Exploiting Chordality in Optimization Algorithms for Model Predictive Control

In this chapter we show that chordal structure can be used to devise efficient optimization methods for many common model predictive control problems. The chordal structure is used both for computing search directions efficiently as well as for distributing all the other computations in an interior-point method for solving the problem. The chordal structure can stem both from the sequential nature of the problem as well as from distributed formulations of the problem related to scenario trees or other formulations. The framework enables efficient parallel computations.Comment: arXiv admin note: text overlap with arXiv:1502.0638

On the relationship between bilevel decomposition algorithms and direct interior-point methods

Engineers have been using bilevel decomposition algorithms to solve certain nonconvex large-scale optimization problems arising in engineering design projects. These algorithms transform the large-scale problem into a bilevel program with one upperlevel problem (the master problem) and several lower-level problems (the subproblems). Unfortunately, there is analytical and numerical evidence that some of these commonly used bilevel decomposition algorithms may fail to converge even when the starting point is very close to the minimizer. In this paper, we establish a relationship between a particular bilevel decomposition algorithm, which only performs one iteration of an interior-point method when solving the subproblems, and a direct interior-point method, which solves the problem in its original (integrated) form. Using this relationship, we formally prove that the bilevel decomposition algorithm converges locally at a superlinear rate. The relevance of our analysis is that it bridges the gap between the incipient local convergence theory of bilevel decomposition algorithms and the mature theory of direct interior-point methods

On solving large-scale multistage stochastic optimization problems with a new specialized interior-point approach

© 2023 The Authors. Published by Elsevier B.VA novel approach based on a specialized interior-point method (IPM) is presented for solving largescale stochastic multistage continuous optimization problems, which represent the uncertainty in strategic multistage and operational two-stage scenario trees. This new solution approach considers a splitvariable formulation of the strategic and operational structures. The specialized IPM solves the normal equations by combining Cholesky factorizations with preconditioned conjugate gradients, doing so for, respectively, the constraints of the stochastic formulation and those that equate the split-variables. We show that, for multistage stochastic problems, the preconditioner (i) is a block-diagonal matrix composed of as many shifted tridiagonal matrices as the number of nested strategic-operational two-stage trees, thus allowing the efficient solution of systems of equations; (ii) its complexity in a multistage stochastic problem is equivalent to that of a very large-scale two-stage problem. A broad computational experience is reported for large multistage stochastic supply network design (SND) and revenue management (RM) problems. Some of the most difficult instances of SND had 5 stages, 839 million linear variables, 13 million quadratic variables, 21 million constraints, and 3750 scenario tree nodes; while those of RM had 8 stages, 278 million linear variables, 100 million constraints, and 100,000 scenario tree nodes. For those problems, the proposed approach obtained the solution in 1.1 days using 174 gigabytes of memory for SND, and in 1.7 days using 83 gigabytes for RM; while CPLEX v20.1 required more than 53 days and 531 gigabytes for SND, and more than 19 days and 410 gigabytes for RM.J. Castro was supported by the MCIN/AEI/FEDER grant RTI2018-097580-B-I00. L.E. Escudero was supported by the MCIN/AEI/10.13039/501100011033 grant PID2021-122640OB-I00. J.F. Monge was supported by the MCIN/AEI/10.13039/501100011033/ERDF grants PID2019-105952GB-I00 and PID2021-122344NB-I00, and by PROMETEO/2021/063 grant funded by the government of the Valencia Community, Spain.Peer ReviewedPostprint (published version

Distributed algorithms for nonlinear tree-sparse problems

[no abstract