Convex relaxations of functions are used to provide bounding information to deterministic global optimization methods for nonconvex systems. To be useful, these relaxations must converge rapidly to the original system as the considered domain shrinks. This article examines the convergence rates of convex outer approximations for functions and nonlinear programs (NLPs), constructed using affine subtangents of an existing convex relaxation scheme. It is shown that these outer approximations inherit rapid second-order pointwise convergence from the original scheme under certain assumptions. To support this analysis, the notion of second-order pointwise convergence is extended to constrained optimization problems, and general sufficient conditions for guaranteeing this convergence are developed. The implications are discussed. An implementation of subtangent-based relaxations of NLPs in Julia is discussed and is applied to example problems for illustration.

Scheduling and feed quality optimization for processing solid concentrates in the copper refining industry may be formulated as a large-scale, discrete-time, nonconvex mixedinteger nonlinear program (MINLP) by including logistics operations and ad-hoc blending constraints. However, to solve this complex problem, the full space MINLP for the blending of solid concentrates of copper and the scheduling of their logistics is partitioned into a mixed-integer linear program (MILP) and a nonlinear program (NLP). The solution strategy considers the relax-and-fix rolling horizon with nearby time window overlaps and the use of multiple MILP solutions applied in a two-step MILP− NLP procedure. Two models are proposed for the flowsheet balances: a split fraction model and a process network model. The results indicate that the split fraction model yields near optimal solutions with a large computational effort, whereas the process network can generate several feasible solutions faster. We present a motivating example and an industrial problem with MILP to NLP gaps close to 0%.

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