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Abstract We present a deterministic global optimization method for nonlinear programming formulations constrained by stiff systems of ordinary differential equation (ODE) initial value problems (IVPs). The examples arise from dynamic optimization problems exhibiting both fast and slow transient phenomena commonly encountered in model‐based systems engineering applications. The proposed approach utilizes unconditionally stable implicit integration methods to reformulate the ODE‐constrained problem into a nonconvex nonlinear program (NLP) with implicit functions embedded. This problem is then solved to global optimality in finite time using a spatial branch‐and‐bound framework utilizing convex/concave relaxations of implicit functions constructed by a method which fully exploits problem sparsity. The algorithms were implemented in the Julia programming language within the EAGO.jl package and demonstrated on five illustrative examples with varying complexity relevant in process systems engineering. The developed methods enable the guaranteed global solution of dynamic optimization problems with stiff ODE–IVPs embedded.
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Extended McCormick relaxation rules for handling empty arguments representing infeasibility
McCormick’s relaxation technique is one of the most versatile and commonly used methods for computing the convex relaxations necessary for deterministic global optimization. The core of the method is a set of rules for propagating relaxations through basic arithmetic operations. Computationally, each rule operates on four-tuples describing each input argument in terms of a lower bound value, an upper bound value, a convex relaxation value, and a concave relaxation value. We call such tuples McCormick objects. This paper extends McCormick’s rules to accommodate input objects that are empty (i.e., the convex relaxation value lies above the concave, or both relaxation values lie outside the bounds). Empty McCormick objects provide a natural way to represent infeasibility and are readily generated by McCormick-based domain reduction techniques. The standard McCormick rules are strictly undefined for empty inputs and applying them anyway can yield relaxations that are non-convex/concave on infeasible parts of their domains. In contrast, our extended rules always produce relaxations that are well-defined and convex/concave on their entire domain. This capability has important applications in reduced-space global optimization, global dynamic optimization, and domain reduction.
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- Award ID(s):
- 1949747
- PAR ID:
- 10431468
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Journal of Global Optimization
- Volume:
- 87
- ISSN:
- 0925-5001
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10898-023-01315-7
