Consistent Approximations in Composite Optimization

• URL: http://arxiv.org/abs/2201.05250v1
• Date: Thu, 13 Jan 2022 23:57:08 GMT
• Title: Consistent Approximations in Composite Optimization
• Authors: Johannes O. Royset
• Abstract summary: We develop a framework for consistent approximations of optimization problems. The framework is developed for a broad class of optimizations. A programming analysis method illustrates extended nonlinear programming solutions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, with even small approximations of components of a problem translating into large errors in the solutions. We specify conditions under which approximations are well behaved in the sense of minimizers, stationary points, and level-sets and this leads to a framework of consistent approximations. The framework is developed for a broad class of composite problems, which are neither convex nor smooth. We demonstrate the framework using examples from stochastic optimization, neural-network based machine learning, distributionally robust optimization, penalty and augmented Lagrangian methods, interior-point methods, homotopy methods, smoothing methods, extended nonlinear programming, difference-of-convex programming, and multi-objective optimization. An enhanced proximal method illustrates the algorithmic possibilities. A quantitative analysis supplements the development by furnishing rates of convergence.

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