Benders decomposition with adaptive oracles for large scale optimization

Abstract

This paper proposes an algorithm to efficiently solve large optimization problems which exhibit a column bounded block-diagonal structure, where subproblems differ in right-hand side and cost coefficients. Similar problems are often tackled using cutting-plane algorithms, which allow for an iterative and decomposed solution of the problem. When solving subproblems is computationally expensive and the set of subproblems is large, cutting-plane algorithms may slow down severely. In this context we propose two novel adaptive oracles that yield inexact information, potentially much faster than solving the subproblem. The first adaptive oracle is used to generate inexact but valid cutting planes, and the second adaptive oracle gives a valid upper bound of the true optimal objective. These two oracles progressively “adapt” towards the true exact oracle if provided with an increasing number of exact solutions, stored throughout the iterations. These adaptive oracles are embedded within a Benders-type algorithm able to handle inexact information. We compare the Benders with adaptive oracles against a standard Benders algorithm on a stochastic investment planning problem. The proposed algorithm shows the capability to substantially reduce the computational effort to obtain an $$\epsilon $$ ϵ -optimal solution: an illustrative case is 31.9 times faster for a $$1.00\%$$ 1.00 % convergence tolerance and 15.4 times faster for a $$0.01\%$$ 0.01 % tolerance.

Decomposition methods for multi-horizon stochastic programming

“…In the Benders reduced master problem, we solve for all strategic nodes, and the operational problems are the Benders subproblems. In addition, if W O ist is the same in all nodes, and the operational problem has certain properties, one can improve Benders decomposition by avoiding solving all operational problems at each iteration, such as the adaptive Benders decomposition (Mazzi et al, 2020;Zhang et al, 2022). These approaches also utilise the property of MHSP that Benders subproblems are independent.…”

“…Fewer decomposition methods for MHSP have been proposed. (Mazzi et al, 2020) proposed adaptive Benders decomposition to solve large scale optimisation problems with column bounded block-diagonal structure, where subproblems differ in right-hand side and cost coefficients. MHSP belongs to this class of optimisation problems if it is formulated using a node formulation and the operational scenarios are identical in all nodes.…”

“…They apply the adaptive Benders decomposition to solve a stochastic investment planning problem, and show that the computational time reduces significantly. The limitation of (Mazzi et al, 2020) is that adaptive Benders cannot solve problems where operational scenarios are different in each node. (Zhang et al, 2022) proposed stabilised adaptive Benders decomposition to solve MHSP, and apply it to solve a large-scale power system planning problem.…”

Modelling and analysis of offshore energy hubs

“…The model includes a long-term investment planning horizon and a short-term operational horizon. The integrated investment planning and operational model is partially based upon the LP model developed in [27].…”

Stochastic Hybrid Approximation for Uncertainty Management in Gas-Electric Systems

“…Third, this algorithm requires solving the second stage problem for every scenario ω at each iteration, as it is not known a priori which of the scenarios will provide cuts that improve the approximation. There exists variations on the algorithm that can merge cuts to make the first stage problem more efficient [36] or identify subproblems that provide improving cuts [37] if the subproblems are convex. However, here we only consider the basic algorithm.…”