Optimization: Using Models, Validating Models, Solutions, Answers

Kallrath, Josef

doi:10.1007/978-3-030-73237-0_1

Cited by 1 publication

(1 citation statement)

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“…Mathematical optimisation approaches closely rely on solving to optimality a mathematical optimisation problem -the set of mathematical relationships representing the real-world problem [26]. [26] classifies mathematical optimisation problems into four groups such as linear programming problems, mixed-integer linear programming (MILP) problems, nonlinear programming problems, and mixed-integer nonlinear programming (MINLP) problems. Mixed-integer problems involve decision variables that can have both continuous and discrete domains.…”

Section: Introductionmentioning

confidence: 99%

A novel dual-decomposition method based on $p$-Lagrangian relaxation

Belyak¹,

Oliveira²

2023

Preprint

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In this paper, we propose the novel p-branch-and-bound method for solving two-stage stochastic programming problems whose deterministic equivalents are represented by mixed-integer quadratically constrained quadratic programming (MIQCQP) models. The precision of the solution generated by the p-branch-and-bound method can be arbitrarily adjusted by altering the value of the precision factor p. The proposed method combines two key techniques. The first one, named p-Lagrangian decomposition, generates a mixed-integer relaxation of a dual problem with a separable structure for a primal MIQCQP problem. The second one is a version of the classical dual decomposition approach that is applied to solve the Lagrangian dual problem and ensures that integrality and nonanticipativity conditions are met in the optimal solution. The p-branchand-bound method's efficiency has been tested on randomly generated instances and demonstrated superior performance over commercial solver Gurobi. This paper also presents a comparative analysis of the p-branchand-bound method efficiency considering two alternative solution methods for the dual problems as a subroutine. These are the proximal bundle method and Frank-Wolfe progressive hedging. The latter algorithm relies on the interpolation of linearisation steps similar to those taken in the Frank-Wolfe method as an inner loop in the classic progressive heading.

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Section: Introductionmentioning

confidence: 99%