:…”

“…Calculation of , can be accomplished by minimizing (14) with respect , using fixed , over all the scenarios. Per [26], if the problem is restricted to take only binary values, we can calculate , as follows at every iteration:…”

“…where ∈ (1,2]. A detailed explanation for this acceleration factor is given in Chapter 4 of [26]. The third critical factor to be considered is how we can accelerate the PBGS solution in view that an individual scenario may have to be solved many times, each time corresponding to one PBGS iteration.…”

“…2) If an exact solution is attained, how can we find a highquality lower bound capable of measuring the quality of the exact solution efficiently? To address the above questions, this paper applies a novel Penalty-Based Gauss-Seidel (PBGS) algorithm [26] with an exact augmented Lagrangian representation to solve S-NCUC within a scenario-based decomposition framework. To improve the computational efficiency of PBGS, we propose an accelerating technique that skips solving scenarios as long as certain conditions are satisfied.…”