Non-convex nested Benders decomposition

Abstract

We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations of the MINLP, obtained by piecewise linear relaxations of nonlinear functions. Those MILPs are solved to global optimality using an enhancement of nested Benders decomposition, in which regularization, dynamically refined binary approximations of the state variables and Lagrangian cut techniques are combined to generate Lipschitz continuous non-convex approximations of the value functions. Those approximations are then used to decide whether the approximating MILP has to be dynamically refined and in order to compute feasible solutions for the original MINLP. We prove that NC-NBD converges to an $$\varepsilon $$ ε -optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size.

Decision‐dependent distributionally robust integrated generation, transmission, and storage expansion planning: An enhanced Benders decomposition approach

“…The classical Benders decomposition algorithm cannot solve the problem () directly, as the exact dual of the MILP problem $$\mathcal {Q}(\mathbf {x}, \xi _{s})$$ cannot be derived. Considering the first‐stage decision variable vector $$\mathbf {x}$$ is a binary vector, the nested Benders decomposition algorithm can be used to derive the expansion plan in finite iterations [38, 39]. However, the original and relaxed $$\mathcal {Q}(\mathbf {x}, \xi _{s})$$ should be solved in sequential to obtain the exact value function and optimal/feasible cuts for a given combination of $$\mathbf {x}$$ and $$\xi _{s}$$, which is computationally expensive.…”

A Benders decomposition framework for the optimization of disjunctive superstructures with ordered discrete decisions

“…To partially overcome the optimality issue, a promising approach that has been previously used in the literature consists of correcting previously generated cuts as the iteration proceeds, such that all previously generated cuts remain valid. This dynamic refinement in the Benders cuts has allowed to obtain proven optimal solutions for MIPs and MINLPs 28,45 . To take advantage of this feature, a refinement strategy is proposed in this work.…”

“…This dynamic refinement in the Benders cuts has allowed to obtain proven optimal solutions for MIPs and MINLPs. 28,45 To take advantage of this feature, a refinement strategy is proposed in this work. The refinement procedure guarantees that, when the Benders algorithm stops, the solution obtained is a locally optimal solution.…”

“…Different Benders methods have been proposed to solve nonlinear programming problems involving both continuous and integer decisions. [25][26][27][28][29] All benders-based strategies follow the same principle, that is, they implement a master-subproblem decomposition strategy that decouples a group of complicating variables from the formulation to facilitate the iterative optimization of the problem. Early works in the field proposed a Generalized Benders Decomposition (GBD) methodology applicable to convex MINLP problems.…”

“…Similar to the proposed LD-BD method, other works in the field have focused on developing Benders-based methods that exploit special structures within specific problems. Some applications of Bendersbased algorithms for MINLP problems include the optimization of distillation sequences 29 ; the solution of two-stage and multi-stage stochastic MINLP problems, [26][27][28] and the optimal integration of decision layers, for example, simultaneous scheduling and control, process design and operation under stochastic uncertainty, and integrated planning, scheduling and dynamics. 17,32,33 While these methods build upon GBD, the proposed LD-BD strategy incorporates concepts of discrete convex analysis within the LBBD theory to introduce a new type of Benders-based method.…”